id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-18200 | Filter Property of Dirac Delta Function | Let $\map \delta x$ denote the Dirac delta function.
Let $g$ be a continuous real function.
Then:
:$\ds \int_{-\infty}^{+ \infty} \map {\delta} {x - s} \map g x \rd x = \map g s$ | We have that:
:$\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin{cases} 0 & : x < -\epsilon + s \\ \dfrac 1 {2 \epsilon} & : -\epsilon + s \le x \le \epsilon + s \\ 0 & : x > \epsilon + s \end{cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_{-\infty}^{+ \... | Let $\map \delta x$ denote the [[Definition:Dirac Delta Function|Dirac delta function]].
Let $g$ be a [[Definition:Continuous Real Function|continuous real function]].
Then:
:$\ds \int_{-\infty}^{+ \infty} \map {\delta} {x - s} \map g x \rd x = \map g s$ | We have that:
:$\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin{cases} 0 & : x < -\epsilon + s \\ \dfrac 1 {2 \epsilon} & : -\epsilon + s \le x \le \epsilon + s \\ 0 & : x > \epsilon + s \end{cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_{-\infty... | Filter Property of Dirac Delta Function | https://proofwiki.org/wiki/Filter_Property_of_Dirac_Delta_Function | https://proofwiki.org/wiki/Filter_Property_of_Dirac_Delta_Function | [
"Dirac Delta Function",
"Convolution Integrals"
] | [
"Definition:Dirac Delta Function",
"Definition:Continuous Real Function"
] | [
"Integral of Constant/Definite",
"Darboux's Theorem",
"Definition:Maximal/Element",
"Definition:Minimal/Element",
"Squeeze Theorem",
"Definition:Dirac Delta Function"
] |
proofwiki-18201 | Derivative Operator on Continuously Differentiable Function Space with Supremum Norm is not Continuous | Let $I = \closedint 0 1$ be a closed real interval.
Let $\map C I$ be the real-valued, continuous on $I$ function space.
Let $\map {C^1} I$ be the continuously differentiable function space.
Let $x \in \map {C^1} I$ be a continuously differentiable real-valued function.
Let $D : \map {C^1} I \to \map C I$ be the deriva... | {{AimForCont}} $D$ is continuous.
We have that the Derivative Operator is Linear Mapping.
By Continuity of Linear Transformation between Normed Vector Spaces:
:$\exists M \in \R_{> 0} : \forall x \in \map {C^1} I : \norm {\map D x}_\infty \le M \norm x_\infty$
Suppose $x = t^n$ with $n \in \N$.
Then:
:$\norm x_\infty =... | Let $I = \closedint 0 1$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\map C I$ be the [[Definition:Space of Real-Valued Functions Continuous on Closed Interval|real-valued, continuous on $I$ function space]].
Let $\map {C^1} I$ be the [[Definition:Space of Continuous Functions of Differentiabi... | {{AimForCont}} $D$ is [[Definition:Continuous Mapping (Normed Vector Space)/Space|continuous]].
We have that the [[Derivative Operator is Linear Mapping]].
By [[Continuity of Linear Transformation between Normed Vector Spaces]]:
:$\exists M \in \R_{> 0} : \forall x \in \map {C^1} I : \norm {\map D x}_\infty \le M \n... | Derivative Operator on Continuously Differentiable Function Space with Supremum Norm is not Continuous | https://proofwiki.org/wiki/Derivative_Operator_on_Continuously_Differentiable_Function_Space_with_Supremum_Norm_is_not_Continuous | https://proofwiki.org/wiki/Derivative_Operator_on_Continuously_Differentiable_Function_Space_with_Supremum_Norm_is_not_Continuous | [
"Operator Theory",
"Continuous Mappings",
"Differentiability Classes"
] | [
"Definition:Real Interval/Closed",
"Definition:Space of Real-Valued Functions Continuous on Closed Interval",
"Definition:Space of Continuous Functions of Differentiability Class k",
"Definition:Continuously Differentiable/Real-Valued Function",
"Definition:Derivative Operator",
"Definition:Supremum Norm"... | [
"Definition:Continuous Mapping (Normed Vector Space)/Space",
"Derivative Operator is Linear Mapping",
"Continuity of Linear Transformation/Normed Vector Space",
"Definition:Finite Extended Real Number",
"Definition:Contradiction",
"Definition:Continuous Mapping (Normed Vector Space)/Space"
] |
proofwiki-18202 | Derivative Operator on Continuously Differentiable Function Space with C^1 Norm is Continuous | Let $I = \closedint 0 1$ be a closed real interval.
Let $\map C I$ be the real-valued, continuous on $I$ function space.
Let $\map {C^1} I$ be the continuously differentiable function space.
Let $x \in \map {C^1} I$ be a continuously differentiable real-valued function.
Let $D : \map {C^1} I \to \map \CC I$ be the deri... | {{begin-eqn}}
{{eqn | l = \norm {D x}_\infty
| r = \norm {x'}_\infty
}}
{{eqn | o = \le
| r = \norm {x'}_\infty + \norm x_\infty
}}
{{eqn | r = \norm x_{1, \infty}
}}
{{end-eqn}}
We have that the Derivative Operator is Linear Mapping.
By definition and Continuity of Linear Transformation between Normed V... | Let $I = \closedint 0 1$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\map C I$ be the [[Definition:Space of Real-Valued Functions Continuous on Closed Interval|real-valued, continuous on $I$ function space]].
Let $\map {C^1} I$ be the [[Definition:Space of Continuous Functions of Differentiabi... | {{begin-eqn}}
{{eqn | l = \norm {D x}_\infty
| r = \norm {x'}_\infty
}}
{{eqn | o = \le
| r = \norm {x'}_\infty + \norm x_\infty
}}
{{eqn | r = \norm x_{1, \infty}
}}
{{end-eqn}}
We have that the [[Derivative Operator is Linear Mapping]].
By definition and [[Continuity of Linear Transformation between ... | Derivative Operator on Continuously Differentiable Function Space with C^1 Norm is Continuous | https://proofwiki.org/wiki/Derivative_Operator_on_Continuously_Differentiable_Function_Space_with_C^1_Norm_is_Continuous | https://proofwiki.org/wiki/Derivative_Operator_on_Continuously_Differentiable_Function_Space_with_C^1_Norm_is_Continuous | [
"Operator Theory",
"Continuous Mappings",
"Differentiability Classes"
] | [
"Definition:Real Interval/Closed",
"Definition:Space of Real-Valued Functions Continuous on Closed Interval",
"Definition:Space of Continuous Functions of Differentiability Class k",
"Definition:Continuously Differentiable/Real-Valued Function",
"Definition:Derivative/Function With Values in Normed Space",
... | [
"Derivative Operator is Linear Mapping",
"Continuity of Linear Transformation/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)/Space"
] |
proofwiki-18203 | Mean Value Theorem for Holomorphic Functions | Let $D$ be a region.
Let $f : D \to \C$ be a holomorphic function.
Let $z \in D$.
Let $r$ be such that $\map {B_r} z \subseteq D$.
Then:
:$\ds \map f z = \frac 1 {2 \pi} \int_0^{2 \pi} \map f {z + r e^{i \theta} } \rd \theta$ | By Cauchy's Integral Formula, we have:
:$\ds \map f z = \frac 1 {2 \pi i} \oint_{\partial \map {B_r} z} \frac {\map f t} {t - z} \rd t$
where $\partial \map {B_r} z$ is the boundary of $\map {B_r} z$.
That is, $\partial \map {B_r} z$ is the circle of radius $r$, centred at $z$.
Note that we can parameterise $\partial... | Let $D$ be a [[Definition:Region (Complex Analysis)|region]].
Let $f : D \to \C$ be a [[Definition:Holomorphic Function|holomorphic function]].
Let $z \in D$.
Let $r$ be such that $\map {B_r} z \subseteq D$.
Then:
:$\ds \map f z = \frac 1 {2 \pi} \int_0^{2 \pi} \map f {z + r e^{i \theta} } \rd \theta$ | By [[Cauchy's Integral Formula]], we have:
:$\ds \map f z = \frac 1 {2 \pi i} \oint_{\partial \map {B_r} z} \frac {\map f t} {t - z} \rd t$
where $\partial \map {B_r} z$ is the [[Definition:Boundary (Topology)|boundary]] of $\map {B_r} z$.
That is, $\partial \map {B_r} z$ is the [[Definition:Circle|circle]] of [[Def... | Mean Value Theorem for Holomorphic Functions | https://proofwiki.org/wiki/Mean_Value_Theorem_for_Holomorphic_Functions | https://proofwiki.org/wiki/Mean_Value_Theorem_for_Holomorphic_Functions | [
"Mean Value Theorem",
"Holomorphic Functions"
] | [
"Definition:Region/Complex",
"Definition:Holomorphic Function"
] | [
"Cauchy's Integral Formula",
"Definition:Boundary (Topology)",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Contour/Parameterization/Complex Plane",
"Definition:Contour Integral/Complex",
"Derivative of Exponential Function",
"Category:Mean Value Theorem",... |
proofwiki-18204 | Fourier Transform of Exponential Function | Let $\map f x$ be defined as the real exponential function where the absolute value of the input is used in the exponent and the exponent is scaled by a factor of $-2 \pi a$:
:$\map f x = e^{-2 \pi a \size x}: \R \to \R$
Then:
:$\map {\hat f} s = \dfrac 1 \pi \dfrac a {a^2 + s^2}$
where $\map {\hat f} s$ is the Four... | By the definition of a Fourier transform:
{{begin-eqn}}
{{eqn | l = \map {\hat f} s
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map f x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} e^{-2 \pi a \size x} \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^0 e^{-2 \pi i x s} e^{-2 \pi a \s... | Let $\map f x$ be defined as the [[Definition:Exponential Function/Real|real exponential function]] where the [[Definition:Absolute Value/Definition 1|absolute value]] of the input is used in the [[Definition:Exponent|exponent]] and the [[Definition:Exponent|exponent]] is scaled by a factor of $-2 \pi a$:
:$\map f x =... | By the definition of a [[Definition:Fourier Transform of Real Function|Fourier transform]]:
{{begin-eqn}}
{{eqn | l = \map {\hat f} s
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map f x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} e^{-2 \pi a \size x} \rd x
| c =
}}
{{eqn | r... | Fourier Transform of Exponential Function | https://proofwiki.org/wiki/Fourier_Transform_of_Exponential_Function | https://proofwiki.org/wiki/Fourier_Transform_of_Exponential_Function | [
"Examples of Fourier Transforms"
] | [
"Definition:Exponential Function/Real",
"Definition:Absolute Value/Definition 1",
"Definition:Power (Algebra)/Exponent",
"Definition:Power (Algebra)/Exponent",
"Definition:Fourier Transform/Real Function"
] | [
"Definition:Fourier Transform/Real Function",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Exponent Combination Laws/Product of Powers",
"Primitive of Exponential of a x"
] |
proofwiki-18205 | Bisectors of Angles between Two Straight Lines/General Form | Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed in general form as:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | lll = \LL_2:
| l = l_2 x + m_2 y + n_2
| r = 0
}}
{{end-eqn}}
The angle bisectors of the angles formed at t... | First we convert $\LL_1$ and $\LL_2$ into normal form:
{{begin-eqn}}
{{eqn | l = \dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} }
| r = 0
}}
{{eqn | l = \dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }
| r = 0
}}
{{eqn | ll= \leadsto
| l = x \cos \alpha + y \sin \alpha
| r = ... | Let $\LL_1$ and $\LL_2$ be [[Definition:Straight Line|straight lines]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] $\CC$, expressed in [[Equation of Straight Line in Plane/General Equation|general form]] as:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | ll... | First we convert $\LL_1$ and $\LL_2$ into [[Equation of Straight Line in Plane/Normal Form|normal form]]:
{{begin-eqn}}
{{eqn | l = \dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} }
| r = 0
}}
{{eqn | l = \dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }
| r = 0
}}
{{eqn | ll= \leadsto
... | Bisectors of Angles between Two Straight Lines/General Form | https://proofwiki.org/wiki/Bisectors_of_Angles_between_Two_Straight_Lines/General_Form | https://proofwiki.org/wiki/Bisectors_of_Angles_between_Two_Straight_Lines/General_Form | [
"Angle Bisectors"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Angle Bisector",
"Definition:Angle",
"Definition:Point",
"Definition:Intersection (Geometry)"
] | [
"Equation of Straight Line in Plane/Normal Form",
"Bisectors of Angles between Two Straight Lines/Normal Form",
"Definition:Angle Bisector",
"Definition:Angle",
"Definition:Point",
"Definition:Intersection (Geometry)"
] |
proofwiki-18206 | Quadratic Representation of Pair of Straight Lines | Consider the general quadratic equation in $2$ variables:
:$(1): \quad a x^2 + b x y + c y^2 + d x + e y + f = 0$
Then $(1)$ is the locus of $2$ straight lines in the Cartesian plane {{iff}} it can be expressed in the form:
:$\paren {l_1 x + m_1 y + n_1} \paren {l_2 x + m_2 y + n_2} = 0$
where $l_1$, $m_1$, $n_1$, $l_2... | === Sufficient Condition ===
Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed in general form as:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | lll = \LL_2:
| l = l_2 x + m_2 y + n_2
| r = 0
}}
{{end-eqn}}
Let $\tuple {x, ... | Consider the general [[Definition:Quadratic Equation|quadratic equation]] in $2$ [[Definition:Independent Variable|variables]]:
:$(1): \quad a x^2 + b x y + c y^2 + d x + e y + f = 0$
Then $(1)$ is the [[Definition:Locus|locus]] of $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|C... | === Sufficient Condition ===
Let $\LL_1$ and $\LL_2$ be [[Definition:Straight Line|straight lines]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] $\CC$, expressed in [[Equation of Straight Line in Plane/General Equation|general form]] as:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = l_1 x + m_1 y + n... | Quadratic Representation of Pair of Straight Lines | https://proofwiki.org/wiki/Quadratic_Representation_of_Pair_of_Straight_Lines | https://proofwiki.org/wiki/Quadratic_Representation_of_Pair_of_Straight_Lines | [
"Quadratic Representation of Pair of Straight Lines",
"Straight Lines",
"Quadratic Equations"
] | [
"Definition:Quadratic Equation",
"Definition:Independent Variable",
"Definition:Locus",
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Real Number"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line"
] |
proofwiki-18207 | Pointwise Limit of Increasing Functions is Increasing | Let $X \subseteq \R$.
Let $\sequence {f_n}$ be a sequence of real functions $X \to \R$ converging pointwise to a function $f : X \to \R$.
Let $f_n$ be a increasing function for each $n$.
Then $f$ is increasing. | Suppose that $f$ is not increasing.
That is:
:there exists $x, y \in X$ such that $x < y$ and $\map f x > \map f y$.
Let:
:$r = \map f x - \map f y > 0$
Since $\sequence {f_n}$ converges to $f$ converging pointwise, the sequence $\sequence {\map {f_n} x}$ converges to $\map f x$.
That is, there exists $N_1 \in \N$ s... | Let $X \subseteq \R$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]] $X \to \R$ [[Definition:Pointwise Convergence|converging pointwise]] to a [[Definition:Real Function|function]] $f : X \to \R$.
Let $f_n$ be a [[Definition:Increasing Function|increasing f... | Suppose that $f$ is not [[Definition:Increasing Function|increasing]].
That is:
:there exists $x, y \in X$ such that $x < y$ and $\map f x > \map f y$.
Let:
:$r = \map f x - \map f y > 0$
Since $\sequence {f_n}$ converges to $f$ [[Definition:Pointwise Convergence|converging pointwise]], the [[Definition:Real Seq... | Pointwise Limit of Increasing Functions is Increasing | https://proofwiki.org/wiki/Pointwise_Limit_of_Increasing_Functions_is_Increasing | https://proofwiki.org/wiki/Pointwise_Limit_of_Increasing_Functions_is_Increasing | [
"Real Analysis"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Pointwise Convergence",
"Definition:Real Function",
"Definition:Increasing/Real Function",
"Definition:Increasing/Real Function"
] | [
"Definition:Increasing/Real Function",
"Definition:Pointwise Convergence",
"Definition:Real Sequence",
"Definition:Convergence",
"Definition:Convergence",
"Definition:Increasing/Real Function",
"Definition:Increasing/Real Function",
"Category:Real Analysis"
] |
proofwiki-18208 | Characteristic of Quadratic Equation that Represents Two Straight Lines | Consider the '''quadratic equation in $2$ variables''':
:$(1): \quad a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
where $x$ and $y$ are independent variables.
Then $(1)$ represents $2$ straight lines {{iff}} its discriminant equals zero:
:$a b c + 2 f g h - a f^2 - b g^2 - c h^2 = 0$
This can also be expressed in t... | Suppose that $a \ne 0$.
We have:
{{begin-eqn}}
{{eqn | l = a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = a^2 x^2 + a b y^2 + 2 a h x y + 2 a g x + 2 a f y + c
| r = 0
| c = multiplying by $a$
}}
{{eqn | ll= \leadsto
| l = \paren {a x + h y +... | Consider the '''[[Definition:Quadratic Equation in Two Variables|quadratic equation in $2$ variables]]''':
:$(1): \quad a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
where $x$ and $y$ are [[Definition:Independent Variable|independent variables]].
Then $(1)$ represents $2$ [[Definition:Straight Line|straight lin... | Suppose that $a \ne 0$.
We have:
{{begin-eqn}}
{{eqn | l = a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = a^2 x^2 + a b y^2 + 2 a h x y + 2 a g x + 2 a f y + c
| r = 0
| c = multiplying by $a$
}}
{{eqn | ll= \leadsto
| l = \paren {a x + h y... | Characteristic of Quadratic Equation that Represents Two Straight Lines | https://proofwiki.org/wiki/Characteristic_of_Quadratic_Equation_that_Represents_Two_Straight_Lines | https://proofwiki.org/wiki/Characteristic_of_Quadratic_Equation_that_Represents_Two_Straight_Lines | [
"Quadratic Equations",
"Straight Lines"
] | [
"Definition:Quadratic Equation in Two Variables",
"Definition:Independent Variable",
"Definition:Line/Straight Line",
"Definition:Discriminant of Quadratic Equation in Two Variables",
"Definition:Zero (Number)",
"Definition:Determinant"
] | [
"Definition:Zero (Number)",
"Difference of Two Squares",
"Equation of Straight Line in Plane/General Equation",
"Definition:Zero (Number)",
"Definition:Quadratic Equation in Two Variables",
"Determinant/Examples/Order 3",
"Determinant/Examples/Order 2"
] |
proofwiki-18209 | Entire Function Bounded by Polynomial is Polynomial | Let $f : \C \to \C$ be an entire function such that:
:$\cmod {\map f z} \le M {\cmod z}^k$
for all $z \in \C$, for some $k \in \N$ and real $M > 0$.
Then $f$ is a polynomial of degree at most $k$. | Let $r > 0$ be a real number.
Let $D = \map {B_r} 0$ be the open ball with centre $0$ of radius $r$.
By Holomorphic Function is Analytic, we have that:
:$\ds \map f z = \sum_{n \mathop = 0}^\infty a_n z^n$
for all $z \in \C$, where:
:$\ds a_n = \frac 1 {2 \pi i} \oint_{\partial D} \frac {\map f t} {t^{n + 1} } \rd t$... | Let $f : \C \to \C$ be an [[Definition:Entire Function|entire function]] such that:
:$\cmod {\map f z} \le M {\cmod z}^k$
for all $z \in \C$, for some $k \in \N$ and [[Definition:Real Number|real]] $M > 0$.
Then $f$ is a [[Definition:Polynomial|polynomial]] of [[Definition:Degree of Polynomial|degree]] at most $k... | Let $r > 0$ be a [[Definition:Real Number|real number]].
Let $D = \map {B_r} 0$ be the [[Definition:Open Ball|open ball]] with [[Definition:Center of Open Ball|centre]] $0$ of [[Definition:Radius of Open Ball|radius]] $r$.
By [[Holomorphic Function is Analytic]], we have that:
:$\ds \map f z = \sum_{n \mathop = 0}^... | Entire Function Bounded by Polynomial is Polynomial | https://proofwiki.org/wiki/Entire_Function_Bounded_by_Polynomial_is_Polynomial | https://proofwiki.org/wiki/Entire_Function_Bounded_by_Polynomial_is_Polynomial | [
"Complex Analysis"
] | [
"Definition:Entire Function",
"Definition:Real Number",
"Definition:Polynomial",
"Definition:Degree of Polynomial"
] | [
"Definition:Real Number",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Holomorphic Function is Analytic",
"Estimation Lemma for Contour Integrals",
"Estimation Lemma for Contour Integrals",
"Definition:Polynomial",
"Definition:Degree of Polynomial",
"Categ... |
proofwiki-18210 | Formula for Total Variation of Continuously Differentiable Function | Let $a, b$ be real numbers with $a < b$.
Let $f : \closedint a b \to \R$ be a continuously differentiable function.
Let $\map {V_} {\closedint a b}$ be the total variation of $f$ on $\closedint a b$.
Then:
:$\ds V_f = \int_a^b \size {\map {f'} x} \rd x$ | For each finite subdivision $P$ of $\closedint a b$, write:
:$P = \set {x_0, x_1, \ldots, x_n }$
with:
:$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$
We first show that:
:$\ds \map {V_f} {\closedint a b} \le \int_a^b \size {\map {f'} x} \rd x$
We then show that:
:$\ds \int_a^b \size {\map {f'} x} \rd x \le \m... | Let $a, b$ be [[Definition:Real Number|real numbers]] with $a < b$.
Let $f : \closedint a b \to \R$ be a [[Definition:Continuously Differentiable Real Function|continuously differentiable]] [[Definition:Real Function|function]].
Let $\map {V_} {\closedint a b}$ be the [[Definition:Total Variation of Real Function on ... | For each [[Definition:Finite Subdivision|finite subdivision]] $P$ of $\closedint a b$, write:
:$P = \set {x_0, x_1, \ldots, x_n }$
with:
:$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$
We first show that:
:$\ds \map {V_f} {\closedint a b} \le \int_a^b \size {\map {f'} x} \rd x$
We then show that:
:$\ds \int_... | Formula for Total Variation of Continuously Differentiable Function | https://proofwiki.org/wiki/Formula_for_Total_Variation_of_Continuously_Differentiable_Function | https://proofwiki.org/wiki/Formula_for_Total_Variation_of_Continuously_Differentiable_Function | [
"Total Variation of Real Function"
] | [
"Definition:Real Number",
"Definition:Continuously Differentiable/Real Function",
"Definition:Real Function",
"Definition:Total Variation/Real Function/Closed Bounded Interval"
] | [
"Definition:Subdivision of Interval/Finite",
"Differentiable Function is Continuous",
"Definition:Continuous Function",
"Restriction of Continuous Mapping is Continuous",
"Definition:Continuous Function",
"Definition:Continuous Function",
"Fundamental Theorem of Calculus",
"Definition:Subdivision of I... |
proofwiki-18211 | Homogeneous Quadratic Equation represents Two Straight Lines through Origin | Let $E$ be a homogeneous quadratic equation in two variables:
:$E: \quad a x^2 + 2 h x y + b y^2 = 0$
Then $E$ represents $2$ straight lines in the Cartesian plane:
:$y = \dfrac {h \pm \sqrt {h^2 - a b} } b x$ | From Characteristic of Quadratic Equation that Represents Two Straight Lines, $E$ represents $2$ straight lines in the Cartesian plane {{iff}}
:$a b c + 2 f g h - a f^2 - b g^2 - c h^2 = 0$
where in this case $c = f = g = 0$, giving:
:$a b \times 0 + 2 \times 0 \times 0 \times h - a \times 0^2 - b \times 0^2 - 0 \times... | Let $E$ be a [[Definition:Homogeneous Quadratic Equation|homogeneous quadratic equation]] in [[Definition:Quadratic Equation in Two Variables|two variables]]:
:$E: \quad a x^2 + 2 h x y + b y^2 = 0$
Then $E$ represents $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane... | From [[Characteristic of Quadratic Equation that Represents Two Straight Lines]], $E$ represents $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane]] {{iff}}
:$a b c + 2 f g h - a f^2 - b g^2 - c h^2 = 0$
where in this case $c = f = g = 0$, giving:
:$a b \times 0 + 2 ... | Homogeneous Quadratic Equation represents Two Straight Lines through Origin | https://proofwiki.org/wiki/Homogeneous_Quadratic_Equation_represents_Two_Straight_Lines_through_Origin | https://proofwiki.org/wiki/Homogeneous_Quadratic_Equation_represents_Two_Straight_Lines_through_Origin | [
"Quadratic Equations",
"Straight Lines"
] | [
"Definition:Homogeneous Quadratic Equation",
"Definition:Quadratic Equation in Two Variables",
"Definition:Line/Straight Line",
"Definition:Cartesian Plane"
] | [
"Characteristic of Quadratic Equation that Represents Two Straight Lines",
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Solution to Quadratic Equation",
"Definition:Point",
"Definition:Line/Straight Line"
] |
proofwiki-18212 | Homogeneous Quadratic Equation representing Imaginary Straight Lines | Let $E$ be a homogeneous quadratic equation in two variables:
:$E: \quad a x^2 + 2 h x y + b y^2$
Let $h^2 - a b < 0$.
Then
Then $E$ represents a conjugate pair of imaginary straight lines in the Cartesian plane: | From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, $E$ represents $2$ straight lines in the Cartesian plane:
:$y = \dfrac {h \pm \sqrt {h^2 - a b} } b x$
But when $h^2 - a b$, $\sqrt {h^2 - a b}$ is not defined as a real number.
Instead we have:
:$y = \dfrac {h \pm i \sqrt {h^2 - a b} } b... | Let $E$ be a [[Definition:Homogeneous Quadratic Equation|homogeneous quadratic equation]] in [[Definition:Quadratic Equation in Two Variables|two variables]]:
:$E: \quad a x^2 + 2 h x y + b y^2$
Let $h^2 - a b < 0$.
Then
Then $E$ represents a [[Definition:Conjugate Pair of Imaginary Straight Lines|conjugate pair]]... | From [[Homogeneous Quadratic Equation represents Two Straight Lines through Origin]], $E$ represents $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane]]:
:$y = \dfrac {h \pm \sqrt {h^2 - a b} } b x$
But when $h^2 - a b$, $\sqrt {h^2 - a b}$ is not defined as a [[Defin... | Homogeneous Quadratic Equation representing Imaginary Straight Lines | https://proofwiki.org/wiki/Homogeneous_Quadratic_Equation_representing_Imaginary_Straight_Lines | https://proofwiki.org/wiki/Homogeneous_Quadratic_Equation_representing_Imaginary_Straight_Lines | [
"Quadratic Equations",
"Straight Lines"
] | [
"Definition:Homogeneous Quadratic Equation",
"Definition:Quadratic Equation in Two Variables",
"Definition:Conjugate Pair of Imaginary Straight Lines",
"Definition:Imaginary Straight Line",
"Definition:Cartesian Plane"
] | [
"Homogeneous Quadratic Equation represents Two Straight Lines through Origin",
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Real Number"
] |
proofwiki-18213 | Homogeneous Quadratic Equation representing Coincident Straight Lines | Let $E$ be a homogeneous quadratic equation in two variables:
:$E: \quad a x^2 + 2 h x y + b y^2$
Let $h^2 - a b = 0$.
Then
Then $E$ represents $2$ straight lines in the Cartesian plane which completely coincide: | From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, $E$ represents $2$ straight lines in the Cartesian plane:
:$y = \dfrac {h \pm \sqrt {h^2 - a b} } b x$
But when $h^2 - a b = 0$, we get:
:$y = \dfrac h b x$
{{qed}} | Let $E$ be a [[Definition:Homogeneous Quadratic Equation|homogeneous quadratic equation]] in [[Definition:Quadratic Equation in Two Variables|two variables]]:
:$E: \quad a x^2 + 2 h x y + b y^2$
Let $h^2 - a b = 0$.
Then
Then $E$ represents $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartes... | From [[Homogeneous Quadratic Equation represents Two Straight Lines through Origin]], $E$ represents $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane]]:
:$y = \dfrac {h \pm \sqrt {h^2 - a b} } b x$
But when $h^2 - a b = 0$, we get:
:$y = \dfrac h b x$
{{qed}} | Homogeneous Quadratic Equation representing Coincident Straight Lines | https://proofwiki.org/wiki/Homogeneous_Quadratic_Equation_representing_Coincident_Straight_Lines | https://proofwiki.org/wiki/Homogeneous_Quadratic_Equation_representing_Coincident_Straight_Lines | [
"Quadratic Equations",
"Straight Lines"
] | [
"Definition:Homogeneous Quadratic Equation",
"Definition:Quadratic Equation in Two Variables",
"Definition:Line/Straight Line",
"Definition:Cartesian Plane"
] | [
"Homogeneous Quadratic Equation represents Two Straight Lines through Origin",
"Definition:Line/Straight Line",
"Definition:Cartesian Plane"
] |
proofwiki-18214 | Angle Between Two Straight Lines described by Homogeneous Quadratic Equation | Let $\LL_1$ and $\LL_2$ represent $2$ straight lines in the Cartesian plane which are represented by a homogeneous quadratic equation $E$ in two variables:
:$E: \quad a x^2 + 2 h x y + b y^2$
Then the angle $\psi$ between $\LL_1$ and $\LL_2$ is given by:
:$\tan \psi = \dfrac {2 \sqrt {h^2 - a b} } {a + b}$ | Let us rewrite $E$ as follows:
:$b y^2 + 2 h x y + a x^2 = b \paren {y - \mu_1 x} \paren {y - \mu_2 x}$
Thus from Homogeneous Quadratic Equation represents Two Straight Lines through Origin:
:$\LL_1$ and $\LL_2$ are represented by the equations $y = \mu_1 x$ and $y = \mu_2 x$ respectively.
From Sum of Roots of Quadrati... | Let $\LL_1$ and $\LL_2$ represent $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane]] which are represented by a [[Definition:Homogeneous Quadratic Equation|homogeneous quadratic equation]] $E$ in [[Definition:Quadratic Equation in Two Variables|two variables]]:
:$E: \... | Let us rewrite $E$ as follows:
:$b y^2 + 2 h x y + a x^2 = b \paren {y - \mu_1 x} \paren {y - \mu_2 x}$
Thus from [[Homogeneous Quadratic Equation represents Two Straight Lines through Origin]]:
:$\LL_1$ and $\LL_2$ are represented by the equations $y = \mu_1 x$ and $y = \mu_2 x$ respectively.
From [[Sum of Roots o... | Angle Between Two Straight Lines described by Homogeneous Quadratic Equation | https://proofwiki.org/wiki/Angle_Between_Two_Straight_Lines_described_by_Homogeneous_Quadratic_Equation | https://proofwiki.org/wiki/Angle_Between_Two_Straight_Lines_described_by_Homogeneous_Quadratic_Equation | [
"Straight Lines"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Homogeneous Quadratic Equation",
"Definition:Quadratic Equation in Two Variables",
"Definition:Angle"
] | [
"Homogeneous Quadratic Equation represents Two Straight Lines through Origin",
"Sum of Roots of Quadratic Equation",
"Product of Roots of Quadratic Equation",
"Angle between Straight Lines in Plane",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-18215 | Condition for Homogeneous Quadratic Equation to describe Perpendicular Straight Lines | Let $\LL_1$ and $\LL_2$ represent $2$ straight lines in the Cartesian plane which are represented by a homogeneous quadratic equation $E$ in two variables.
Let $\LL_1$ and $\LL_2$ be perpendicular.
Then $E$ is of the form:
:$a x^2 + 2 h x y - a y^2$ | From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, $E$ is of the form:
:$a x^2 + 2 h x y + b y^2$
From Angle Between Two Straight Lines described by Homogeneous Quadratic Equation, the angle $\psi$ between $\LL_1$ and $\LL_2$ is given by:
:$\tan \psi = \dfrac {2 \sqrt {h^2 - a b} } {a + b... | Let $\LL_1$ and $\LL_2$ represent $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane]] which are represented by a [[Definition:Homogeneous Quadratic Equation|homogeneous quadratic equation]] $E$ in [[Definition:Quadratic Equation in Two Variables|two variables]].
Let $\... | From [[Homogeneous Quadratic Equation represents Two Straight Lines through Origin]], $E$ is of the form:
:$a x^2 + 2 h x y + b y^2$
From [[Angle Between Two Straight Lines described by Homogeneous Quadratic Equation]], the [[Definition:Angle|angle]] $\psi$ between $\LL_1$ and $\LL_2$ is given by:
:$\tan \psi = \df... | Condition for Homogeneous Quadratic Equation to describe Perpendicular Straight Lines | https://proofwiki.org/wiki/Condition_for_Homogeneous_Quadratic_Equation_to_describe_Perpendicular_Straight_Lines | https://proofwiki.org/wiki/Condition_for_Homogeneous_Quadratic_Equation_to_describe_Perpendicular_Straight_Lines | [
"Straight Lines",
"Perpendiculars"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Homogeneous Quadratic Equation",
"Definition:Quadratic Equation in Two Variables",
"Definition:Right Angle/Perpendicular"
] | [
"Homogeneous Quadratic Equation represents Two Straight Lines through Origin",
"Angle Between Two Straight Lines described by Homogeneous Quadratic Equation",
"Definition:Angle"
] |
proofwiki-18216 | Homogeneous Quadratic Equation for Straight Lines Parallel to those Passing through Origin | Let $\LL_1$ and $\LL_2$ represent $2$ straight lines in the Cartesian plane which are represented by a quadratic equation $E$ in two variables:
:$a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
Then $\LL_1$ and $\LL_2$ are parallel respectively to the $2$ straight lines through the origin represented by the homogeneou... | From Characteristic of Quadratic Equation that Represents Two Straight Lines we have the conditions in which $E$ does indeed represent $2$ straight lines.
Let $E$ be written as:
{{begin-eqn}}
{{eqn | l = b \paren {y - \mu_1 x - b_1} \paren {y - \mu_2 x - b_2}
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l... | Let $\LL_1$ and $\LL_2$ represent $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane]] which are represented by a [[Definition:Quadratic Equation in Two Variables|quadratic equation $E$ in two variables]]:
:$a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
Then $\LL_1$... | From [[Characteristic of Quadratic Equation that Represents Two Straight Lines]] we have the conditions in which $E$ does indeed represent $2$ [[Definition:Straight Line|straight lines]].
Let $E$ be written as:
{{begin-eqn}}
{{eqn | l = b \paren {y - \mu_1 x - b_1} \paren {y - \mu_2 x - b_2}
| r = 0
| c =... | Homogeneous Quadratic Equation for Straight Lines Parallel to those Passing through Origin | https://proofwiki.org/wiki/Homogeneous_Quadratic_Equation_for_Straight_Lines_Parallel_to_those_Passing_through_Origin | https://proofwiki.org/wiki/Homogeneous_Quadratic_Equation_for_Straight_Lines_Parallel_to_those_Passing_through_Origin | [
"Straight Lines"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Quadratic Equation in Two Variables",
"Definition:Parallel",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Homogeneous Quadratic Equation"
] | [
"Characteristic of Quadratic Equation that Represents Two Straight Lines",
"Definition:Line/Straight Line",
"Homogeneous Quadratic Equation represents Two Straight Lines through Origin",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Homogeneous Quadratic Equation",
... |
proofwiki-18217 | Angle Between Two Straight Lines described by Quadratic Equation | Let $\LL_1$ and $\LL_2$ represent $2$ straight lines in the Cartesian plane which are represented by a quadratic equation $E$ in two variables:
:$a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
Then the angle $\psi$ between $\LL_1$ and $\LL_2$ is given by:
:$\tan \psi = \dfrac {2 \sqrt {h^2 - a b} } {a + b}$ | From Homogeneous Quadratic Equation for Straight Lines Parallel to those Passing through Origin, $\LL_1$ and $\LL_2$ are parallel respectively to the $2$ straight lines through the origin $\LL'_1$ and $\LL'_2$ represented by the homogeneous quadratic equation:
:$a x^2 + 2 h x y + b y^2$
From Angle Between Two Straight ... | Let $\LL_1$ and $\LL_2$ represent $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane]] which are represented by a [[Definition:Quadratic Equation in Two Variables|quadratic equation $E$ in two variables]]:
:$a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
Then the [[... | From [[Homogeneous Quadratic Equation for Straight Lines Parallel to those Passing through Origin]], $\LL_1$ and $\LL_2$ are [[Definition:Parallel|parallel]] respectively to the $2$ [[Definition:Straight Line|straight lines]] through the [[Definition:Origin|origin]] $\LL'_1$ and $\LL'_2$ represented by the [[Definition... | Angle Between Two Straight Lines described by Quadratic Equation | https://proofwiki.org/wiki/Angle_Between_Two_Straight_Lines_described_by_Quadratic_Equation | https://proofwiki.org/wiki/Angle_Between_Two_Straight_Lines_described_by_Quadratic_Equation | [
"Straight Lines"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Quadratic Equation in Two Variables",
"Definition:Angle"
] | [
"Homogeneous Quadratic Equation for Straight Lines Parallel to those Passing through Origin",
"Definition:Parallel",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Homogeneous Quadratic Equation",
"Angle Between Two Straight Lines described by Homogeneous Quadratic Equ... |
proofwiki-18218 | Condition for Quadratic Equation to describe Perpendicular Straight Lines | Let $\LL_1$ and $\LL_2$ represent $2$ straight lines in the Cartesian plane which are represented by a quadratic equation $E$ in two variables:
:$a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
Let $\LL_1$ and $\LL_2$ be perpendicular.
Then:
:$a + b = 0$
That is, $E$ is of the form:
:$a \paren {x^2 - y^2} + 2 h x y + ... | From Homogeneous Quadratic Equation for Straight Lines Parallel to those Passing through Origin, $\LL_1$ and $\LL_2$ are parallel respectively to the $2$ straight lines through the origin $\LL'_1$ and $\LL'_2$ represented by the homogeneous quadratic equation:
:$a x^2 + 2 h x y + b y^2$
As $\LL_1$ and $\LL_2$ are paral... | Let $\LL_1$ and $\LL_2$ represent $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane]] which are represented by a [[Definition:Quadratic Equation in Two Variables|quadratic equation $E$ in two variables]]:
:$a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
Let $\LL_1$... | From [[Homogeneous Quadratic Equation for Straight Lines Parallel to those Passing through Origin]], $\LL_1$ and $\LL_2$ are [[Definition:Parallel|parallel]] respectively to the $2$ [[Definition:Straight Line|straight lines]] through the [[Definition:Origin|origin]] $\LL'_1$ and $\LL'_2$ represented by the [[Definition... | Condition for Quadratic Equation to describe Perpendicular Straight Lines | https://proofwiki.org/wiki/Condition_for_Quadratic_Equation_to_describe_Perpendicular_Straight_Lines | https://proofwiki.org/wiki/Condition_for_Quadratic_Equation_to_describe_Perpendicular_Straight_Lines | [
"Straight Lines",
"Perpendiculars"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Quadratic Equation in Two Variables",
"Definition:Right Angle/Perpendicular"
] | [
"Homogeneous Quadratic Equation for Straight Lines Parallel to those Passing through Origin",
"Definition:Parallel",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Homogeneous Quadratic Equation",
"Definition:Parallel",
"Definition:Right Angle/Perpendicular",
"Cond... |
proofwiki-18219 | Quadratic Equation for Parallel Straight Lines | Let $\LL_1$ and $\LL_2$ represent $2$ straight lines in the Cartesian plane which are represented by a quadratic equation $E$ in two variables:
:$a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
Let $\LL_1$ and $\LL_2$ be parallel.
Then:
:$h^2 - a b = 0$ | From Homogeneous Quadratic Equation representing Coincident Straight Lines, $\LL_1$ and $\LL_2$ are parallel respectively to the $2$ coincident straight lines through the origin $\LL'_1$ and $\LL'_2$ represented by the homogeneous quadratic equation:
:$a x^2 + 2 h x y + b y^2$
where:
:$h^2 - a b = 0$
Hence $\LL_1$ and ... | Let $\LL_1$ and $\LL_2$ represent $2$ [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane]] which are represented by a [[Definition:Quadratic Equation in Two Variables|quadratic equation $E$ in two variables]]:
:$a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
Let $\LL_1$... | From [[Homogeneous Quadratic Equation representing Coincident Straight Lines]], $\LL_1$ and $\LL_2$ are [[Definition:Parallel|parallel]] respectively to the $2$ coincident [[Definition:Straight Line|straight lines]] through the [[Definition:Origin|origin]] $\LL'_1$ and $\LL'_2$ represented by the [[Definition:Homogeneo... | Quadratic Equation for Parallel Straight Lines | https://proofwiki.org/wiki/Quadratic_Equation_for_Parallel_Straight_Lines | https://proofwiki.org/wiki/Quadratic_Equation_for_Parallel_Straight_Lines | [
"Straight Lines",
"Parallel Lines"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Quadratic Equation in Two Variables",
"Definition:Parallel"
] | [
"Homogeneous Quadratic Equation representing Coincident Straight Lines",
"Definition:Parallel",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Homogeneous Quadratic Equation",
"Definition:Homogeneous Quadratic Equation"
] |
proofwiki-18220 | Harmonic Conjugacy is Symmetric | Let $AB$ and $PQ$ be line segments on a straight line.
Let $P$ and $Q$ be '''harmonic conjugates''' with respect to $A$ and $B$.
Then $A$ and $B$ are '''harmonic conjugates''' with respect to $P$ and $Q$ . | By definition of harmonic conjugates, $\tuple {AB, PQ}$ is a harmonic range.
:400px
We have:
{{begin-eqn}}
{{eqn | l = \dfrac {AP} {PB}
| r = -\dfrac {AQ} {QB}
| c = {{Defof|Harmonic Range}}
}}
{{eqn | ll= \leadsto
| l = -\dfrac {PA} {PB}
| r = -\paren {-\dfrac {AQ} {BQ} }
| c = reversing ... | Let $AB$ and $PQ$ be [[Definition:Line Segment|line segments]] on a [[Definition:Straight Line|straight line]].
Let $P$ and $Q$ be '''[[Definition:Harmonic Conjugates|harmonic conjugates]]''' with respect to $A$ and $B$.
Then $A$ and $B$ are '''[[Definition:Harmonic Conjugates|harmonic conjugates]]''' with respect t... | By definition of [[Definition:Harmonic Conjugates|harmonic conjugates]], $\tuple {AB, PQ}$ is a [[Definition:Harmonic Range|harmonic range]].
:[[File:Harmonic-range.png|400px]]
We have:
{{begin-eqn}}
{{eqn | l = \dfrac {AP} {PB}
| r = -\dfrac {AQ} {QB}
| c = {{Defof|Harmonic Range}}
}}
{{eqn | ll= \lea... | Harmonic Conjugacy is Symmetric | https://proofwiki.org/wiki/Harmonic_Conjugacy_is_Symmetric | https://proofwiki.org/wiki/Harmonic_Conjugacy_is_Symmetric | [
"Harmonic Ranges"
] | [
"Definition:Line/Segment",
"Definition:Line/Straight Line",
"Definition:Harmonic Conjugates",
"Definition:Harmonic Conjugates"
] | [
"Definition:Harmonic Conjugates",
"Definition:Harmonic Range",
"File:Harmonic-range.png",
"Definition:Direction",
"Definition:Harmonic Range",
"Definition:Harmonic Conjugates"
] |
proofwiki-18221 | Condition for Pairs of Lines through Origin to be Harmonic Conjugates | Consider $4$ lines $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$ through the origin $O$ whose equations embedded in the Cartesian plane are as follows:
{{begin-eqn}}
{{eqn | ll= (\LL_1):
| l = y
| r = \lambda x
}}
{{eqn | ll= (\LL_2):
| l = y
| r = \mu x
}}
{{eqn | ll= (\LL_3):
| l = y
| r =... | === Sufficient Condition ===
Let $\set {\LL_1, \LL_2}$ and $\set {\LL_3, \LL_4}$ be harmonic conjugates as asserted.
A straight line in the plane which does not pass through $O$ will either:
:intersect all four of $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$
or:
:be parallel to one such straight line and intersect the other t... | Consider $4$ [[Definition:Straight Line|lines]] $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$ through the [[Definition:Origin|origin]] $O$ whose [[Definition:Equation|equations]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]] are as follows:
{{begin-eqn}}
{{eqn | ll= (\LL_1):
| l = y
| r = \lambda ... | === Sufficient Condition ===
Let $\set {\LL_1, \LL_2}$ and $\set {\LL_3, \LL_4}$ be [[Definition:Harmonic Conjugates of Harmonic Pencil|harmonic conjugates]] as asserted.
A [[Definition:Straight Line|straight line]] in [[Definition:The Plane|the plane]] which does not pass through $O$ will either:
:[[Definition:Inter... | Condition for Pairs of Lines through Origin to be Harmonic Conjugates | https://proofwiki.org/wiki/Condition_for_Pairs_of_Lines_through_Origin_to_be_Harmonic_Conjugates | https://proofwiki.org/wiki/Condition_for_Pairs_of_Lines_through_Origin_to_be_Harmonic_Conjugates | [
"Harmonic Ranges"
] | [
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Equation",
"Definition:Cartesian Plane",
"Definition:Doubleton",
"Definition:Line/Straight Line",
"Definition:Harmonic Conjugates/Harmonic Pencil"
] | [
"Definition:Harmonic Conjugates/Harmonic Pencil",
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Intersection (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line",
"Definition:Intersection (Geometry)",
"Definition:Line/Straight Line",... |
proofwiki-18222 | Nth Derivative of Exponential of Minus One over x | Let $n \in \N$ be a natural number.
Let $\map {P_n} x$ be a real polynomial of degree $n$.
Then:
:$\ds \dfrac {\d^n} {\d x^n} \map \exp {- \frac 1 x} = \frac {\map {P_n} x}{x^{2n}} \map \exp {- \frac 1 x}$
where $\map {P_n} x$ satisfies the following recurrence relation:
:$\map {P_{n + 1}} x = x^2 \dfrac \d {\d x} \map... | Proof by induction: | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $\map {P_n} x$ be a [[Definition:Polynomial over Real Numbers|real polynomial]] of [[Definition:Degree of Polynomial|degree $n$]].
Then:
:$\ds \dfrac {\d^n} {\d x^n} \map \exp {- \frac 1 x} = \frac {\map {P_n} x}{x^{2n}} \map \exp {- \frac 1 x}$
... | Proof by [[Principle of Mathematical Induction|induction]]: | Nth Derivative of Exponential of Minus One over x | https://proofwiki.org/wiki/Nth_Derivative_of_Exponential_of_Minus_One_over_x | https://proofwiki.org/wiki/Nth_Derivative_of_Exponential_of_Minus_One_over_x | [
"Proofs by Induction"
] | [
"Definition:Natural Numbers",
"Definition:Polynomial/Real Numbers",
"Definition:Degree of Polynomial",
"Definition:Recursive Sequence/Recurrence Relation"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-18223 | Condition for Pairs of Lines through Origin to be Harmonic Conjugates/Homogeneous Quadratic Equation Form | Consider the two homogeneous quadratic equations:
{{begin-eqn}}
{{eqn | n = E1
| l = a_1 x^2 + 2 h_1 x y + b_1 y^2
| r = 0
}}
{{eqn | n = E2
| l = a_2 x^2 + 2 h_2 x y + b_2 y^2
| r = 0
}}
{{end-eqn}}
each representing two straight lines through the origin.
Then the two straight lines represented... | From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, $(\text E1)$ and $(\text E2)$ represent straight lines through the origin {{iff}}:
{{begin-eqn}}
{{eqn | l = h_1^2 - a_1 b_1
| o = >
| r = 0
}}
{{eqn | l = h_2^2 - a_2 b_2
| o = >
| r = 0
}}
{{end-eqn}}
Let the two... | Consider the two [[Definition:Homogeneous Quadratic Equation|homogeneous quadratic equations]]:
{{begin-eqn}}
{{eqn | n = E1
| l = a_1 x^2 + 2 h_1 x y + b_1 y^2
| r = 0
}}
{{eqn | n = E2
| l = a_2 x^2 + 2 h_2 x y + b_2 y^2
| r = 0
}}
{{end-eqn}}
each representing two [[Definition:Straight Line... | From [[Homogeneous Quadratic Equation represents Two Straight Lines through Origin]], $(\text E1)$ and $(\text E2)$ represent [[Definition:Straight Line|straight lines]] through the [[Definition:Origin|origin]] {{iff}}:
{{begin-eqn}}
{{eqn | l = h_1^2 - a_1 b_1
| o = >
| r = 0
}}
{{eqn | l = h_2^2 - a_2 b_... | Condition for Pairs of Lines through Origin to be Harmonic Conjugates/Homogeneous Quadratic Equation Form | https://proofwiki.org/wiki/Condition_for_Pairs_of_Lines_through_Origin_to_be_Harmonic_Conjugates/Homogeneous_Quadratic_Equation_Form | https://proofwiki.org/wiki/Condition_for_Pairs_of_Lines_through_Origin_to_be_Harmonic_Conjugates/Homogeneous_Quadratic_Equation_Form | [
"Harmonic Ranges"
] | [
"Definition:Homogeneous Quadratic Equation",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Line/Straight Line",
"Definition:Harmonic Conjugates/Harmonic Pencil",
"Definition:Line/Straight Line"
] | [
"Homogeneous Quadratic Equation represents Two Straight Lines through Origin",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Line/Straight Line",
"Definition:Equation",
"Definition:Line/Straight Line",
"Definition:Equation",
"Condition for Pairs of Lines through O... |
proofwiki-18224 | Angle Bisectors are Harmonic Conjugates | Let $\LL_1$ and $\LL_2$ be straight lines which intersect at $O$.
Let $\LL_3$ and $\LL_4$ be the angle bisectors of the angles formed at the point of intersection of $\LL_1$ and $\LL_2$.
Then $\LL_3$ and $\LL_4$ are harmonic conjugates with respect to $\LL_1$ and $\LL_2$. | Consider a straight line parallel to $\LL_4$ which intersects $\LL_1$, $\LL_2$ and $\LL_3$ at $L$, $M$ and $N$ respectively.
From Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular, $\LL_3$ is perpendicular to $\LL_4$.
Hence as $LM$ is parallel to $\LL_4$, $LM$ is perpendicular to $O... | Let $\LL_1$ and $\LL_2$ be [[Definition:Straight Line|straight lines]] which [[Definition:Intersection (Geometry)|intersect]] at $O$.
Let $\LL_3$ and $\LL_4$ be the [[Definition:Angle Bisector|angle bisectors]] of the [[Definition:Angle|angles]] formed at the [[Definition:Point|point]] of [[Definition:Intersection (Ge... | Consider a [[Definition:Straight Line|straight line]] [[Definition:Parallel Lines|parallel]] to $\LL_4$ which [[Definition:Intersection (Geometry)|intersects]] $\LL_1$, $\LL_2$ and $\LL_3$ at $L$, $M$ and $N$ respectively.
From [[Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular]],... | Angle Bisectors are Harmonic Conjugates | https://proofwiki.org/wiki/Angle_Bisectors_are_Harmonic_Conjugates | https://proofwiki.org/wiki/Angle_Bisectors_are_Harmonic_Conjugates | [
"Angle Bisectors",
"Harmonic Ranges"
] | [
"Definition:Line/Straight Line",
"Definition:Intersection (Geometry)",
"Definition:Angle Bisector",
"Definition:Angle",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Harmonic Conjugates/Harmonic Pencil"
] | [
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Intersection (Geometry)",
"Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular",
"Definition:Right Angle/Perpendicular",
"Definition:Parallel (Geometry)/Lines",
"Definition:Right Angle... |
proofwiki-18225 | Bisectors of Angles between Two Straight Lines/Homogeneous Quadratic Equation Form | Consider the homogeneous quadratic equation:
:$(1): \quad a x^2 + 2 h x y + b y^2 = 0$
representing two straight lines through the origin.
Then the homogeneous quadratic equation which represents the angle bisectors of the angles formed at their point of intersection is given by:
:$h x^2 - \paren {a - b} x y - h y^2 = ... | From Angle Bisectors are Harmonic Conjugates, the two angle bisectors are harmonic conjugates of the straight lines represented by $(1)$.
From Condition for Homogeneous Quadratic Equation to describe Perpendicular Straight Lines, these angle bisectors can be described by the homogeneous quadratic equation:
:$x^2 + 2 \l... | Consider the [[Definition:Homogeneous Quadratic Equation|homogeneous quadratic equation]]:
:$(1): \quad a x^2 + 2 h x y + b y^2 = 0$
representing two [[Definition:Straight Line|straight lines]] through the [[Definition:Origin|origin]].
Then the [[Definition:Homogeneous Quadratic Equation|homogeneous quadratic equati... | From [[Angle Bisectors are Harmonic Conjugates]], the two [[Definition:Angle Bisector|angle bisectors]] are [[Definition:Harmonic Conjugates of Harmonic Pencil|harmonic conjugates]] of the [[Definition:Straight Line|straight lines]] represented by $(1)$.
From [[Condition for Homogeneous Quadratic Equation to describe ... | Bisectors of Angles between Two Straight Lines/Homogeneous Quadratic Equation Form | https://proofwiki.org/wiki/Bisectors_of_Angles_between_Two_Straight_Lines/Homogeneous_Quadratic_Equation_Form | https://proofwiki.org/wiki/Bisectors_of_Angles_between_Two_Straight_Lines/Homogeneous_Quadratic_Equation_Form | [
"Harmonic Ranges",
"Bisectors of Angles between Two Straight Lines"
] | [
"Definition:Homogeneous Quadratic Equation",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Homogeneous Quadratic Equation",
"Definition:Angle Bisector",
"Definition:Angle",
"Definition:Point",
"Definition:Intersection (Geometry)"
] | [
"Angle Bisectors are Harmonic Conjugates",
"Definition:Angle Bisector",
"Definition:Harmonic Conjugates/Harmonic Pencil",
"Definition:Line/Straight Line",
"Condition for Homogeneous Quadratic Equation to describe Perpendicular Straight Lines",
"Definition:Angle Bisector",
"Definition:Homogeneous Quadrat... |
proofwiki-18226 | Equation of Tangent to Circle Centered at Origin | Let $\CC$ be a circle whose radius is $r$ and whose center is at the origin of a Cartesian plane.
Let $P = \tuple {x_1, y_1}$ be a point on $\CC$.
Let $\TT$ be a tangent to $\CC$ passing through $P$.
Then $\TT$ can be defined by the equation:
:$x x_1 + y y_1 = r^2$ | From Equation of Straight Line Tangent to Circle we have that for a general circle of radius $r$ and center $\tuple {a, b}$:
:$y - y_1 = \dfrac {a - x_1} {y_1 - b} \paren {x - x_1}$
is the equation of a tangent $\TT$ to $\CC$ passing through $\tuple {x_1, y_1}$.
Setting the center to $\tuple {0, 0}$:
{{begin-eqn}}
{{eq... | Let $\CC$ be a [[Definition:Circle|circle]] whose [[Definition:Radius of Circle|radius]] is $r$ and whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P = \tuple {x_1, y_1}$ be a [[Definition:Point|point]] on $\CC$.
Let $\TT$ b... | From [[Equation of Straight Line Tangent to Circle]] we have that for a general [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ and [[Definition:Center of Circle|center]] $\tuple {a, b}$:
:$y - y_1 = \dfrac {a - x_1} {y_1 - b} \paren {x - x_1}$
is the equation of a [[Definition:Tangent to C... | Equation of Tangent to Circle Centered at Origin/Proof 1 | https://proofwiki.org/wiki/Equation_of_Tangent_to_Circle_Centered_at_Origin | https://proofwiki.org/wiki/Equation_of_Tangent_to_Circle_Centered_at_Origin/Proof_1 | [
"Equation of Tangent to Circle Centered at Origin",
"Circles",
"Tangents"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Tangent Line/Circle",
"Definition:Equation"
] | [
"Equation of Straight Line Tangent to Circle",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Tangent Line/Circle",
"Definition:Circle/Center"
] |
proofwiki-18227 | Equation of Tangent to Circle Centered at Origin | Let $\CC$ be a circle whose radius is $r$ and whose center is at the origin of a Cartesian plane.
Let $P = \tuple {x_1, y_1}$ be a point on $\CC$.
Let $\TT$ be a tangent to $\CC$ passing through $P$.
Then $\TT$ can be defined by the equation:
:$x x_1 + y y_1 = r^2$ | From the slope-intercept form of a line, the equation of a line passing through $P$ is:
:$y - y_1 = \mu \paren {x - x_1}$
If this line passes through another point $\tuple {x_2, y_2}$ on $\CC$, the slope of the line is given by:
:$\mu = \dfrac {y_2 - y_1} {x_2 - x_1}$
Because $P$ and $Q$ both lie on $\CC$, we have:
{{b... | Let $\CC$ be a [[Definition:Circle|circle]] whose [[Definition:Radius of Circle|radius]] is $r$ and whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P = \tuple {x_1, y_1}$ be a [[Definition:Point|point]] on $\CC$.
Let $\TT$ b... | From the [[Equation of Straight Line in Plane/Slope-Intercept Form|slope-intercept form]] of a [[Definition:Straight Line|line]], the equation of a [[Definition:Straight Line|line]] passing through $P$ is:
:$y - y_1 = \mu \paren {x - x_1}$
If this [[Definition:Straight Line|line]] passes through another [[Definition:P... | Equation of Tangent to Circle Centered at Origin/Proof 2 | https://proofwiki.org/wiki/Equation_of_Tangent_to_Circle_Centered_at_Origin | https://proofwiki.org/wiki/Equation_of_Tangent_to_Circle_Centered_at_Origin/Proof_2 | [
"Equation of Tangent to Circle Centered at Origin",
"Circles",
"Tangents"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Tangent Line/Circle",
"Definition:Equation"
] | [
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line",
"Definition:Point",
"Definition:Slope/Straight Line",
"Definition:Line/Straight Line",
"Definition:Limit of Real Function",
"Definition:Slope/... |
proofwiki-18228 | Equation of Normal to Circle Centered at Origin | Let $\CC$ be a circle whose radius is $r$ and whose center is at the origin of a Cartesian plane.
Let $P = \tuple {x_1, y_1}$ be a point on $\CC$.
Let $\NN$ be a normal to $\CC$ passing through $P$.
Then $\NN$ can be defined by the equation:
:$y_1 x - x_1 y = 0$ | Let $\TT$ be the tangent to $\CC$ passing through $P$.
From Equation of Tangent to Circle Centered at Origin, $\TT$ can be described using the equation:
:$x x_1 + y y_1 = r^2$
expressible as:
:$y - y_1 = -\dfrac {x_1} {y_1} \paren {x - x_1}$
where the slope of $\TT$ is $-\dfrac {x_1} {y_1}$.
By definition, the normal i... | Let $\CC$ be a [[Definition:Circle|circle]] whose [[Definition:Radius of Circle|radius]] is $r$ and whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P = \tuple {x_1, y_1}$ be a [[Definition:Point|point]] on $\CC$.
Let $\NN$ b... | Let $\TT$ be the [[Definition:Tangent to Circle|tangent]] to $\CC$ passing through $P$.
From [[Equation of Tangent to Circle Centered at Origin]], $\TT$ can be described using the [[Definition:Equation|equation]]:
:$x x_1 + y y_1 = r^2$
expressible as:
:$y - y_1 = -\dfrac {x_1} {y_1} \paren {x - x_1}$
where the [[... | Equation of Normal to Circle Centered at Origin | https://proofwiki.org/wiki/Equation_of_Normal_to_Circle_Centered_at_Origin | https://proofwiki.org/wiki/Equation_of_Normal_to_Circle_Centered_at_Origin | [
"Circles",
"Normals to Curves"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Normal to Curve",
"Definition:Equation"
] | [
"Definition:Tangent Line/Circle",
"Equation of Tangent to Circle Centered at Origin",
"Definition:Equation",
"Definition:Slope/Straight Line",
"Definition:Normal to Curve",
"Definition:Right Angle/Perpendicular",
"Definition:Tangent Line/Circle",
"Condition for Straight Lines in Plane to be Perpendicu... |
proofwiki-18229 | Normal to Circle passes through Center | A normal $\NN$ to a circle $\CC$ passes through the center of $\CC$. | Let $\CC$ be positioned in a Cartesian plane with its center at the origin.
Let $\NN$ pass through the point $\tuple {x_1, y_1}$.
From Equation of Normal to Circle Centered at Origin, $\NN$ has the equation:
:$y_1 x - x_1 y = 0$
or:
:$y = \dfrac {y_1} {x_1} x$
From the Equation of Straight Line in Plane: Slope-Intercep... | A [[Definition:Normal to Curve|normal]] $\NN$ to a [[Definition:Circle|circle]] $\CC$ passes through the [[Definition:Center of Circle|center]] of $\CC$. | Let $\CC$ be positioned in a [[Definition:Cartesian Plane|Cartesian]] plane with its [[Definition:Center of Circle|center]] at the [[Definition:Origin|origin]].
Let $\NN$ pass through the [[Definition:Point|point]] $\tuple {x_1, y_1}$.
From [[Equation of Normal to Circle Centered at Origin]], $\NN$ has the equation:
... | Normal to Circle passes through Center | https://proofwiki.org/wiki/Normal_to_Circle_passes_through_Center | https://proofwiki.org/wiki/Normal_to_Circle_passes_through_Center | [
"Circles",
"Normals to Curves"
] | [
"Definition:Normal to Curve",
"Definition:Circle",
"Definition:Circle/Center"
] | [
"Definition:Cartesian Plane",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Point",
"Equation of Normal to Circle Centered at Origin",
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Definition:Line/Straight Line",
"Definition:Coordinate System/Origin",
"... |
proofwiki-18230 | Equation of Chord of Contact on Circle Centered at Origin | Let $\CC$ be a circle whose radius is $r$ and whose center is at the origin of a Cartesian plane.
Let $P = \tuple {x_0, y_0}$ be a point which is outside the boundary of $\CC$.
Let $UV$ be the chord of contact on $\CC$ with respect to $P$.
Then $UV$ can be defined by the equation:
:$x x_0 + y y_0 = r^2$ | Let $\TT_1$ and $\TT_2$ be a tangents to $\CC$ passing through $P$.
Let:
:$\TT_1$ touch $\CC$ at $U = \tuple {x_1, y_1}$
:$\TT_2$ touch $\CC$ at $V = \tuple {x_2, y_2}$
Then the chord of contact on $\CC$ with respect to $P$ is defined as $UV$.
:480px
From Equation of Tangent to Circle Centered at Origin, $\TT_1$ is exp... | Let $\CC$ be a [[Definition:Circle|circle]] whose [[Definition:Radius of Circle|radius]] is $r$ and whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P = \tuple {x_0, y_0}$ be a [[Definition:Point|point]] which is outside the [... | Let $\TT_1$ and $\TT_2$ be a [[Definition:Tangent to Circle|tangents]] to $\CC$ passing through $P$.
Let:
:$\TT_1$ [[Definition:Tangent to Circle|touch]] $\CC$ at $U = \tuple {x_1, y_1}$
:$\TT_2$ [[Definition:Tangent to Circle|touch]] $\CC$ at $V = \tuple {x_2, y_2}$
Then the [[Definition:Chord of Contact on Circle|c... | Equation of Chord of Contact on Circle Centered at Origin | https://proofwiki.org/wiki/Equation_of_Chord_of_Contact_on_Circle_Centered_at_Origin | https://proofwiki.org/wiki/Equation_of_Chord_of_Contact_on_Circle_Centered_at_Origin | [
"Circles",
"Chords of Contact"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Boundary (Geometry)",
"Definition:Chord of Contact/Circle",
"Definition:Equation"
] | [
"Definition:Tangent Line/Circle",
"Definition:Tangent Line/Circle",
"Definition:Tangent Line/Circle",
"Definition:Chord of Contact/Circle",
"File:Equation-of-Polar.png",
"Equation of Tangent to Circle Centered at Origin",
"Equation of Tangent to Circle Centered at Origin",
"Definition:Line/Straight Li... |
proofwiki-18231 | Harmonic Property of Pole and Polar/Circle | Let $\CC$ be a circle whose radius is $r$ and whose center is at the origin of a Cartesian plane.
Let $P$ be an arbitrary point in the Cartesian plane.
Let $\LL$ be a straight line through $P$ which intersects $\CC$ at points $U$ and $V$.
Let $Q$ be the point where $\LL$ intersects the polar of $P$.
Then $\tuple {PQ, U... | From Equation of Circle center Origin, we have that the equation of $\CC$ is:
:$x^2 + y^2 = r^2$
Let $P = \tuple {x_1, y_1}$.
Let $Q = \tuple {x, y}$ be a point on $\LL$.
Let $V$ divide $PQ$ in the ratio $k : 1$.
Then the coordinates of $V$ are:
:$V = \tuple {\dfrac {k x + x_1} {k + 1}, \dfrac {k y + y_1} {k + 1} }$
Su... | Let $\CC$ be a [[Definition:Circle|circle]] whose [[Definition:Radius of Circle|radius]] is $r$ and whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P$ be an arbitrary [[Definition:Point|point]] in the [[Definition:Cartesian P... | From [[Equation of Circle center Origin]], we have that the equation of $\CC$ is:
:$x^2 + y^2 = r^2$
Let $P = \tuple {x_1, y_1}$.
Let $Q = \tuple {x, y}$ be a [[Definition:Point|point]] on $\LL$.
Let $V$ divide $PQ$ in the [[Definition:Ratio|ratio]] $k : 1$.
Then the [[Definition:Cartesian Coordinates|coordinates]... | Harmonic Property of Pole and Polar/Circle | https://proofwiki.org/wiki/Harmonic_Property_of_Pole_and_Polar/Circle | https://proofwiki.org/wiki/Harmonic_Property_of_Pole_and_Polar/Circle | [
"Harmonic Property of Pole and Polar"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Line/Straight Line",
"Definition:Intersection (Geometry)",
"Definition:Point",
"Definiti... | [
"Equation of Circle center Origin",
"Definition:Point",
"Definition:Ratio",
"Definition:Cartesian Coordinate System",
"Definition:Root of Equation",
"Definition:Quadratic Equation",
"Definition:Polar of Point/Circle",
"File:Harmonic-property-of-polar.png",
"Definition:Polar of Point/Circle",
"Defi... |
proofwiki-18232 | Polar of Point is Perpendicular to Line through Center | Let $\CC$ be a circle.
Let $P$ be a point.
Let $\LL$ be the polar of $P$ with respect to $\CC$.
Then $\LL$ is perpendicular to the straight line through $P$ and the center of $\CC$. | Let $\CC$ be positioned so as for its center to be at the origin of a Cartesian plane.
Let $P$ be located at $\tuple {x_0, y_0}$.
From Equation of Straight Line in Plane, $P$ can be described as:
:$y = \dfrac {y_0} {x_0} x$
and so has slope $\dfrac {y_0} {x_0}$.
By definition of polar, $\LL$ has the equation:
:$x x_0 +... | Let $\CC$ be a [[Definition:Circle|circle]].
Let $P$ be a [[Definition:Point|point]].
Let $\LL$ be the [[Definition:Polar of Point wrt Circle|polar]] of $P$ with respect to $\CC$.
Then $\LL$ is [[Definition:Perpendicular|perpendicular]] to the [[Definition:Straight Line|straight line]] through $P$ and the [[Definit... | Let $\CC$ be positioned so as for its [[Definition:Center of Circle|center]] to be at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P$ be located at $\tuple {x_0, y_0}$.
From [[Equation of Straight Line in Plane]], $P$ can be described as:
:$y = \dfrac {y_0} {x_0} x$
and s... | Polar of Point is Perpendicular to Line through Center | https://proofwiki.org/wiki/Polar_of_Point_is_Perpendicular_to_Line_through_Center | https://proofwiki.org/wiki/Polar_of_Point_is_Perpendicular_to_Line_through_Center | [
"Polars of Points",
"Perpendiculars"
] | [
"Definition:Circle",
"Definition:Point",
"Definition:Polar of Point/Circle",
"Definition:Right Angle/Perpendicular",
"Definition:Line/Straight Line",
"Definition:Circle/Center"
] | [
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane",
"Definition:Slope/Straight Line",
"Definition:Polar of Point/Circle",
"Definition:Slope/Straight Line",
"Condition for Straight Lines in Plane to be Perpendicular"
] |
proofwiki-18233 | Product of Distances of Polar and Pole from Center of Circle | Let $\CC$ be a circle of radius $r$ whose center is at $O$.
Let $P$ be a point.
Let $\LL_1$ be the polar of $P$ with respect to $\CC$.
Let $\LL_2$ be the line $OP$.
Let $N$ be the point of intersection of $\LL_1$ and $\LL_2$.
Then:
:$ON \cdot OP = r^2$ | Let $U$ and $V$ be the points where $OP$ intersects $\CC$.
:400px
From Harmonic Property of Pole and Polar wrt Circle, $\tuple {UV, NP}$ form a harmonic range.
That is:
{{begin-eqn}}
{{eqn | l = \dfrac {VN} {NU}
| r = -\dfrac {VP} {PN}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {ON} {OV}
| r = ... | Let $\CC$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ whose [[Definition:Center of Circle|center]] is at $O$.
Let $P$ be a [[Definition:Point|point]].
Let $\LL_1$ be the [[Definition:Polar of Point wrt Circle|polar]] of $P$ with respect to $\CC$.
Let $\LL_2$ be the [[Definition:St... | Let $U$ and $V$ be the [[Definition:Point|points]] where $OP$ [[Definition:Intersection (Geometry)|intersects]] $\CC$.
:[[File:Distance-from-center-of-polar.png|400px]]
From [[Harmonic Property of Pole and Polar wrt Circle]], $\tuple {UV, NP}$ form a [[Definition:Harmonic Range|harmonic range]].
That is:
{{begin-... | Product of Distances of Polar and Pole from Center of Circle | https://proofwiki.org/wiki/Product_of_Distances_of_Polar_and_Pole_from_Center_of_Circle | https://proofwiki.org/wiki/Product_of_Distances_of_Polar_and_Pole_from_Center_of_Circle | [
"Polars of Points"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Point",
"Definition:Polar of Point/Circle",
"Definition:Line/Straight Line",
"Definition:Point",
"Definition:Intersection (Geometry)"
] | [
"Definition:Point",
"Definition:Intersection (Geometry)",
"File:Distance-from-center-of-polar.png",
"Harmonic Property of Pole and Polar/Circle",
"Definition:Harmonic Range",
"Definition:Line/Midpoint",
"Definition:Circle/Diameter"
] |
proofwiki-18234 | Coordinates of Pole of Given Polar | Let $\CC$ be a circle of radius $r$ whose center is at the origin of a Cartesian plane.
Let $\LL$ be a straight line whose equation is given as:
:$l x + m y + n = 0$
Then the pole $P$ of $\LL$ with respect to $\CC$ is:
:$P = \tuple {-\dfrac l n r^2, -\dfrac m n r^2}$ | From Equation of Circle center Origin, we have that the equation of $\CC$ is:
:$x^2 + y^2 = r^2$
Let $P = \tuple {x_0, y_0}$.
By definition of polar:
:$x x_0 + y y_0 = r^2$
Comparing this with the equation for $\LL$:
:$\dfrac {x_0} l = \dfrac {y_0} m = \dfrac {r^2} {-n}$
The result follows.
{{qed}} | Let $\CC$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $\LL$ be a [[Definition:Straight Line|straight line]] whose [[Equation of Straight Line ... | From [[Equation of Circle center Origin]], we have that the equation of $\CC$ is:
:$x^2 + y^2 = r^2$
Let $P = \tuple {x_0, y_0}$.
By definition of [[Definition:Polar of Point wrt Circle|polar]]:
:$x x_0 + y y_0 = r^2$
Comparing this with the equation for $\LL$:
:$\dfrac {x_0} l = \dfrac {y_0} m = \dfrac {r^2} {-n}$... | Coordinates of Pole of Given Polar | https://proofwiki.org/wiki/Coordinates_of_Pole_of_Given_Polar | https://proofwiki.org/wiki/Coordinates_of_Pole_of_Given_Polar | [
"Polars of Points"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Line/Straight Line",
"Equation of Straight Line in Plane/General Equation",
"Definition:Polar of Point/Circle/Pole"
] | [
"Equation of Circle center Origin",
"Definition:Polar of Point/Circle"
] |
proofwiki-18235 | Coordinates of Pole of Given Polar/Homogeneous Coordinates | The pole $P$ of $\LL$ with respect to $\CC$, given in homogeneous Cartesian coordinates is:
:$P = \tuple {l, m, -\dfrac n {r^2} }$ | From Coordinates of Pole of Given Polar, $P$ can be expressed in conventional Cartesian coordinates as:
:$P = \tuple {-\dfrac l n r^2, -\dfrac m n r^2}$
Hence in homogeneous Cartesian coordinates:
:$P = \tuple {-\dfrac l n r^2, -\dfrac m n r^2, 1}$
From Multiples of Homogeneous Cartesian Coordinates represent Same Poin... | The [[Definition:Pole of Polar wrt Circle|pole]] $P$ of $\LL$ with respect to $\CC$, given in [[Definition:Homogeneous Cartesian Coordinates|homogeneous Cartesian coordinates]] is:
:$P = \tuple {l, m, -\dfrac n {r^2} }$ | From [[Coordinates of Pole of Given Polar]], $P$ can be expressed in conventional [[Definition:Cartesian Coordinates|Cartesian coordinates]] as:
:$P = \tuple {-\dfrac l n r^2, -\dfrac m n r^2}$
Hence in [[Definition:Homogeneous Cartesian Coordinates|homogeneous Cartesian coordinates]]:
:$P = \tuple {-\dfrac l n r^2, ... | Coordinates of Pole of Given Polar/Homogeneous Coordinates | https://proofwiki.org/wiki/Coordinates_of_Pole_of_Given_Polar/Homogeneous_Coordinates | https://proofwiki.org/wiki/Coordinates_of_Pole_of_Given_Polar/Homogeneous_Coordinates | [
"Polars of Points"
] | [
"Definition:Polar of Point/Circle/Pole",
"Definition:Homogeneous Cartesian Coordinates"
] | [
"Coordinates of Pole of Given Polar",
"Definition:Cartesian Coordinate System",
"Definition:Homogeneous Cartesian Coordinates",
"Multiples of Homogeneous Cartesian Coordinates represent Same Point"
] |
proofwiki-18236 | Reciprocal Property of Pole and Polar | Let $\CC$ be a circle.
Let $P$ and $Q$ be points in the plane of $\CC$.
Let $\PP$ and $\QQ$ be the polars of $P$ and $Q$ with respect to $\CC$ respectively.
Let $Q$ lie on $\PP$ with respect to $\CC$.
Then $P$ lies on $\QQ$.
That is:
:if $P$ lies on the polar of $Q$, then $Q$ lies on the polar of $P$
:if the pole of $\... | Let $\CC$ be a circle of radius $r$ whose center is at the origin of a Cartesian plane.
Let $P = \tuple {x_0, y_0}$.
Let $Q = \tuple {x_1, y_1}$.
The polar of $P$ is given by:
:$x x_0 + y y_0 = r^2$
The polar of $Q$ is given by:
:$x x_1 + y y_1 = r^2$
Let $Q$ lie on the polar of $P$.
Then $Q$ satisfies the equation:
:$... | Let $\CC$ be a [[Definition:Circle|circle]].
Let $P$ and $Q$ be [[Definition:Point|points]] in the [[Definition:The Plane|plane]] of $\CC$.
Let $\PP$ and $\QQ$ be the [[Definition:Polar of Point wrt Circle|polars]] of $P$ and $Q$ with respect to $\CC$ respectively.
Let $Q$ lie on $\PP$ with respect to $\CC$.
Then ... | Let $\CC$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P = \tuple {x_0, y_0}$.
Let $Q = \tuple {x_1, y_1}$.
The [[Definition:Polar of Point ... | Reciprocal Property of Pole and Polar | https://proofwiki.org/wiki/Reciprocal_Property_of_Pole_and_Polar | https://proofwiki.org/wiki/Reciprocal_Property_of_Pole_and_Polar | [
"Polars of Points",
"Conjugate Points",
"Conjugate Lines"
] | [
"Definition:Circle",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle/Pole",
"Definition:Polar of Point/Circle/Pole"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle"
] |
proofwiki-18237 | Derivative of Differentiable Function on Open Interval is Baire Function | Let $I = \openint a b$ be a open interval.
Let $f : I \to \R$ be a differentiable function.
Let $f' : I \to \R$ be the derivative of $f$.
Then $f'$ is a Baire function. | For each $n \in \N$, let $f_n : I \to \R$ be a function with:
:$\ds \map {f_n} x = n \paren {\map f {\min \set {x + \frac 1 n, \frac {b + x} 2} } - \map f x}$
for each $x$.
Note that from:
:Minimum of Finitely Many Continuous Real Functions is Continuous
:Composite of Continuous Mappings is Continuous
:Combined Sum Ru... | Let $I = \openint a b$ be a [[Definition:Open Interval|open interval]].
Let $f : I \to \R$ be a [[Definition:Differentiable Real Function|differentiable function]].
Let $f' : I \to \R$ be the [[Definition:Derivative of Real Function|derivative]] of $f$.
Then $f'$ is a [[Definition:Baire Function|Baire function]]. | For each $n \in \N$, let $f_n : I \to \R$ be a [[Definition:Real Function|function]] with:
:$\ds \map {f_n} x = n \paren {\map f {\min \set {x + \frac 1 n, \frac {b + x} 2} } - \map f x}$
for each $x$.
Note that from:
:[[Minimum of Finitely Many Continuous Real Functions is Continuous]]
:[[Composite of Continuous Map... | Derivative of Differentiable Function on Open Interval is Baire Function | https://proofwiki.org/wiki/Derivative_of_Differentiable_Function_on_Open_Interval_is_Baire_Function | https://proofwiki.org/wiki/Derivative_of_Differentiable_Function_on_Open_Interval_is_Baire_Function | [
"Baire Functions"
] | [
"Definition:Interval/Ordered Set/Open",
"Definition:Differentiable Mapping/Real Function",
"Definition:Derivative/Real Function",
"Definition:Baire Function"
] | [
"Definition:Real Function",
"Minimum of Finitely Many Continuous Real Functions is Continuous",
"Composite of Continuous Mappings is Continuous",
"Combination Theorem for Continuous Functions/Real/Combined Sum Rule",
"Definition:Continuous Real Function",
"Definition:Pointwise Convergence",
"Definition:... |
proofwiki-18238 | Riemann Integrable Dirac Function does not Exist | Let $\delta : \R \to \R$ be a real function.
Let $\phi : \R \to \R$ be a smooth function vanishing outside $\closedint a b$.
Let $a \in \R_{> 0}$ be a real number.
Suppose $\delta$ is Riemann integrable on $\closedint {-a} a$.
Suppose for every $\phi$ we have that:
:$\ds \int_{-a}^a \map \delta x \map \phi x = \map \ph... | {{AimForCont}} $\delta$ exists.
Let $\phi$ be a test function of the following form:
:$\map \phi x = \begin {cases} \map \exp {\dfrac 1 {x^2 - 1} } & : \size x < 1 \\ 0 & : \size x \ge 1 \end {cases}$
Let $n \in \N$ be a natural number.
Let $\phi_n : \R \to \R$ be a real function such that:
:$\forall n \in N : \forall ... | Let $\delta : \R \to \R$ be a [[Definition:Real Function|real function]].
Let $\phi : \R \to \R$ be a [[Definition:Smooth Function|smooth function]] vanishing outside $\closedint a b$.
Let $a \in \R_{> 0}$ be a [[Definition:Real Number|real number]].
Suppose $\delta$ is [[Definition:Riemann Integrable Function|Riema... | {{AimForCont}} $\delta$ [[Definition:Existential Quantifier|exists]].
Let $\phi$ be a [[Test Function/Examples/Exponential of One over x Squared minus One|test function]] of the following form:
:$\map \phi x = \begin {cases} \map \exp {\dfrac 1 {x^2 - 1} } & : \size x < 1 \\ 0 & : \size x \ge 1 \end {cases}$
Let $n ... | Riemann Integrable Dirac Function does not Exist | https://proofwiki.org/wiki/Riemann_Integrable_Dirac_Function_does_not_Exist | https://proofwiki.org/wiki/Riemann_Integrable_Dirac_Function_does_not_Exist | [
"Dirac Delta Function"
] | [
"Definition:Real Function",
"Definition:Smooth Real Function",
"Definition:Real Number",
"Definition:Definite Integral/Riemann",
"Definition:Existential Quantifier"
] | [
"Definition:Existential Quantifier",
"Test Function/Examples/Exponential of One over x Squared minus One",
"Definition:Natural Numbers",
"Definition:Real Function",
"Definition:Smooth Real Function",
"Riemann Integrable Function is Bounded",
"Definition:Limit of Real Function",
"Definition:Contradicti... |
proofwiki-18239 | Baire Function may not be Continuous | Let $X \subseteq \R$.
Let $f : X \to \R$ be a Baire function.
Then $f$ is not necessarily continuous. | Let:
:$X = \closedint 0 1$
For each $n$, define the function $f_n : \closedint 0 1 \to \R$ to have:
:$\map {f_n} x = x^n$
for each $x \in \closedint 0 1$.
Note that for $0 \le x < 1$, we have:
:$x^n \to 0$
Note also that:
:$\map {f_n} 1 = 1$
so:
:$\map {f_n} 1 \to 1$
Define $f : \closedint 0 1 \to \R$ by:
:$\ds \ma... | Let $X \subseteq \R$.
Let $f : X \to \R$ be a [[Definition:Baire Function|Baire function]].
Then $f$ is not necessarily [[Definition:Continuous Real Function|continuous]]. | Let:
:$X = \closedint 0 1$
For each $n$, define the [[Definition:Real Function|function]] $f_n : \closedint 0 1 \to \R$ to have:
:$\map {f_n} x = x^n$
for each $x \in \closedint 0 1$.
Note that for $0 \le x < 1$, we have:
:$x^n \to 0$
Note also that:
:$\map {f_n} 1 = 1$
so:
:$\map {f_n} 1 \to 1$
Define $f : \clo... | Baire Function may not be Continuous | https://proofwiki.org/wiki/Baire_Function_may_not_be_Continuous | https://proofwiki.org/wiki/Baire_Function_may_not_be_Continuous | [
"Baire Functions"
] | [
"Definition:Baire Function",
"Definition:Continuous Real Function"
] | [
"Definition:Real Function",
"Definition:Pointwise Convergence",
"Definition:Continuous Real Function",
"Definition:Baire Function",
"Definition:Continuous Real Function"
] |
proofwiki-18240 | Principal Value of One over x is Distribution | Let $\phi \in \map \DD \R$ be a test function.
Let $T : \map \DD \R \to \C$ be a mapping such that:
:$\forall \phi \in \map \DD \R : \map T \phi = \PV \frac {\map \phi x} x \rd x : = \lim_{\epsilon \mathop \to 0} \int_{\size x \mathop > \epsilon} \frac {\map \phi x} x \rd x$
where $\PV$ denotes the Cauchy principal val... | Let $\phi \in \map \DD \R$ be a test function with a support on $\closedint {-a} a$.
Then:
{{begin-eqn}}
{{eqn | l = \map \phi x
| r = \map \phi 0 + \int_0^x \dfrac {\d \map \phi x} {\d x} \rd x
}}
{{eqn | r = \map \phi 0 + x \int_0^1 \dfrac {\d \map \phi {t x} } {\map \d {t x} } \rd t
}}
{{end-eqn}}
Furthermore:... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
Let $T : \map \DD \R \to \C$ be a [[Definition:Mapping|mapping]] such that:
:$\forall \phi \in \map \DD \R : \map T \phi = \PV \frac {\map \phi x} x \rd x : = \lim_{\epsilon \mathop \to 0} \int_{\size x \mathop > \epsilon} \frac {\map \phi x}... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]] with a [[Definition:Support of Schwartz Distribution|support]] on $\closedint {-a} a$.
Then:
{{begin-eqn}}
{{eqn | l = \map \phi x
| r = \map \phi 0 + \int_0^x \dfrac {\d \map \phi x} {\d x} \rd x
}}
{{eqn | r = \map \phi 0 + x \int_0^1 ... | Principal Value of One over x is Distribution | https://proofwiki.org/wiki/Principal_Value_of_One_over_x_is_Distribution | https://proofwiki.org/wiki/Principal_Value_of_One_over_x_is_Distribution | [
"Examples of Schwartz Distributions",
"Functional Analysis"
] | [
"Definition:Test Function",
"Definition:Mapping",
"Definition:Cauchy Principal Value",
"Definition:Schwartz Distribution"
] | [
"Definition:Test Function",
"Definition:Support of Schwartz Distribution",
"Definition:Support of Schwartz Distribution"
] |
proofwiki-18241 | Minimum of Finitely Many Continuous Real Functions is Continuous | Let $n \ge 2$ be a natural number.
Let $X \subseteq \R$.
Let $f_1, f_2, \ldots, f_n$ be functions $X \to \R$.
Define the function $m : X \to \R$ by:
:$\ds \map m x = \min_i \map {f_i} x$
for all $x \in X$.
Then $m$ is continuous. | We proceed by induction.
For all natural numbers $n \ge 2$, let $\map P n$ be the proposition:
:for every collection of $n$ functions $f_1, f_2, \ldots, f_n : X \to \R$, $m$ is continuous. | Let $n \ge 2$ be a [[Definition:Natural Number|natural number]].
Let $X \subseteq \R$.
Let $f_1, f_2, \ldots, f_n$ be [[Definition:Real Function|functions]] $X \to \R$.
Define the [[Definition:Real Function|function]] $m : X \to \R$ by:
:$\ds \map m x = \min_i \map {f_i} x$
for all $x \in X$.
Then $m$ is [[Defini... | We proceed by [[Principle of Mathematical Induction|induction]].
For all [[Definition:Natural Number|natural numbers]] $n \ge 2$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:for every [[Definition:Collection|collection]] of $n$ [[Definition:Real Function|functions]] $f_1, f_2, \ldots, f_n : X \to \R... | Minimum of Finitely Many Continuous Real Functions is Continuous | https://proofwiki.org/wiki/Minimum_of_Finitely_Many_Continuous_Real_Functions_is_Continuous | https://proofwiki.org/wiki/Minimum_of_Finitely_Many_Continuous_Real_Functions_is_Continuous | [
"Continuous Functions",
"Proofs by Induction"
] | [
"Definition:Natural Numbers",
"Definition:Real Function",
"Definition:Real Function",
"Definition:Continuous Real Function"
] | [
"Principle of Mathematical Induction",
"Definition:Natural Numbers",
"Definition:Proposition",
"Definition:Collection",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function",
"Definition:Continuous Real Functio... |
proofwiki-18242 | Set of Discontinuities of Baire Function is Meager | Let $X \subseteq \R$.
Let $f : X \to \R$ be a Baire function.
Let $D$ be the set of points for which $f$ is discontinuous.
Let $d$ be the Euclidean metric on $\R$.
Then $D$ is meager in the metric space $\struct {X, d}$. | Since $f$ is a Baire function, there exists a sequence $\sequence {f_n}$ of continuous functions such that:
:$\sequence {f_n}$ converges pointwise to $f$.
For each $\epsilon > 0$, define:
:$\map F \epsilon = \set {x \in X : \map {\omega_f} x > \epsilon}$
where $\map {\omega_f} x$ denotes the oscillation of $f$ at $x$.
... | Let $X \subseteq \R$.
Let $f : X \to \R$ be a [[Definition:Baire Function|Baire function]].
Let $D$ be the [[Definition:Set|set]] of points for which $f$ is [[Definition:Discontinuous Real Function at Point|discontinuous]].
Let $d$ be the [[Definition:Euclidean Metric|Euclidean metric]] on $\R$.
Then $D$ is [[De... | Since $f$ is a [[Definition:Baire Function|Baire function]], there exists a [[Definition:Sequence|sequence]] $\sequence {f_n}$ of [[Definition:Continuous Real Function|continuous functions]] such that:
:$\sequence {f_n}$ [[Definition:Pointwise Convergence|converges pointwise]] to $f$.
For each $\epsilon > 0$, define:
... | Set of Discontinuities of Baire Function is Meager | https://proofwiki.org/wiki/Set_of_Discontinuities_of_Baire_Function_is_Meager | https://proofwiki.org/wiki/Set_of_Discontinuities_of_Baire_Function_is_Meager | [
"Meager Spaces"
] | [
"Definition:Baire Function",
"Definition:Set",
"Definition:Discontinuous Mapping/Real Function/Point",
"Definition:Euclidean Metric",
"Definition:Meager Space",
"Definition:Metric Space"
] | [
"Definition:Baire Function",
"Definition:Sequence",
"Definition:Continuous Real Function",
"Definition:Pointwise Convergence",
"Definition:Oscillation/Real Space/Oscillation at Point",
"Real Function is Continuous at Point iff Oscillation is Zero",
"Definition:Discontinuous Mapping/Real Function/Point",... |
proofwiki-18243 | Condition for Lines to be Conjugate | Let $\CC$ be a circle of radius $r$ whose center is at the origin of a Cartesian plane.
Let $\PP$ and $\QQ$ be conjugate lines with respect to $\CC$:
{{begin-eqn}}
{{eqn | ll= \PP:
| l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | ll= \QQ:
| l = l_2 x + m_2 y + n_2
| r = 0
}}
{{end-eqn}}
Then:
:$l_1 ... | By definition of conjugate lines, $\PP$ and $\QQ$ are the polars of points $P$ and $Q$ respectively, such that $P$ lies on $\QQ$ and $Q$ lies on $\PP$.
From Coordinates of Pole of Given Polar, $P$ is given by:
:$P = \tuple {-\dfrac {l_1} {n_1} r^2, -\dfrac {m_1} {n_1} r^2}$
We have that $P$ lies on $\QQ$.
Substituting ... | Let $\CC$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $\PP$ and $\QQ$ be [[Definition:Conjugate Lines|conjugate lines]] with respect to $\CC$:... | By definition of [[Definition:Conjugate Lines|conjugate lines]], $\PP$ and $\QQ$ are the [[Definition:Polar of Point wrt Circle|polars]] of [[Definition:Point|points]] $P$ and $Q$ respectively, such that $P$ lies on $\QQ$ and $Q$ lies on $\PP$.
From [[Coordinates of Pole of Given Polar]], $P$ is given by:
:$P = \tuple... | Condition for Lines to be Conjugate | https://proofwiki.org/wiki/Condition_for_Lines_to_be_Conjugate | https://proofwiki.org/wiki/Condition_for_Lines_to_be_Conjugate | [
"Conjugate Lines"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Conjugate Lines"
] | [
"Definition:Conjugate Lines",
"Definition:Polar of Point/Circle",
"Definition:Point",
"Coordinates of Pole of Given Polar"
] |
proofwiki-18244 | Distributional Derivative of Heaviside Step Function | Let $H : \R \to \closedint 0 1$ be the Heaviside step function.
Let $T \in \map {\DD'} \R$ be a Schwartz distribution corresponding to $H$.
Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Then the distributional derivative of $T$ is $\delta$. | Let $\phi \in \map \DD \R$ be a test function.
Then:
{{begin-eqn}}
{{eqn | l = \map {T'} \phi
| r = -\map T {\phi'}
| c = {{Defof|Distributional Derivative}}
}}
{{eqn | r = -\int_{-\infty}^\infty \map H x \map {\phi'} x \rd x
}}
{{eqn | r = -\int_0^\infty \map {\phi'} x \rd x
}}
{{eqn | r = \map \phi 0 - \m... | Let $H : \R \to \closedint 0 1$ be the [[Definition:Heaviside Step Function|Heaviside step function]].
Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]] corresponding to $H$.
Let $\delta \in \map {\DD'} \R$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
Then:
{{begin-eqn}}
{{eqn | l = \map {T'} \phi
| r = -\map T {\phi'}
| c = {{Defof|Distributional Derivative}}
}}
{{eqn | r = -\int_{-\infty}^\infty \map H x \map {\phi'} x \rd x
}}
{{eqn | r = -\int_0^\infty \map {\phi'} x \rd x
... | Distributional Derivative of Heaviside Step Function | https://proofwiki.org/wiki/Distributional_Derivative_of_Heaviside_Step_Function | https://proofwiki.org/wiki/Distributional_Derivative_of_Heaviside_Step_Function | [
"Examples of Distributional Derivatives",
"Examples of Schwartz Distributions"
] | [
"Definition:Heaviside Step Function",
"Definition:Schwartz Distribution",
"Definition:Dirac Delta Distribution",
"Definition:Distributional Derivative"
] | [
"Definition:Test Function",
"Value of Compactly Supported Function outside its Support",
"Definition:Bounded Set/Real Numbers"
] |
proofwiki-18245 | Differentiable Function as Distribution | Let $T \in \map {\DD'} \R$ be a Schwartz distribution.
Let $f : \R \to \R$ be a continuously differentiable real function.
Suppose $T_f$ is a Schwartz distribution identified with $f$.
Then $T_f' = T_{f'}$. | Let $\phi \in \map \DD \R$ be a test function with a support on $\closedint a b$.
Then:
{{begin-eqn}}
{{eqn | l = \map {T'_f} \phi
| r = -\map {T_f} {\phi'}
| c = {{Defof|Distributional Derivative}}
}}
{{eqn | r = -\int_{-\infty}^\infty \map f x \map {\phi'} x \rd x
}}
{{eqn | r = -\int_a^b \map f x \map {\... | Let $T \in \map {\DD'} \R$ be a [[Definition:Schwartz Distribution|Schwartz distribution]].
Let $f : \R \to \R$ be a [[Definition:Continuously Differentiable|continuously differentiable]] [[Definition:Real Function|real function]].
Suppose $T_f$ is a [[Definition:Schwartz Distribution|Schwartz distribution]] identifi... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]] with a [[Definition:Support of Continuous Mapping|support]] on $\closedint a b$.
Then:
{{begin-eqn}}
{{eqn | l = \map {T'_f} \phi
| r = -\map {T_f} {\phi'}
| c = {{Defof|Distributional Derivative}}
}}
{{eqn | r = -\int_{-\infty}^\i... | Differentiable Function as Distribution | https://proofwiki.org/wiki/Differentiable_Function_as_Distribution | https://proofwiki.org/wiki/Differentiable_Function_as_Distribution | [
"Examples of Schwartz Distributions",
"Functional Analysis"
] | [
"Definition:Schwartz Distribution",
"Definition:Continuously Differentiable",
"Definition:Real Function",
"Definition:Schwartz Distribution"
] | [
"Definition:Test Function",
"Definition:Support of Continuous Mapping",
"Integral of Compactly Supported Function",
"Integration by Parts/Definite Integral"
] |
proofwiki-18246 | Intersections of Line joining Conjugate Points with Circle form Harmonic Range | Let $\CC$ be a circle.
Let $P$ and $Q$ be conjugate points {{WRT}} $\CC$.
Let $A$ and $B$ be the points of intersection of $PQ$ with $\CC$.
Then $A$ and $B$ are harmonic conjugates {{WRT}} $P$ and $Q$. | By definition of conjugate points, the polar of $P$ passes through $Q$.
Hence by Harmonic Property of Pole and Polar wrt Circle, $A$ and $B$ are harmonic conjugates {{WRT}} $P$ and $Q$.
{{Qed}} | Let $\CC$ be a [[Definition:Circle|circle]].
Let $P$ and $Q$ be [[Definition:Conjugate Points (Geometry)|conjugate points]] {{WRT}} $\CC$.
Let $A$ and $B$ be the [[Definition:Point|points]] of [[Definition:Intersection (Geometry)|intersection]] of $PQ$ with $\CC$.
Then $A$ and $B$ are [[Definition:Harmonic Conjugat... | By definition of [[Definition:Conjugate Points (Geometry)|conjugate points]], the [[Definition:Polar of Point wrt Circle|polar]] of $P$ passes through $Q$.
Hence by [[Harmonic Property of Pole and Polar wrt Circle]], $A$ and $B$ are [[Definition:Harmonic Conjugates of Harmonic Range|harmonic conjugates]] {{WRT}} $P$ a... | Intersections of Line joining Conjugate Points with Circle form Harmonic Range | https://proofwiki.org/wiki/Intersections_of_Line_joining_Conjugate_Points_with_Circle_form_Harmonic_Range | https://proofwiki.org/wiki/Intersections_of_Line_joining_Conjugate_Points_with_Circle_form_Harmonic_Range | [
"Conjugate Points"
] | [
"Definition:Circle",
"Definition:Conjugate Points (Geometry)",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Harmonic Conjugates/Harmonic Range"
] | [
"Definition:Conjugate Points (Geometry)",
"Definition:Polar of Point/Circle",
"Harmonic Property of Pole and Polar/Circle",
"Definition:Harmonic Conjugates/Harmonic Range"
] |
proofwiki-18247 | Conjugate Lines are Harmonic Conjugates with respect to Tangents from Point of Intersection | Let $\CC$ be a circle.
Let $\PP$ and $\QQ$ be conjugate lines {{WRT}} $\CC$.
Let $\PP$ and $\QQ$ intersect at $O$.
Let $OS$ and $OT$ be the tangents to $\CC$ from $O$.
Then $\PP$ and $\QQ$ are harmonic conjugates {{WRT}} $OS$ and $OT$. | Let $P$ and $Q$ be the poles of $\PP$ and $\QQ$ {{WRT}} $\CC$.
Because $O$ lies on both $\PP$ and $\QQ$, its polar passes through both $P$ and $Q$.
That is, the polar of $O$ is $PQ$ itself.
:420px
The polar of $O$, by definition, is the chord of contact of $OS$ and $OT$ with $\CC$.
From Intersections of Line joining Co... | Let $\CC$ be a [[Definition:Circle|circle]].
Let $\PP$ and $\QQ$ be [[Definition:Conjugate Lines|conjugate lines]] {{WRT}} $\CC$.
Let $\PP$ and $\QQ$ [[Definition:Intersection (Geometry)|intersect]] at $O$.
Let $OS$ and $OT$ be the [[Definition:Tangent to Circle|tangents]] to $\CC$ from $O$.
Then $\PP$ and $\QQ$ a... | Let $P$ and $Q$ be the [[Definition:Pole of Polar wrt Circle|poles]] of $\PP$ and $\QQ$ {{WRT}} $\CC$.
Because $O$ lies on both $\PP$ and $\QQ$, its [[Definition:Polar of Point wrt Circle|polar]] passes through both $P$ and $Q$.
That is, the [[Definition:Polar of Point wrt Circle|polar]] of $O$ is $PQ$ itself.
:[[F... | Conjugate Lines are Harmonic Conjugates with respect to Tangents from Point of Intersection | https://proofwiki.org/wiki/Conjugate_Lines_are_Harmonic_Conjugates_with_respect_to_Tangents_from_Point_of_Intersection | https://proofwiki.org/wiki/Conjugate_Lines_are_Harmonic_Conjugates_with_respect_to_Tangents_from_Point_of_Intersection | [
"Conjugate Lines"
] | [
"Definition:Circle",
"Definition:Conjugate Lines",
"Definition:Intersection (Geometry)",
"Definition:Tangent Line/Circle",
"Definition:Harmonic Conjugates/Harmonic Pencil"
] | [
"Definition:Polar of Point/Circle/Pole",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle",
"File:Conjugate-lines-intersect-Tangents.png",
"Definition:Polar of Point/Circle",
"Definition:Chord of Contact/Circle",
"Intersections of Line joining Conjugate Points with Circle form Harmo... |
proofwiki-18248 | Triangle Conjugacy is Mutual | Let $\CC$ be a circle.
Let $\triangle PQR$ be a triangle.
Let $\triangle P'Q'R'$ be such that:
:$P'$ is the pole of $QR$
:$Q'$ is the pole of $PR$
:$R'$ is the pole of $PQ$
{{WRT}} $\CC$.
Then:
:$P$ is the pole of $Q'R'$
:$Q$ is the pole of $P'R'$
:$R$ is the pole of $P'Q'$
{{WRT}} $\CC$.
That is, $\triangle PQR$ and $... | We have that:
:the polar of $P'$ is $QR$
:the polar of $Q'$ is $PR$
and so both polars pass through $R$.
Therefore:
:the polar of $R$ is $P'Q'$.
Similarly:
:the polar of $P$ is $Q'R'$
and:
:the polar of $Q$ is $P'R'$.
{{qed}} | Let $\CC$ be a [[Definition:Circle|circle]].
Let $\triangle PQR$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $\triangle P'Q'R'$ be such that:
:$P'$ is the [[Definition:Pole of Polar wrt Circle|pole]] of $QR$
:$Q'$ is the [[Definition:Pole of Polar wrt Circle|pole]] of $PR$
:$R'$ is the [[Definition:Pole of ... | We have that:
:the [[Definition:Polar of Point wrt Circle|polar]] of $P'$ is $QR$
:the [[Definition:Polar of Point wrt Circle|polar]] of $Q'$ is $PR$
and so both [[Definition:Polar of Point wrt Circle|polars]] pass through $R$.
Therefore:
:the [[Definition:Polar of Point wrt Circle|polar]] of $R$ is $P'Q'$.
Similar... | Triangle Conjugacy is Mutual | https://proofwiki.org/wiki/Triangle_Conjugacy_is_Mutual | https://proofwiki.org/wiki/Triangle_Conjugacy_is_Mutual | [
"Conjugate Triangles"
] | [
"Definition:Circle",
"Definition:Triangle (Geometry)",
"Definition:Polar of Point/Circle/Pole",
"Definition:Polar of Point/Circle/Pole",
"Definition:Polar of Point/Circle/Pole",
"Definition:Polar of Point/Circle/Pole",
"Definition:Polar of Point/Circle/Pole",
"Definition:Polar of Point/Circle/Pole",
... | [
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle"
] |
proofwiki-18249 | Orthocenter of Self-Conjugate Triangle | Let $\CC$ be a circle.
Let $\triangle PQR$ be a '''self-conjugate triangle with respect to $\CC$'''.
Then the orthocenter of $\triangle PQR$ is the center of $\CC$. | By definition of self-conjugate triangle:
:$PR$ is the polar of $Q$
:$QR$ is the polar of $P$
and from Self-Conjugate Triangle needs Two Sides to be Specified:
:$PQ$ is the polar of $R$
all with respect to $\CC$.
Let $O$ be the center of $\CC$.
Then from Polar of Point is Perpendicular to Line through Center:
:$PQ \per... | Let $\CC$ be a [[Definition:Circle|circle]].
Let $\triangle PQR$ be a '''[[Definition:Self-Conjugate Triangle|self-conjugate triangle]] with respect to $\CC$'''.
Then the [[Definition:Orthocenter|orthocenter]] of $\triangle PQR$ is the [[Definition:Center of Circle|center]] of $\CC$. | By definition of [[Definition:Self-Conjugate Triangle|self-conjugate triangle]]:
:$PR$ is the [[Definition:Polar of Point wrt Circle|polar]] of $Q$
:$QR$ is the [[Definition:Polar of Point wrt Circle|polar]] of $P$
and from [[Self-Conjugate Triangle needs Two Sides to be Specified]]:
:$PQ$ is the [[Definition:Polar of... | Orthocenter of Self-Conjugate Triangle | https://proofwiki.org/wiki/Orthocenter_of_Self-Conjugate_Triangle | https://proofwiki.org/wiki/Orthocenter_of_Self-Conjugate_Triangle | [
"Conjugate Triangles",
"Orthocenters of Triangles"
] | [
"Definition:Circle",
"Definition:Self-Conjugate Triangle",
"Definition:Orthocenter",
"Definition:Circle/Center"
] | [
"Definition:Self-Conjugate Triangle",
"Definition:Polar of Point/Circle",
"Definition:Polar of Point/Circle",
"Self-Conjugate Triangle needs Two Sides to be Specified",
"Definition:Polar of Point/Circle",
"Definition:Circle/Center",
"Polar of Point is Perpendicular to Line through Center",
"Definition... |
proofwiki-18250 | Hat-Check Distribution Gives Rise to Probability Mass Function | Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $X$ have the hat-check distribution with parameter $n$ (where $n > 0$).
Then $X$ gives rise to a probability mass function. | By definition:
:$\Img X = \set {0, 1, \ldots, n}$
:$\ds \map \Pr {X = k} = \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$
Then:
{{begin-eqn}}
{{eqn | l = \map \Pr \Omega
| r = \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] on a [[Definition:Probability Space|probability space]] $\struct {\Omega, \Sigma, \Pr}$.
Let $X$ have the [[Definition:Hat-Check Distribution|hat-check distribution with parameter $n$]] (where $n > 0$).
Then $X$ gives rise to a [[Definiti... | By [[Definition:Hat-Check Distribution|definition]]:
:$\Img X = \set {0, 1, \ldots, n}$
:$\ds \map \Pr {X = k} = \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$
Then:
{{begin-eqn}}
{{eqn | l = \map \Pr \Omega
| r = \sum_{k \mathop = 0}^n \dfrac 1 {\paren {n - k }!} \sum_{s \math... | Hat-Check Distribution Gives Rise to Probability Mass Function | https://proofwiki.org/wiki/Hat-Check_Distribution_Gives_Rise_to_Probability_Mass_Function | https://proofwiki.org/wiki/Hat-Check_Distribution_Gives_Rise_to_Probability_Mass_Function | [
"Hat-Check Distribution",
"Hat-Check Problem"
] | [
"Definition:Random Variable/Discrete",
"Definition:Probability Space",
"Definition:Hat-Check Distribution",
"Definition:Probability Mass Function"
] | [
"Definition:Hat-Check Distribution",
"Sum over k of r Choose k by -1^r-k by Polynomial",
"Category:Hat-Check Distribution",
"Category:Hat-Check Problem"
] |
proofwiki-18251 | Expectation of Hat-Check Distribution | Let $X$ be a discrete random variable with the Hat-Check distribution with parameter $n$.
Then the expectation of $X$ is given by:
:$\expect X = n - 1$ | From the definition of expectation:
:$\ds \expect X = \sum_{x \mathop \in \Img X} x \map \Pr {X = x}$
By definition of hat-check distribution:
:$\ds \expect X = \sum_{k \mathop = 0}^n k \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}$
Then:
{{begin-eqn}}
{{eqn | l = \expect X
| r = ... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with the [[Definition:Hat-Check Distribution|Hat-Check distribution with parameter $n$]].
Then the [[Definition:Expectation|expectation]] of $X$ is given by:
:$\expect X = n - 1$ | From the definition of [[Definition:Expectation|expectation]]:
:$\ds \expect X = \sum_{x \mathop \in \Img X} x \map \Pr {X = x}$
By definition of [[Definition:Hat-Check Distribution|hat-check distribution]]:
:$\ds \expect X = \sum_{k \mathop = 0}^n k \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren ... | Expectation of Hat-Check Distribution | https://proofwiki.org/wiki/Expectation_of_Hat-Check_Distribution | https://proofwiki.org/wiki/Expectation_of_Hat-Check_Distribution | [
"Expectation",
"Hat-Check Distribution",
"Hat-Check Problem"
] | [
"Definition:Random Variable/Discrete",
"Definition:Hat-Check Distribution",
"Definition:Expectation"
] | [
"Definition:Expectation",
"Definition:Hat-Check Distribution",
"Hat-Check Distribution Gives Rise to Probability Mass Function"
] |
proofwiki-18252 | Variance of Hat-Check Distribution | Let $X$ be a discrete random variable with the Hat-Check distribution with parameter $n$.
Then the variance of $X$ is given by:
:$\var X = 1$ | From the definition of Variance as Expectation of Square minus Square of Expectation:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
From Expectation of Function of Discrete Random Variable:
:$\ds \expect {X^2} = \sum_{x \mathop \in \Omega_X} x^2 \, \map \Pr {X = x}$
So:
{{begin-eqn}}
{{eqn | l = \expect {X^2}
... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with the [[Definition:Hat-Check Distribution|Hat-Check distribution with parameter $n$]].
Then the [[Definition:Variance of Discrete Random Variable|variance]] of $X$ is given by:
:$\var X = 1$ | From the definition of [[Variance as Expectation of Square minus Square of Expectation]]:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
From [[Expectation of Function of Discrete Random Variable]]:
:$\ds \expect {X^2} = \sum_{x \mathop \in \Omega_X} x^2 \, \map \Pr {X = x}$
So:
{{begin-eqn}}
{{eqn | l = \expect ... | Variance of Hat-Check Distribution | https://proofwiki.org/wiki/Variance_of_Hat-Check_Distribution | https://proofwiki.org/wiki/Variance_of_Hat-Check_Distribution | [
"Variance",
"Hat-Check Distribution",
"Hat-Check Problem"
] | [
"Definition:Random Variable/Discrete",
"Definition:Hat-Check Distribution",
"Definition:Variance/Discrete"
] | [
"Variance as Expectation of Square minus Square of Expectation",
"Expectation of Function of Discrete Random Variable",
"Hat-Check Distribution Gives Rise to Probability Mass Function",
"Hat-Check Distribution Gives Rise to Probability Mass Function",
"Expectation of Hat-Check Distribution",
"Expectation ... |
proofwiki-18253 | Jump Rule | Let $f : \R \to \R$ be a piecewise continuously differentiable real function with a discontinuity at $c \in \R$.
Suppose the limits $\map f {c^+}, \map f {c^-}, \map {f'} {c^+}, \map {f'} {c^-}$ exist.
Let $T \in \map {\DD'} \R$ be a Schwartz distribution associated with $f$.
Then in the distributional sense we have th... | Let $\phi \in \map \DD {\R}$ be a test function with its support on $\closedint a b \subset \R$.
Let $c \in \closedint a b$.
Then:
{{begin-eqn}}
{{eqn | l = \map {T_f'} \phi
| r = -\map {T_f} {\phi'}
| c = {{Defof|Distributional Derivative}}
}}
{{eqn | r = -\int_a^b \map f x \map {\phi'} x \rd x
}}
{{eqn | ... | Let $f : \R \to \R$ be a [[Definition:Piecewise Continuously Differentiable Function|piecewise continuously differentiable]] [[Definition:Real Function|real function]] with a [[Definition:Jump Discontinuity|discontinuity]] at $c \in \R$.
Suppose the [[Definition:Limit of Real Function|limits]] $\map f {c^+}, \map f {c... | Let $\phi \in \map \DD {\R}$ be a [[Definition:Test Function|test function]] with its [[Definition:Support of Schwartz Distribution|support]] on $\closedint a b \subset \R$.
Let $c \in \closedint a b$.
Then:
{{begin-eqn}}
{{eqn | l = \map {T_f'} \phi
| r = -\map {T_f} {\phi'}
| c = {{Defof|Distributional... | Jump Rule | https://proofwiki.org/wiki/Jump_Rule | https://proofwiki.org/wiki/Jump_Rule | [
"Distributional Derivatives",
"Piecewise Continuously Differentiable Functions"
] | [
"Definition:Piecewise Continuously Differentiable Function",
"Definition:Real Function",
"Definition:Discontinuity (Real Analysis)/Jump",
"Definition:Limit of Real Function",
"Definition:Schwartz Distribution",
"Definition:Distributional Derivative",
"Definition:Dirac Delta Distribution"
] | [
"Definition:Test Function",
"Definition:Support of Schwartz Distribution",
"Integration by Parts/Definite Integral",
"Definition:Schwartz Distribution",
"Definition:Linear Transformation"
] |
proofwiki-18254 | Skewness of Hat-Check Distribution | Let $X$ be a discrete random variable with a Hat-Check distribution with parameter $n$. ($n \gt 2$)
Then the skewness $\gamma_1$ of $X$ is given by:
:$\gamma_1 = -1$ | From Skewness in terms of Non-Central Moments:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
We have, by Expectation of Hat-Check Distribution:
:$\expect X = n - 1$
By Variance of Hat-Check Distribution:
... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with a [[Definition:Hat-Check Distribution|Hat-Check distribution with parameter $n$]]. ($n \gt 2$)
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is given by:
:$\gamma_1 = -1$ | From [[Skewness in terms of Non-Central Moments]]:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
We have, by [[Expectation of Hat-Check Dist... | Skewness of Hat-Check Distribution | https://proofwiki.org/wiki/Skewness_of_Hat-Check_Distribution | https://proofwiki.org/wiki/Skewness_of_Hat-Check_Distribution | [
"Skewness",
"Hat-Check Distribution"
] | [
"Definition:Random Variable/Discrete",
"Definition:Hat-Check Distribution",
"Definition:Skewness"
] | [
"Skewness in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Hat-Check Distribution",
"Variance of Hat-Check Distribution",
"Hat-Check Distribution Gives Rise to Probability Mass Function",
"Hat-Check Distribution Gives Rise to Probability Mass F... |
proofwiki-18255 | Characteristic of Interior Point of Circle whose Center is Origin | Let $\CC$ be a circle of radius $r$ whose center is at the origin $O$ of a Cartesian plane.
Let $P = \tuple {x, y}$ be a point in the plane of $\CC$.
Then $P$ is in the interior of $\CC$ {{iff}}:
:$x^2 + y^2 - r^2 < 0$ | Let $d$ be the distance of $P$ from $O$.
{{begin-eqn}}
{{eqn | l = d
| r = \sqrt {\paren {x - 0}^2 + \paren {y - 0}^2}
| c = Distance Formula
}}
{{eqn | ll= \leadsto
| l = d^2
| r = x^2 + y^2
| c =
}}
{{end-eqn}}
Then by definition of interior of $\CC$:
:$P$ is in the interior of $\CC$ {{... | Let $\CC$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] $O$ of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P = \tuple {x, y}$ be a [[Definition:Point|point]] in the [[Definition:Plane|plane]... | Let $d$ be the [[Definition:Distance between Points|distance]] of $P$ from $O$.
{{begin-eqn}}
{{eqn | l = d
| r = \sqrt {\paren {x - 0}^2 + \paren {y - 0}^2}
| c = [[Distance Formula]]
}}
{{eqn | ll= \leadsto
| l = d^2
| r = x^2 + y^2
| c =
}}
{{end-eqn}}
Then by definition of [[Defini... | Characteristic of Interior Point of Circle whose Center is Origin | https://proofwiki.org/wiki/Characteristic_of_Interior_Point_of_Circle_whose_Center_is_Origin | https://proofwiki.org/wiki/Characteristic_of_Interior_Point_of_Circle_whose_Center_is_Origin | [
"Circles"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Plane Surface",
"Definition:Interior of Circle"
] | [
"Definition:Distance between Points",
"Distance Formula",
"Definition:Interior of Circle",
"Definition:Interior of Circle"
] |
proofwiki-18256 | Length of Tangent from Point to Circle center Origin | Let $\CC$ be a circle of radius $r$ whose center is at the origin $O$ of a Cartesian plane.
Let $P = \tuple {x, y}$ be a point in the plane of $\CC$ in the exterior of $\CC$.
Let $PT$ be a tangent to $\CC$ from $P$ such that $T$ is the point of tangency.
Then the length of $PT$ is given by:
:$PT^2 = x^2 + y^2 - r^2$ | Let $\NN$ be the normal to $\CC$ at the point $T$.
From Normal to Circle passes through Center, $\NN$ passes through $O$.
By definition of the normal to $\CC$, $\NN$ is perpendicular to $PT$.
Hence $OT$, $PT$ and $OP$ form a right triangle whose hypotenuse is $OP$.
As $OT$ is a line segment coinciding with a radius of ... | Let $\CC$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] $O$ of a [[Definition:Cartesian Plane|Cartesian plane]].
Let $P = \tuple {x, y}$ be a [[Definition:Point|point]] in the [[Definition:Plane|plane]... | Let $\NN$ be the [[Definition:Normal to Curve|normal]] to $\CC$ at the [[Definition:Point|point]] $T$.
From [[Normal to Circle passes through Center]], $\NN$ passes through $O$.
By definition of the [[Definition:Normal to Curve|normal]] to $\CC$, $\NN$ is [[Definition:Perpendicular|perpendicular]] to $PT$.
Hence $OT... | Length of Tangent from Point to Circle center Origin | https://proofwiki.org/wiki/Length_of_Tangent_from_Point_to_Circle_center_Origin | https://proofwiki.org/wiki/Length_of_Tangent_from_Point_to_Circle_center_Origin | [
"Circles",
"Tangents"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Plane Surface",
"Definition:Exterior of Circle",
"Definition:Tangent Line/Circle",
"Definition:Point",
"Definition:Tan... | [
"Definition:Normal to Curve",
"Definition:Point",
"Normal to Circle passes through Center",
"Definition:Normal to Curve",
"Definition:Right Angle/Perpendicular",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Line/Straight Line Segm... |
proofwiki-18257 | Equation of Tangents to Circle from Point | Let $\CC$ be a circle embedded in the Cartesian plane of radius $r$ with its center located at the origin.
Let $P = \tuple {x_0, y_0}$ be a point in the plane of $\CC$ which is outside $\CC$.
The tangents to $\CC$ which pass through $P$ can be described using the equation:
:$\paren {x x_0 + y y_0 - r^2}^2 = \paren {x^2... | From Equation of Circle center Origin, $\CC$ can be described as:
:$x^2 + y^2 = r^2$
Let $\LL$ be an arbitrary straight line through $P$ which intersects $\CC$ at $U$ and $V$.
Let $Q = \tuple{x, y}$ be an arbitrary point on $\LL$.
Let $k$ be the position-ratio of one of the points $U$ and $V$ with respect to $P$ and $Q... | Let $\CC$ be a [[Definition:Circle|circle]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]] of [[Definition:Radius of Circle|radius]] $r$ with its [[Definition:Center of Circle|center]] located at the [[Definition:Origin|origin]].
Let $P = \tuple {x_0, y_0}$ be a [[Definition:Point|point]] in the [[Defi... | From [[Equation of Circle center Origin]], $\CC$ can be described as:
:$x^2 + y^2 = r^2$
Let $\LL$ be an arbitrary [[Definition:Straight Line|straight line]] through $P$ which [[Definition:Intersection (Geometry)|intersects]] $\CC$ at $U$ and $V$.
Let $Q = \tuple{x, y}$ be an arbitrary [[Definition:Point|point]] on $... | Equation of Tangents to Circle from Point | https://proofwiki.org/wiki/Equation_of_Tangents_to_Circle_from_Point | https://proofwiki.org/wiki/Equation_of_Tangents_to_Circle_from_Point | [
"Circles",
"Tangents to Circles"
] | [
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Point",
"Definition:Plane Surface",
"Definition:Exterior of Circle",
"Definition:Tangent Line/Circle",
"Definition:Equation"
] | [
"Equation of Circle center Origin",
"Definition:Line/Straight Line",
"Definition:Intersection (Geometry)",
"Definition:Point",
"Definition:Position-Ratio of Point",
"Definition:Point",
"Definition:Joachimsthal's Equation/Circle",
"Definition:Quadratic Equation",
"Definition:Tangent Line/Circle",
"... |
proofwiki-18258 | Condition of Tangency to Circle whose Center is Origin | Let $\CC$ be a circle embedded in the Cartesian plane of radius $r$ with its center located at the origin.
Let $\LL$ be a straight line in the plane of $\CC$ whose equation is given by:
:$(1): \quad l x + m y + n = 0$
such that $l \ne 0$.
Then $\LL$ is tangent to $\CC$ {{iff}}:
:$\paren {l^2 + m^2} r^2 = n^2$ | From Equation of Circle center Origin, $\CC$ can be described as:
:$(2): \quad x^2 + y^2 = r^2$
Let $\LL$ intersect with $\CC$.
To find where this happens, we find $x$ and $y$ which satisfy both $(1)$ and $(2)$.
So:
{{begin-eqn}}
{{eqn | n = 1
| l = l x + m y + n
| r = 0
| c = Equation for $\LL$
}}
{{... | Let $\CC$ be a [[Definition:Circle|circle]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]] of [[Definition:Radius of Circle|radius]] $r$ with its [[Definition:Center of Circle|center]] located at the [[Definition:Origin|origin]].
Let $\LL$ be a [[Definition:Straight Line|straight line]] in the [[Defini... | From [[Equation of Circle center Origin]], $\CC$ can be described as:
:$(2): \quad x^2 + y^2 = r^2$
Let $\LL$ [[Definition:Intersection (Geometry)|intersect]] with $\CC$.
To find where this happens, we find $x$ and $y$ which satisfy both $(1)$ and $(2)$.
So:
{{begin-eqn}}
{{eqn | n = 1
| l = l x + m y + n
... | Condition of Tangency to Circle whose Center is Origin | https://proofwiki.org/wiki/Condition_of_Tangency_to_Circle_whose_Center_is_Origin | https://proofwiki.org/wiki/Condition_of_Tangency_to_Circle_whose_Center_is_Origin | [
"Circles",
"Tangents"
] | [
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Line/Straight Line",
"Definition:Plane Surface",
"Equation of Straight Line in Plane/General Equation",
"Definition:Tangent Line/Circle"
] | [
"Equation of Circle center Origin",
"Definition:Intersection (Geometry)",
"Definition:Quadratic Equation",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Tangent Line/Circle",
"Definition:Point",
"Definition:Root of Polynomial",
"Solution to Quadratic Equation",
"Definition:... |
proofwiki-18259 | Product of Smooth Function and Dirac Delta Distribution | Let $a \in \R^d$ be a vector in real Euclidean space.
Let $\delta_a \in \map {\DD'} {\R^d}$ be the Dirac delta distribution.
Let $\alpha \in \map {C^\infty} {\R^d}$ be a smooth real function.
Then in the Schwartz distributional sense we have that:
:$\alpha \delta_a = \map \alpha a \delta_a$ | Let $\phi \in \map \DD {\R^d}$ be a test function.
Then:
{{begin-eqn}}
{{eqn | l = \map {\alpha \delta_a} \phi
| r = \map {\delta_a} {\alpha \phi}
| c = {{Defof|Multiplication of Schwartz Distribution by Smooth Function}}
}}
{{eqn | r = \map {\paren {\alpha \phi} } a
| c = {{Defof|Dirac Delta Distribu... | Let $a \in \R^d$ be a [[Definition:Vector (Real Euclidean Space)|vector]] in [[Definition:Real Euclidean Space|real Euclidean space]].
Let $\delta_a \in \map {\DD'} {\R^d}$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Let $\alpha \in \map {C^\infty} {\R^d}$ be a [[Definition:Smooth Functio... | Let $\phi \in \map \DD {\R^d}$ be a [[Definition:Test Function|test function]].
Then:
{{begin-eqn}}
{{eqn | l = \map {\alpha \delta_a} \phi
| r = \map {\delta_a} {\alpha \phi}
| c = {{Defof|Multiplication of Schwartz Distribution by Smooth Function}}
}}
{{eqn | r = \map {\paren {\alpha \phi} } a
| c... | Product of Smooth Function and Dirac Delta Distribution | https://proofwiki.org/wiki/Product_of_Smooth_Function_and_Dirac_Delta_Distribution | https://proofwiki.org/wiki/Product_of_Smooth_Function_and_Dirac_Delta_Distribution | [
"Dirac Delta Distribution",
"Schwartz Distributions",
"Functional Analysis"
] | [
"Definition:Vector/Real Euclidean Space",
"Definition:Euclidean Space/Real",
"Definition:Dirac Delta Distribution",
"Definition:Smooth Real Function",
"Definition:Real Function",
"Definition:Schwartz Distribution"
] | [
"Definition:Test Function"
] |
proofwiki-18260 | Circle is Curve of Second Degree | The circle is a curve of degree $2$. | From Equation of Circle in Cartesian Plane, a circle can be expressed in the form:
:$x^2 + y^2 - 2 a x - 2 b y + a^2 + b^2 - R^2 = 0$
where $\tuple {a, b}$ is the center and $R$ is the radius.
This is a quadratic equation in $2$ variables.
Hence the result by definition of degree of curve.
{{qed}}
{{explain|Sommerville... | The [[Definition:Circle|circle]] is a [[Definition:Degree of Curve|curve of degree $2$]]. | From [[Equation of Circle in Cartesian Plane]], a [[Definition:Circle|circle]] can be expressed in the form:
:$x^2 + y^2 - 2 a x - 2 b y + a^2 + b^2 - R^2 = 0$
where $\tuple {a, b}$ is the [[Definition:Center of Circle|center]] and $R$ is the [[Definition:Radius of Circle|radius]].
This is a [[Definition:Quadratic E... | Circle is Curve of Second Degree | https://proofwiki.org/wiki/Circle_is_Curve_of_Second_Degree | https://proofwiki.org/wiki/Circle_is_Curve_of_Second_Degree | [
"Circles"
] | [
"Definition:Circle",
"Definition:Degree of Curve"
] | [
"Equation of Circle/Cartesian",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Quadratic Equation in Two Variables",
"Definition:Degree of Curve"
] |
proofwiki-18261 | Characteristic of Quadratic Equation that Represents Circle | A '''quadratic equation in $2$ variables''':
:$a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
describes a circle embedded in the Cartesian plane {{iff}}:
:$(1): \quad h = 0$
:$(2): \quad a = b \ne 0$ | === Necessary Condition ===
Consider the equation of a circle:
:$A \paren {x^2 + y^2} + B x + C y + D = 0$
which is the equation of a circle with radius $R$ and center $\tuple {a, b}$, where:
:$R = \dfrac 1 {2 A} \sqrt {B^2 + C^2 - 4 A D}$
:$\tuple {a, b} = \tuple {\dfrac {-B} {2 A}, \dfrac {-C} {2 A} }$
provided:
:$A ... | A '''[[Definition:Quadratic Equation in Two Variables|quadratic equation in $2$ variables]]''':
:$a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
describes a [[Definition:Circle|circle]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]] {{iff}}:
:$(1): \quad h = 0$
:$(2): \quad a = b \ne 0$ | === Necessary Condition ===
Consider the [[Equation of Circle/Cartesian/Formulation 2|equation of a circle]]:
:$A \paren {x^2 + y^2} + B x + C y + D = 0$
which is the [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Circle|circle]] with [[Definition:Radius of Circle|radius]] $R$ and [[Definition... | Characteristic of Quadratic Equation that Represents Circle | https://proofwiki.org/wiki/Characteristic_of_Quadratic_Equation_that_Represents_Circle | https://proofwiki.org/wiki/Characteristic_of_Quadratic_Equation_that_Represents_Circle | [
"Circles"
] | [
"Definition:Quadratic Equation in Two Variables",
"Definition:Circle",
"Definition:Cartesian Plane"
] | [
"Equation of Circle/Cartesian/Formulation 2",
"Definition:Equation of Geometric Figure",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coefficient of Polynomial",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definit... |
proofwiki-18262 | Angle of Intersection of Circles equals Angle between Radii | Let $\CC$ and $\CC'$ be circles whose centers are at $C$ and $C'$ respectively.
Let $\CC$ and $\CC'$ intersect at $A$ and $B$.
The angle of intersection of $\CC$ and $\CC'$ is equal to the angle between the radii to the point of intersection. | From Normal to Circle passes through Center, the straight line passing through the center of a circle is normal to that circle.
Hence the radii $CA$ and $C'A$ are perpendicular to the tangents to $\CC$ and $\CC'$ respectively.
:540px
Thus, with reference to the above diagram, we have that:
:$\angle FAC = \angle DAC'$
a... | Let $\CC$ and $\CC'$ be [[Definition:Circle|circles]] whose [[Definition:Center of Circle|centers]] are at $C$ and $C'$ respectively.
Let $\CC$ and $\CC'$ [[Definition:Intersection (Geometry)|intersect]] at $A$ and $B$.
The [[Definition:Angle of Intersection of Circles|angle of intersection]] of $\CC$ and $\CC'$ is ... | From [[Normal to Circle passes through Center]], the [[Definition:Straight Line|straight line]] passing through the [[Definition:Center of Circle|center]] of a [[Definition:Circle|circle]] is [[Definition:Normal to Curve|normal]] to that [[Definition:Circle|circle]].
Hence the [[Definition:Radius of Circle|radii]] $CA... | Angle of Intersection of Circles equals Angle between Radii | https://proofwiki.org/wiki/Angle_of_Intersection_of_Circles_equals_Angle_between_Radii | https://proofwiki.org/wiki/Angle_of_Intersection_of_Circles_equals_Angle_between_Radii | [
"Circles"
] | [
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Intersection (Geometry)",
"Definition:Angle of Intersection of Circles",
"Definition:Angle",
"Definition:Circle/Radius",
"Definition:Point",
"Definition:Intersection (Geometry)"
] | [
"Normal to Circle passes through Center",
"Definition:Line/Straight Line",
"Definition:Circle/Center",
"Definition:Circle",
"Definition:Normal to Curve",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Right Angle/Perpendicular",
"Definition:Tangent Line/Circle",
"File:Circles-angle-i... |
proofwiki-18263 | Angle of Intersection of Circles is Equal at Both Points | Let $\CC$ and $\CC'$ be circles whose centers are at $C$ and $C'$ respectively.
Let $\CC$ and $\CC'$ intersect at $A$ and $B$.
The angle of intersection of $\CC$ and $\CC'$ at $A$ is equal to the angle of intersection of $\CC$ and $\CC'$ at $B$. | Consider the two triangles $CAC'$ and $CBC'$.
:540px
By definition of radius:
:$CA = CB$ and $C'A = C'B$
{{begin-eqn}}
{{eqn | l = CA
| r = CB
| c = {{Defof|Radius of Circle}}
}}
{{eqn | l = C'A
| r = C'B
| c = {{Defof|Radius of Circle}}
}}
{{eqn | ll= \leadsto
| l = \triangle CAC'
|... | Let $\CC$ and $\CC'$ be [[Definition:Circle|circles]] whose [[Definition:Center of Circle|centers]] are at $C$ and $C'$ respectively.
Let $\CC$ and $\CC'$ [[Definition:Intersection (Geometry)|intersect]] at $A$ and $B$.
The [[Definition:Angle of Intersection of Circles|angle of intersection]] of $\CC$ and $\CC'$ at ... | Consider the two [[Definition:Triangle (Geometry)|triangles]] $CAC'$ and $CBC'$.
:[[File:Circles-angle-intersection.png|540px]]
By definition of [[Definition:Radius of Circle|radius]]:
:$CA = CB$ and $C'A = C'B$
{{begin-eqn}}
{{eqn | l = CA
| r = CB
| c = {{Defof|Radius of Circle}}
}}
{{eqn | l = C'A
... | Angle of Intersection of Circles is Equal at Both Points | https://proofwiki.org/wiki/Angle_of_Intersection_of_Circles_is_Equal_at_Both_Points | https://proofwiki.org/wiki/Angle_of_Intersection_of_Circles_is_Equal_at_Both_Points | [
"Circles"
] | [
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Intersection (Geometry)",
"Definition:Angle of Intersection of Circles",
"Definition:Angle of Intersection of Circles"
] | [
"Definition:Triangle (Geometry)",
"File:Circles-angle-intersection.png",
"Definition:Circle/Radius",
"Triangle Side-Side-Side Congruence",
"Angle of Intersection of Circles equals Angle between Radii"
] |
proofwiki-18264 | Equation of Circle/Cartesian/Formulation 1 | The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\tuple {a, b}$ can be expressed as:
:$\paren {x - a}^2 + \paren {y - b}^2 = R^2$ | Let the point $\tuple {x, y}$ satisfy the equation:
:$(1): \quad \paren {x - a}^2 + \paren {y - b}^2 = R^2$
By the Distance Formula, the distance between this $\tuple {x, y}$ and $\tuple {a, b}$ is:
:$\sqrt {\paren {x - a}^2 + \paren {y - b}^2}$
But from equation $(1)$, this quantity equals $R$.
Therefore the distance ... | The [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Circle|circle]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]] with [[Definition:Radius of Circle|radius]] $R$ and [[Definition:Center of Circle|center]] $\tuple {a, b}$ can be expressed as:
:$\paren {x - a}^2 + \paren {y - b}^2... | Let the [[Definition:Point|point]] $\tuple {x, y}$ satisfy the [[Definition:Equation of Geometric Figure|equation]]:
:$(1): \quad \paren {x - a}^2 + \paren {y - b}^2 = R^2$
By the [[Distance Formula]], the [[Definition:Distance between Points|distance]] between this $\tuple {x, y}$ and $\tuple {a, b}$ is:
:$\sqrt {\pa... | Equation of Circle/Cartesian/Formulation 1 | https://proofwiki.org/wiki/Equation_of_Circle/Cartesian/Formulation_1 | https://proofwiki.org/wiki/Equation_of_Circle/Cartesian/Formulation_1 | [
"Equation of Circle"
] | [
"Definition:Equation of Geometric Figure",
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Circle/Radius",
"Definition:Circle/Center"
] | [
"Definition:Point",
"Definition:Equation of Geometric Figure",
"Distance Formula",
"Definition:Distance between Points",
"Definition:Distance between Points",
"Definition:Point",
"Definition:Equation of Geometric Figure",
"Definition:Circle/Center",
"Definition:Constant",
"Definition:Circle/Radius... |
proofwiki-18265 | Condition for Circles to be Orthogonal | Let $\CC_1$ and $\CC_2$ be circles embedded in a Cartesian plane.
Let $\CC_1$ and $\CC_2$ be described by Equation of Circle in Cartesian Plane as:
{{begin-eqn}}
{{eqn | q = \CC_1
| l = x^2 + y^2 + 2 \alpha_1 x + 2 \beta_1 y + c_1
| r = 0
| c =
}}
{{eqn | q = \CC_2
| l = x^2 + y^2 + 2 \alpha_2 ... | When $\CC_1$ and $\CC_2$ are orthogonal, the distance between their centers forms the hypotenuse of a right triangle whose legs are equal to the radii.
From Equation of Circle in Cartesian Plane: Formulation 3, the radii $r_1$ and $r_2$ of $\CC_1$ and $\CC_2$ respectively are given by:
{{begin-eqn}}
{{eqn | l = c_1
... | Let $\CC_1$ and $\CC_2$ be [[Definition:Circle|circles]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]].
Let $\CC_1$ and $\CC_2$ be described by [[Equation of Circle in Cartesian Plane]] as:
{{begin-eqn}}
{{eqn | q = \CC_1
| l = x^2 + y^2 + 2 \alpha_1 x + 2 \beta_1 y + c_1
| r = 0
| c ... | When $\CC_1$ and $\CC_2$ are [[Definition:Orthogonal Circles|orthogonal]], the [[Definition:Distance between Points|distance]] between their [[Definition:Center of Circle|centers]] forms the [[Definition:Hypotenuse|hypotenuse]] of a [[Definition:Right Triangle|right triangle]] whose [[Definition:Leg of Right Triangle|l... | Condition for Circles to be Orthogonal | https://proofwiki.org/wiki/Condition_for_Circles_to_be_Orthogonal | https://proofwiki.org/wiki/Condition_for_Circles_to_be_Orthogonal | [
"Circles",
"Orthogonal Circles"
] | [
"Definition:Circle",
"Definition:Cartesian Plane",
"Equation of Circle/Cartesian",
"Definition:Orthogonal Curves/Circles"
] | [
"Definition:Orthogonal Curves/Circles",
"Definition:Distance between Points",
"Definition:Circle/Center",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled/Legs",
"Definition:Circle/Radius",
"Equation o... |
proofwiki-18266 | Test Function with Vanishing Partial Derivative | Let $\phi \in \map \DD {\R^2}$ be a test function such that:
:$\tuple {x, y} \stackrel \phi {\longrightarrow} \map \phi {x, y}$
Suppose $\phi$ is a solution to the following partial differential equation:
:$\dfrac {\partial \phi} {\partial x} = 0$
Then $\phi$ is identically $0$. | $\dfrac {\partial \phi} {\partial x} = 0$ implies that:
:$\forall x \in \R : \map \phi {x, y} = \map C y$
where $C : \R \to \C$ is a complex-valued function.
By definition, $\phi$ is a test function.
Hence, $\phi$ must have a compact support $\Omega \subset \R^2$.
Let $\map {B^-_\epsilon} 0 \subset \R^2$ be a closed ba... | Let $\phi \in \map \DD {\R^2}$ be a [[Definition:Test Function|test function]] such that:
:$\tuple {x, y} \stackrel \phi {\longrightarrow} \map \phi {x, y}$
Suppose $\phi$ is a [[Definition:Solution of Differential Equation|solution]] to the following [[Definition:Partial Differential Equation|partial differential eq... | $\dfrac {\partial \phi} {\partial x} = 0$ implies that:
:$\forall x \in \R : \map \phi {x, y} = \map C y$
where $C : \R \to \C$ is a [[Definition:Complex-Valued Function|complex-valued function]].
By definition, $\phi$ is a [[Definition:Test Function|test function]].
Hence, $\phi$ must have a [[Definition:Compact S... | Test Function with Vanishing Partial Derivative | https://proofwiki.org/wiki/Test_Function_with_Vanishing_Partial_Derivative | https://proofwiki.org/wiki/Test_Function_with_Vanishing_Partial_Derivative | [
"Examples of Test Functions",
"Partial Differentiation"
] | [
"Definition:Test Function",
"Definition:Differential Equation/Solution",
"Definition:Differential Equation/Partial"
] | [
"Definition:Complex-Valued Function",
"Definition:Test Function",
"Definition:Compact Space/Euclidean Space",
"Definition:Support of Continuous Mapping/Real-Valued",
"Definition:Closed Ball/Real Euclidean Space",
"Closed Ball in Euclidean Space is Compact",
"Definition:Compact Space/Euclidean Space",
... |
proofwiki-18267 | Construction of Conic Section | Consider a right circular cone $\CC$ with opening angle $2 \alpha$ whose apex is at $O$.
Consider a plane $\PP$ which intersects $\CC$, not passing through $O$, at an angle $\beta$ to the axis of $\CC$.
Then the points of intersection form one of the following:
:a circle
:an ellipse
:a hyperbola
:a parabola. | When $\beta$ is a right angle, the points of intersection form a circle, by definition of transverse section.
{{qed|lemma}}
Otherwise, let the plane $OAA'$ through the axis of $\CC$ perpendicular to $\PP$ intersect $\PP$ in the line $AA'$.
Let $P$ be an arbitrary point on the intersection of $\PP$ with $\CC$.
Let $PM$ ... | Consider a [[Definition:Right Circular Cone|right circular cone]] $\CC$ with [[Definition:Opening Angle|opening angle]] $2 \alpha$ whose [[Definition:Apex of Cone|apex]] is at $O$.
Consider a [[Definition:Plane|plane]] $\PP$ which [[Definition:Intersection (Geometry)|intersects]] $\CC$, not passing through $O$, at an ... | When $\beta$ is a [[Definition:Right Angle|right angle]], the [[Definition:Point|points]] of [[Definition:Intersection (Geometry)|intersection]] form a [[Definition:Circle|circle]], by definition of [[Definition:Transverse Section|transverse section]].
{{qed|lemma}}
Otherwise, let the [[Definition:Plane|plane]] $OAA'... | Construction of Conic Section | https://proofwiki.org/wiki/Construction_of_Conic_Section | https://proofwiki.org/wiki/Construction_of_Conic_Section | [
"Conic Sections"
] | [
"Definition:Right Circular Cone",
"Definition:Right Circular Cone/Opening Angle",
"Definition:Cone (Geometry)/Apex",
"Definition:Plane Surface",
"Definition:Intersection (Geometry)",
"Definition:Angle",
"Definition:Right Circular Cone/Axis",
"Definition:Point",
"Definition:Intersection (Geometry)",
... | [
"Definition:Right Angle",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Circle",
"Definition:Conic Section/Intersection with Cone/Circle/Transverse Section",
"Definition:Plane Surface",
"Definition:Right Circular Cone/Axis",
"Definition:Right Angle/Perpendicular/Plane to Plane"... |
proofwiki-18268 | Dandelin's Theorem | Let $\CC$ be a double napped right circular cone with apex $O$.
Let $\PP$ be a plane which intersects $\CC$ such that:
:$\PP$ does not pass through $O$
:$\PP$ is not perpendicular to the axis of $\CC$.
Let $\EE$ be the conic section arising as the intersection between $\PP$ and $\CC$.
Let $\SS$ and $\SS'$ be the Dandel... | Let $\SS$ and $\SS'$ be the Dandelin spheres with respect to $\PP$.
Let $P$ be a point on $\EE$.
Let $F$ and $F'$ be the points at which $\SS$ and $\SS'$ are tangent to $\PP$ respectively.
Let the generatrix of $\CC$ which passes through $P$ touch $\SS$ and $\SS'$ at $E$ and $E'$ respectively.
Let $\theta$ be half the ... | Let $\CC$ be a [[Definition:Double Napped Cone|double napped]] [[Definition:Right Circular Cone|right circular cone]] with [[Definition:Apex of Cone|apex]] $O$.
Let $\PP$ be a [[Definition:Plane|plane]] which [[Definition:Intersection (Geometry)|intersects]] $\CC$ such that:
:$\PP$ does not pass through $O$
:$\PP$ is ... | Let $\SS$ and $\SS'$ be the [[Definition:Dandelin Spheres|Dandelin spheres]] with respect to $\PP$.
Let $P$ be a [[Definition:Point|point]] on $\EE$.
Let $F$ and $F'$ be the [[Definition:Point|points]] at which $\SS$ and $\SS'$ are [[Definition:Tangent Plane|tangent]] to $\PP$ respectively.
Let the [[Definition:Gene... | Dandelin's Theorem/Directrices/Proof | https://proofwiki.org/wiki/Dandelin's_Theorem | https://proofwiki.org/wiki/Dandelin's_Theorem/Directrices/Proof | [
"Dandelin's Theorem",
"Dandelin Spheres",
"Conic Sections"
] | [
"Definition:Cone (Geometry)/Double Napped Cone",
"Definition:Right Circular Cone",
"Definition:Cone (Geometry)/Apex",
"Definition:Plane Surface",
"Definition:Intersection (Geometry)",
"Definition:Right Angle/Perpendicular/Plane to Plane",
"Definition:Right Circular Cone/Axis",
"Definition:Conic Sectio... | [
"Definition:Dandelin Spheres",
"Definition:Point",
"Definition:Point",
"Definition:Tangent Plane",
"Definition:Right Circular Cone/Generatrix",
"Definition:Right Circular Cone/Opening Angle",
"Definition:Inclination/Straight Line to Plane",
"Definition:Right Circular Cone/Axis",
"Definition:Intersec... |
proofwiki-18269 | Dandelin's Theorem | Let $\CC$ be a double napped right circular cone with apex $O$.
Let $\PP$ be a plane which intersects $\CC$ such that:
:$\PP$ does not pass through $O$
:$\PP$ is not perpendicular to the axis of $\CC$.
Let $\EE$ be the conic section arising as the intersection between $\PP$ and $\CC$.
Let $\SS$ and $\SS'$ be the Dandel... | Let $\SS$ and $\SS'$ be the Dandelin spheres with respect to $\PP$.
Let $P$ be a point on $\EE$.
Let $F$ and $F'$ be the points at which $\SS$ and $\SS'$ are tangent to $\PP$ respectively.
Let the generatrix of $\CC$ which passes through $P$ touch $\SS$ and $\SS'$ at $E$ and $E'$ respectively.
Because $PF$ and $PE$ are... | Let $\CC$ be a [[Definition:Double Napped Cone|double napped]] [[Definition:Right Circular Cone|right circular cone]] with [[Definition:Apex of Cone|apex]] $O$.
Let $\PP$ be a [[Definition:Plane|plane]] which [[Definition:Intersection (Geometry)|intersects]] $\CC$ such that:
:$\PP$ does not pass through $O$
:$\PP$ is ... | Let $\SS$ and $\SS'$ be the [[Definition:Dandelin Spheres|Dandelin spheres]] with respect to $\PP$.
Let $P$ be a [[Definition:Point|point]] on $\EE$.
Let $F$ and $F'$ be the [[Definition:Point|points]] at which $\SS$ and $\SS'$ are [[Definition:Tangent Plane|tangent]] to $\PP$ respectively.
Let the [[Definition:Gene... | Dandelin's Theorem/Foci/Proof | https://proofwiki.org/wiki/Dandelin's_Theorem | https://proofwiki.org/wiki/Dandelin's_Theorem/Foci/Proof | [
"Dandelin's Theorem",
"Dandelin Spheres",
"Conic Sections"
] | [
"Definition:Cone (Geometry)/Double Napped Cone",
"Definition:Right Circular Cone",
"Definition:Cone (Geometry)/Apex",
"Definition:Plane Surface",
"Definition:Intersection (Geometry)",
"Definition:Right Angle/Perpendicular/Plane to Plane",
"Definition:Right Circular Cone/Axis",
"Definition:Conic Sectio... | [
"Definition:Dandelin Spheres",
"Definition:Point",
"Definition:Point",
"Definition:Tangent Plane",
"Definition:Right Circular Cone/Generatrix",
"Definition:Tangent Line",
"Definition:Ellipse",
"File:Dandelins-theorem-ellipse.png",
"Definition:Constant",
"Definition:Ellipse/Equidistance",
"Defini... |
proofwiki-18270 | Dandelin's Theorem/Foci/Proof | Let $\CC$ be a double napped right circular cone with apex $O$.
Let $\PP$ be a plane which intersects $\CC$ such that:
:$\PP$ does not pass through $O$
:$\PP$ is not perpendicular to the axis of $\CC$.
Let $\EE$ be the conic section arising as the intersection between $\PP$ and $\CC$.
Let $\SS$ and $\SS'$ be the Dandel... | Let $\SS$ and $\SS'$ be the Dandelin spheres with respect to $\PP$.
Let $P$ be a point on $\EE$.
Let $F$ and $F'$ be the points at which $\SS$ and $\SS'$ are tangent to $\PP$ respectively.
Let the generatrix of $\CC$ which passes through $P$ touch $\SS$ and $\SS'$ at $E$ and $E'$ respectively.
Because $PF$ and $PE$ are... | Let $\CC$ be a [[Definition:Double Napped Cone|double napped]] [[Definition:Right Circular Cone|right circular cone]] with [[Definition:Apex of Cone|apex]] $O$.
Let $\PP$ be a [[Definition:Plane|plane]] which [[Definition:Intersection (Geometry)|intersects]] $\CC$ such that:
:$\PP$ does not pass through $O$
:$\PP$ is ... | Let $\SS$ and $\SS'$ be the [[Definition:Dandelin Spheres|Dandelin spheres]] with respect to $\PP$.
Let $P$ be a [[Definition:Point|point]] on $\EE$.
Let $F$ and $F'$ be the [[Definition:Point|points]] at which $\SS$ and $\SS'$ are [[Definition:Tangent Plane|tangent]] to $\PP$ respectively.
Let the [[Definition:Gene... | Dandelin's Theorem/Foci/Proof | https://proofwiki.org/wiki/Dandelin's_Theorem/Foci/Proof | https://proofwiki.org/wiki/Dandelin's_Theorem/Foci/Proof | [
"Dandelin's Theorem"
] | [
"Definition:Cone (Geometry)/Double Napped Cone",
"Definition:Right Circular Cone",
"Definition:Cone (Geometry)/Apex",
"Definition:Plane Surface",
"Definition:Intersection (Geometry)",
"Definition:Right Angle/Perpendicular/Plane to Plane",
"Definition:Right Circular Cone/Axis",
"Definition:Conic Sectio... | [
"Definition:Dandelin Spheres",
"Definition:Point",
"Definition:Point",
"Definition:Tangent Plane",
"Definition:Right Circular Cone/Generatrix",
"Definition:Tangent Line",
"Definition:Ellipse",
"File:Dandelins-theorem-ellipse.png",
"Definition:Constant",
"Definition:Ellipse/Equidistance",
"Defini... |
proofwiki-18271 | Dandelin's Theorem/Foci | :$\SS$ and $\SS'$ are tangent to $\PP$ at the foci of $\EE$. | Let $\SS$ and $\SS'$ be the Dandelin spheres with respect to $\PP$.
Let $P$ be a point on $\EE$.
Let $F$ and $F'$ be the points at which $\SS$ and $\SS'$ are tangent to $\PP$ respectively.
Let the generatrix of $\CC$ which passes through $P$ touch $\SS$ and $\SS'$ at $E$ and $E'$ respectively.
Because $PF$ and $PE$ are... | :$\SS$ and $\SS'$ are [[Definition:Tangent Plane|tangent]] to $\PP$ at the [[Definition:Focus of Conic Section|foci]] of $\EE$. | Let $\SS$ and $\SS'$ be the [[Definition:Dandelin Spheres|Dandelin spheres]] with respect to $\PP$.
Let $P$ be a [[Definition:Point|point]] on $\EE$.
Let $F$ and $F'$ be the [[Definition:Point|points]] at which $\SS$ and $\SS'$ are [[Definition:Tangent Plane|tangent]] to $\PP$ respectively.
Let the [[Definition:Gene... | Dandelin's Theorem/Foci/Proof | https://proofwiki.org/wiki/Dandelin's_Theorem/Foci | https://proofwiki.org/wiki/Dandelin's_Theorem/Foci/Proof | [
"Dandelin's Theorem"
] | [
"Definition:Tangent Plane",
"Definition:Conic Section/Focus"
] | [
"Definition:Dandelin Spheres",
"Definition:Point",
"Definition:Point",
"Definition:Tangent Plane",
"Definition:Right Circular Cone/Generatrix",
"Definition:Tangent Line",
"Definition:Ellipse",
"File:Dandelins-theorem-ellipse.png",
"Definition:Constant",
"Definition:Ellipse/Equidistance",
"Defini... |
proofwiki-18272 | Dandelin's Theorem/Directrices | Let $\KK$ and $\KK'$ be the planes in which the ring-contacts of $\CC$ with $\SS$ and $\SS'$ are embedded respectively.
:The intersections of $\KK$ and $\KK'$ with $\PP$ form the directrices of $\EE$. | Let $\SS$ and $\SS'$ be the Dandelin spheres with respect to $\PP$.
Let $P$ be a point on $\EE$.
Let $F$ and $F'$ be the points at which $\SS$ and $\SS'$ are tangent to $\PP$ respectively.
Let the generatrix of $\CC$ which passes through $P$ touch $\SS$ and $\SS'$ at $E$ and $E'$ respectively.
Let $\theta$ be half the ... | Let $\KK$ and $\KK'$ be the [[Definition:Plane|planes]] in which the [[Definition:Ring-Contact|ring-contacts]] of $\CC$ with $\SS$ and $\SS'$ are embedded respectively.
:The [[Definition:Intersection (Geometry)|intersections]] of $\KK$ and $\KK'$ with $\PP$ form the [[Definition:Directrix of Conic Section|directrices]... | Let $\SS$ and $\SS'$ be the [[Definition:Dandelin Spheres|Dandelin spheres]] with respect to $\PP$.
Let $P$ be a [[Definition:Point|point]] on $\EE$.
Let $F$ and $F'$ be the [[Definition:Point|points]] at which $\SS$ and $\SS'$ are [[Definition:Tangent Plane|tangent]] to $\PP$ respectively.
Let the [[Definition:Gene... | Dandelin's Theorem/Directrices/Proof | https://proofwiki.org/wiki/Dandelin's_Theorem/Directrices | https://proofwiki.org/wiki/Dandelin's_Theorem/Directrices/Proof | [
"Dandelin's Theorem"
] | [
"Definition:Plane Surface",
"Definition:Ring-Contact",
"Definition:Intersection (Geometry)",
"Definition:Conic Section/Directrix"
] | [
"Definition:Dandelin Spheres",
"Definition:Point",
"Definition:Point",
"Definition:Tangent Plane",
"Definition:Right Circular Cone/Generatrix",
"Definition:Right Circular Cone/Opening Angle",
"Definition:Inclination/Straight Line to Plane",
"Definition:Right Circular Cone/Axis",
"Definition:Intersec... |
proofwiki-18273 | Dandelin's Theorem/Directrices/Proof | Let $\CC$ be a double napped right circular cone with apex $O$.
Let $\PP$ be a plane which intersects $\CC$ such that:
:$\PP$ does not pass through $O$
:$\PP$ is not perpendicular to the axis of $\CC$.
Let $\EE$ be the conic section arising as the intersection between $\PP$ and $\CC$.
Let $\SS$ and $\SS'$ be the Dandel... | Let $\SS$ and $\SS'$ be the Dandelin spheres with respect to $\PP$.
Let $P$ be a point on $\EE$.
Let $F$ and $F'$ be the points at which $\SS$ and $\SS'$ are tangent to $\PP$ respectively.
Let the generatrix of $\CC$ which passes through $P$ touch $\SS$ and $\SS'$ at $E$ and $E'$ respectively.
Let $\theta$ be half the ... | Let $\CC$ be a [[Definition:Double Napped Cone|double napped]] [[Definition:Right Circular Cone|right circular cone]] with [[Definition:Apex of Cone|apex]] $O$.
Let $\PP$ be a [[Definition:Plane|plane]] which [[Definition:Intersection (Geometry)|intersects]] $\CC$ such that:
:$\PP$ does not pass through $O$
:$\PP$ is ... | Let $\SS$ and $\SS'$ be the [[Definition:Dandelin Spheres|Dandelin spheres]] with respect to $\PP$.
Let $P$ be a [[Definition:Point|point]] on $\EE$.
Let $F$ and $F'$ be the [[Definition:Point|points]] at which $\SS$ and $\SS'$ are [[Definition:Tangent Plane|tangent]] to $\PP$ respectively.
Let the [[Definition:Gene... | Dandelin's Theorem/Directrices/Proof | https://proofwiki.org/wiki/Dandelin's_Theorem/Directrices/Proof | https://proofwiki.org/wiki/Dandelin's_Theorem/Directrices/Proof | [
"Dandelin's Theorem"
] | [
"Definition:Cone (Geometry)/Double Napped Cone",
"Definition:Right Circular Cone",
"Definition:Cone (Geometry)/Apex",
"Definition:Plane Surface",
"Definition:Intersection (Geometry)",
"Definition:Right Angle/Perpendicular/Plane to Plane",
"Definition:Right Circular Cone/Axis",
"Definition:Conic Sectio... | [
"Definition:Dandelin Spheres",
"Definition:Point",
"Definition:Point",
"Definition:Tangent Plane",
"Definition:Right Circular Cone/Generatrix",
"Definition:Right Circular Cone/Opening Angle",
"Definition:Inclination/Straight Line to Plane",
"Definition:Right Circular Cone/Axis",
"Definition:Intersec... |
proofwiki-18274 | Conic Section is Curve of Second Degree | A '''conic section''' is a curve of second degree. | {{ProofWanted|A rigorous approach to this needs more background}} | A '''[[Definition:Conic Section|conic section]]''' is a [[Definition:Curve|curve]] of [[Definition:Degree of Algebraic Curve|second degree]]. | {{ProofWanted|A rigorous approach to this needs more background}} | Conic Section is Curve of Second Degree | https://proofwiki.org/wiki/Conic_Section_is_Curve_of_Second_Degree | https://proofwiki.org/wiki/Conic_Section_is_Curve_of_Second_Degree | [
"Conic Sections"
] | [
"Definition:Conic Section",
"Definition:Line/Curve",
"Definition:Algebraic Curve/Degree"
] | [] |
proofwiki-18275 | Equation of Conic in Cartesian Coordinates is Quadratic | Let $\CC$ be a conic section.
Then $\CC$ can be expressed by an quadratic equation in $2$ variables. | {{ProofWanted|Follows apparently from Conic Section is Curve of Second Degree. Sommerville is not rigorous about defining his terms.}} | Let $\CC$ be a [[Definition:Conic Section|conic section]].
Then $\CC$ can be expressed by an [[Definition:Quadratic Equation in Two Variables|quadratic equation in $2$ variables]]. | {{ProofWanted|Follows apparently from [[Conic Section is Curve of Second Degree]]. Sommerville is not rigorous about defining his terms.}} | Equation of Conic in Cartesian Coordinates is Quadratic | https://proofwiki.org/wiki/Equation_of_Conic_in_Cartesian_Coordinates_is_Quadratic | https://proofwiki.org/wiki/Equation_of_Conic_in_Cartesian_Coordinates_is_Quadratic | [
"Conic Sections"
] | [
"Definition:Conic Section",
"Definition:Quadratic Equation in Two Variables"
] | [
"Conic Section is Curve of Second Degree"
] |
proofwiki-18276 | Intersecting Chord Theorem for Conic Sections | Consider a right circular cone $\CC$ with opening angle $2 \alpha$ whose apex is at $O$.
Consider a slicing plane $\PP$, not passing through $O$, at an angle $\beta$ to the axis of $\CC$.
Let the plane $OAA'$ through the axis of $\CC$ perpendicular to $\PP$ intersect $\PP$ in the line $AA'$.
Let $P$ be an arbitrary poi... | {{ProofWanted|This is part of the muddled Construction of Conic Section in Sommerville. This collection of pages needs to be revisited with respect to a more coherent work than this one.}} | Consider a [[Definition:Right Circular Cone|right circular cone]] $\CC$ with [[Definition:Opening Angle|opening angle]] $2 \alpha$ whose [[Definition:Apex of Cone|apex]] is at $O$.
Consider a [[Definition:Slicing Plane|slicing plane]] $\PP$, not passing through $O$, at an [[Definition:Angle|angle]] $\beta$ to the [[De... | {{ProofWanted|This is part of the muddled [[Construction of Conic Section]] in Sommerville. This collection of pages needs to be revisited with respect to a more coherent work than this one.}} | Intersecting Chord Theorem for Conic Sections | https://proofwiki.org/wiki/Intersecting_Chord_Theorem_for_Conic_Sections | https://proofwiki.org/wiki/Intersecting_Chord_Theorem_for_Conic_Sections | [
"Conic Sections"
] | [
"Definition:Right Circular Cone",
"Definition:Right Circular Cone/Opening Angle",
"Definition:Cone (Geometry)/Apex",
"Definition:Conic Section/Intersection with Cone/Slicing Plane",
"Definition:Angle",
"Definition:Right Circular Cone/Axis",
"Definition:Plane Surface",
"Definition:Right Circular Cone/A... | [
"Construction of Conic Section"
] |
proofwiki-18277 | P-adic Norm Characterisation of Divisibility by Power of p | Let $p \in \N$ be a prime.
Let $\Q$ denote the rational numbers.
Let $\norm{\,\cdot\,}$ denote the $p$-adic norm on $\Q$.
Then:
:$\forall a, b \in \Z: a \equiv b \pmod {p^n} \iff \norm {a - b}_p \le p^{-n}$ | Let $a, b \in \Z$.
We have:
{{begin-eqn}}
{{eqn | l = a \equiv b \pmod {p^n}
| o = \iff
| r = p^n \divides \paren{a - b}
| c = {{defof|Congruence Modulo Integer}}
}}
{{eqn | o = \iff
| r = n \le sup(m : p^m \divides \paren{a - b}
| c = {{defof|Supremum of Set}}
}}
{{eqn | o = \iff
| ... | Let $p \in \N$ be a [[Definition:Prime Number|prime]].
Let $\Q$ denote the [[Definition:Rational Numbers|rational numbers]].
Let $\norm{\,\cdot\,}$ denote the [[Definition:P-adic Norm|$p$-adic norm]] on $\Q$.
Then:
:$\forall a, b \in \Z: a \equiv b \pmod {p^n} \iff \norm {a - b}_p \le p^{-n}$ | Let $a, b \in \Z$.
We have:
{{begin-eqn}}
{{eqn | l = a \equiv b \pmod {p^n}
| o = \iff
| r = p^n \divides \paren{a - b}
| c = {{defof|Congruence Modulo Integer}}
}}
{{eqn | o = \iff
| r = n \le sup(m : p^m \divides \paren{a - b}
| c = {{defof|Supremum of Set}}
}}
{{eqn | o = \iff
|... | P-adic Norm Characterisation of Divisibility by Power of p | https://proofwiki.org/wiki/P-adic_Norm_Characterisation_of_Divisibility_by_Power_of_p | https://proofwiki.org/wiki/P-adic_Norm_Characterisation_of_Divisibility_by_Power_of_p | [] | [
"Definition:Prime Number",
"Definition:Rational Number",
"Definition:P-adic Norm"
] | [
"Power Function on Base Greater than One is Strictly Increasing",
"Inversion Mapping Reverses Ordering in Ordered Group"
] |
proofwiki-18278 | Continuously Differentiable Real Function at Removable Discontinuity | Let $f : \R \to \R$ be a real function.
Let $a \in \R$ be real number.
Let $f$ be continuous on $\R$ and continuously differentiable in $\R \setminus \set a$.
Suppose that $a$ is a removable discontinuity of $f'$.
That is, suppose the limit $\ds \lim_{x \mathop \to a} \map {f'} x$ exists.
Then $f$ is continuously diff... | By assumption, $\ds \lim_{x \mathop \to a} \map {f'} x = L$ exists.
By definition:
:$\forall \epsilon \in \R_{>0} : \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - a} < \delta \implies \size {\map {f'} x - L} < \epsilon$
Let $h := x - a$.
Suppose $0 < h < \delta$.
Consider a closed interval $\closedint 0 h... | Let $f : \R \to \R$ be a [[Definition:Real Function|real function]].
Let $a \in \R$ be [[Definition:Real Number|real number]].
Let $f$ be [[Definition:Continuous Real Function|continuous]] on $\R$ and [[Definition:Continuously Differentiable Real Function|continuously differentiable]] in $\R \setminus \set a$.
Supp... | By assumption, $\ds \lim_{x \mathop \to a} \map {f'} x = L$ exists.
By [[Definition:Limit of Real Function/Definition 1|definition]]:
:$\forall \epsilon \in \R_{>0} : \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - a} < \delta \implies \size {\map {f'} x - L} < \epsilon$
Let $h := x - a$.
Suppose $0 < ... | Continuously Differentiable Real Function at Removable Discontinuity | https://proofwiki.org/wiki/Continuously_Differentiable_Real_Function_at_Removable_Discontinuity | https://proofwiki.org/wiki/Continuously_Differentiable_Real_Function_at_Removable_Discontinuity | [
"Continuity",
"Differentiable Real Functions"
] | [
"Definition:Real Function",
"Definition:Real Number",
"Definition:Continuous Real Function",
"Definition:Continuously Differentiable/Real Function",
"Definition:Discontinuity (Real Analysis)/Removable",
"Definition:Limit of Real Function",
"Definition:Continuously Differentiable/Real Function"
] | [
"Definition:Limit of Real Function/Definition 1",
"Definition:Real Interval/Closed",
"Definition:Continuous Real Function",
"Definition:Continuously Differentiable/Real Function",
"Mean Value Theorem",
"Definition:Real Interval/Closed",
"Definition:Continuous Real Function",
"Definition:Continuously D... |
proofwiki-18279 | Ellipse is Bounded in Plane | Let $E$ be an ellipse embedded in in a Euclidean plane.
Then $E$ is bounded. | Let a Cartesian coordinate system be applied to the Euclidean plane in which $E$ is embedded.
Let $E$ be expressed in reduced form:
{{begin-eqn}}
{{eqn | l = \dfrac {x^2} {a^2} + \dfrac {y^2} {b^2}
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac x a
| r = \sqrt {1 - \dfrac {y^2} {b^2} }
... | Let $E$ be an [[Definition:Ellipse|ellipse]] embedded in in a [[Definition:Real Number Plane with Euclidean Metric|Euclidean plane]].
Then $E$ is [[Definition:Bounded Metric Space|bounded]]. | Let a [[Definition:Cartesian Coordinate System|Cartesian coordinate system]] be applied to the [[Definition:Euclidean Metric on Real Number Plane|Euclidean plane]] in which $E$ is embedded.
Let $E$ be expressed in [[Definition:Reduced Form of Ellipse|reduced form]]:
{{begin-eqn}}
{{eqn | l = \dfrac {x^2} {a^2} + \dfr... | Ellipse is Bounded in Plane | https://proofwiki.org/wiki/Ellipse_is_Bounded_in_Plane | https://proofwiki.org/wiki/Ellipse_is_Bounded_in_Plane | [
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Euclidean Metric/Real Number Plane",
"Definition:Bounded Metric Space"
] | [
"Definition:Cartesian Coordinate System",
"Definition:Euclidean Metric/Real Number Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Quadrilateral/Rectangle",
"Definition:Polygon/Side"
] |
proofwiki-18280 | Circle is Ellipse with Equal Major and Minor Axes | Let $E$ be an ellipse whose major axis is equal to its minor axis.
Then $E$ is a circle. | Let $E$ be embedded in a Cartesian plane in reduced form.
Then from Equation of Ellipse in Reduced Form $E$ can be expressed using the equation:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
where the major axis and minor axis are $a$ and $b$ respectively.
Let $a = b$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac {x^2} {a^2... | Let $E$ be an [[Definition:Ellipse|ellipse]] whose [[Definition:Major Axis of Ellipse|major axis]] is equal to its [[Definition:Minor Axis of Ellipse|minor axis]].
Then $E$ is a [[Definition:Circle|circle]]. | Let $E$ be embedded in a [[Definition:Cartesian Plane|Cartesian plane]] in [[Definition:Reduced Form of Ellipse|reduced form]].
Then from [[Equation of Ellipse in Reduced Form]] $E$ can be expressed using the equation:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
where the [[Definition:Major Axis of Ellipse|major ax... | Circle is Ellipse with Equal Major and Minor Axes | https://proofwiki.org/wiki/Circle_is_Ellipse_with_Equal_Major_and_Minor_Axes | https://proofwiki.org/wiki/Circle_is_Ellipse_with_Equal_Major_and_Minor_Axes | [
"Circles",
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Ellipse/Major Axis",
"Definition:Ellipse/Minor Axis",
"Definition:Circle"
] | [
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Equation of Ellipse in Reduced Form",
"Definition:Ellipse/Major Axis",
"Definition:Ellipse/Minor Axis",
"Equation of Circle center Origin",
"Definition:Circle",
"Definition:Circle/Radius"
] |
proofwiki-18281 | Equation of Tangent to Ellipse in Reduced Form | Let $E$ be an ellipse embedded in a Cartesian plane in reduced form with the equation:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Let $P = \tuple {x_1, y_1}$ be a point on $E$.
The tangent to $E$ at $P$ is given by the equation:
:$\dfrac {x x_1} {a^2} + \dfrac {y y_1} {b^2} = 1$ | From the slope-intercept form of a line, the equation of a line passing through $P$ is:
:$y - y_1 = \mu \paren {x - x_1}$
If this line passes through another point $\tuple {x_2, y_2}$ on $E$, the slope of the line is given by:
:$\mu = \dfrac {y_2 - y_1} {x_2 - x_1}$
Because $P$ and $Q$ both lie on $E$, we have:
{{begin... | Let $E$ be an [[Definition:Ellipse|ellipse]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]] in [[Definition:Reduced Form of Ellipse|reduced form]] with the equation:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Let $P = \tuple {x_1, y_1}$ be a [[Definition:Point|point]] on $E$.
The [[Definition:Tange... | From the [[Equation of Straight Line in Plane/Slope-Intercept Form|slope-intercept form]] of a [[Definition:Straight Line|line]], the equation of a [[Definition:Straight Line|line]] passing through $P$ is:
:$y - y_1 = \mu \paren {x - x_1}$
If this [[Definition:Straight Line|line]] passes through another [[Definition:P... | Equation of Tangent to Ellipse in Reduced Form | https://proofwiki.org/wiki/Equation_of_Tangent_to_Ellipse_in_Reduced_Form | https://proofwiki.org/wiki/Equation_of_Tangent_to_Ellipse_in_Reduced_Form | [
"Ellipses",
"Tangents"
] | [
"Definition:Ellipse",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Definition:Point",
"Definition:Tangent Line"
] | [
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line",
"Definition:Point",
"Definition:Slope/Straight Line",
"Definition:Line/Straight Line",
"Difference of Two Squares",
"Definition:Limit of Real ... |
proofwiki-18282 | Metric on P-adic Numbers Extends Metric on Rationals | Let $p$ be any prime number.
Let $\struct{\Q, \norm {\,\cdot\,}^\Q_p}$ be the rational numbers $\Q$ with the $p$-adic norm $\norm {\,\cdot\,}^\Q_p$.
Let $d^{\Q}_p$ be the $p$-adic metric on the rational numbers.
Let $\struct{\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $d_p$ be the $p$-adic metric on the $p... | From Rational Numbers are Dense Subfield of P-adic Numbers:
:$\norm {\,\cdot\,}_p$ on $\Q_p$ is an extension of $\norm {\,\cdot\,}^Q_p$ on $\Q$.
By definition of $p$-adic metric on the rational numbers:
:$\forall x, y \in \Q : \map {d^\Q_p} {x, y} = \norm{x - y}^\Q_p$
By definition of $p$-adic metric on the $p$-adic nu... | Let $p$ be any [[Definition:Prime Number|prime number]].
Let $\struct{\Q, \norm {\,\cdot\,}^\Q_p}$ be the [[Definition:Rational Number|rational numbers]] $\Q$ with the [[Definition:P-adic Norm on Rational Numbers|$p$-adic norm]] $\norm {\,\cdot\,}^\Q_p$.
Let $d^{\Q}_p$ be the [[Definition:P-adic Metric on Rational N... | From [[Rational Numbers are Dense Subfield of P-adic Numbers]]:
:$\norm {\,\cdot\,}_p$ on $\Q_p$ is an [[Definition:Extension|extension]] of $\norm {\,\cdot\,}^Q_p$ on $\Q$.
By definition of [[Definition:P-adic Metric on Rational Numbers|$p$-adic metric]] on the [[Definition:Rational Number|rational numbers]]:
:$\fora... | Metric on P-adic Numbers Extends Metric on Rationals | https://proofwiki.org/wiki/Metric_on_P-adic_Numbers_Extends_Metric_on_Rationals | https://proofwiki.org/wiki/Metric_on_P-adic_Numbers_Extends_Metric_on_Rationals | [
"P-adic Metrics"
] | [
"Definition:Prime Number",
"Definition:Rational Number",
"Definition:P-adic Norm/Rational Numbers",
"Definition:P-adic Metric/Rational Numbers",
"Definition:Rational Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Metric/P-adic Numbers",
"Definition:Valued Field of P-adic Numb... | [
"Rational Numbers are Dense Subfield of P-adic Numbers",
"Definition:Extension",
"Definition:P-adic Metric/Rational Numbers",
"Definition:Rational Number",
"Definition:P-adic Metric/P-adic Numbers",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Metric/P-adic Numbers",
"Definition:Ext... |
proofwiki-18283 | Poisson Distribution Approximated by Hat-Check Distribution | Let $X$ be a discrete random variable which has the hat-check distribution with parameter $n$.
Then $X$ can be approximated by a Poisson distribution with parameter $\lambda = 1$. | Let $X$ be as described.
Let $k \ge 0$ be fixed.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
| r = \lim_{n \mathop \to \infty} \dfrac 1 {\paren {n - \paren {n - k} }!} \sum_{s \mathop = 0}^{n - k} \dfrac {\paren {-1}^s}... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] which has the [[Definition:Hat-Check Distribution|hat-check distribution]] with parameter $n$.
Then $X$ can be approximated by a [[Definition:Poisson Distribution|Poisson distribution with parameter $\lambda = 1$]]. | Let $X$ be as described.
Let $k \ge 0$ be fixed.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}
| r = \lim_{n \mathop \to \infty} \dfrac 1 {\paren {n - \paren {n - k} }!} \sum_{s \mathop = 0}^{n - k} \dfrac {\paren {-1... | Poisson Distribution Approximated by Hat-Check Distribution | https://proofwiki.org/wiki/Poisson_Distribution_Approximated_by_Hat-Check_Distribution | https://proofwiki.org/wiki/Poisson_Distribution_Approximated_by_Hat-Check_Distribution | [
"Hat-Check Distribution",
"Poisson Distribution"
] | [
"Definition:Random Variable/Discrete",
"Definition:Hat-Check Distribution",
"Definition:Poisson Distribution"
] | [
"Power Series Expansion for Exponential Function"
] |
proofwiki-18284 | Simson Line Theorem | Let $\triangle ABC$ be a triangle.
Let $P$ be a point on the circumcircle of $\triangle ABC$.
Then the feet of the perpendiculars drawn from $P$ to each of the sides of $\triangle ABC$ are collinear.
:300px
This line is called the '''Wallace-Simson line'''. | In the figure above, construct the lines $BP$ and $CP$.
:300px
By the converse of Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles, $EPDB$ is cyclic.
By the converse of Angles in Same Segment of Circle are Equal, $EPCF$ is cyclic.
{{WIP|Looking for the pages for the above converses and the (simple) unlin... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $P$ be a [[Definition:Point|point]] on the [[Definition:Circumcircle|circumcircle]] of $\triangle ABC$.
Then the [[Definition:Foot of Perpendicular|feet]] of the [[Definition:Perpendicular|perpendiculars]] drawn from $P$ to each of the [[Defini... | In the figure above, construct the [[Definition:Line Segment|lines]] $BP$ and $CP$.
:[[File:Simson-line-2.png|300px]]
By the converse of [[Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles]], $EPDB$ is [[Definition:Cyclic Quadrilateral|cyclic]].
By the converse of [[Angles in Same Segment of Circle a... | Simson Line Theorem | https://proofwiki.org/wiki/Simson_Line_Theorem | https://proofwiki.org/wiki/Simson_Line_Theorem | [
"Wallace-Simson Line",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Point",
"Definition:Circumcircle",
"Definition:Right Angle/Perpendicular/Foot",
"Definition:Right Angle/Perpendicular",
"Definition:Polygon/Side",
"Definition:Collinear/Points",
"File:Simson-line-1.png",
"Definition:Wallace-Simson Line"
] | [
"Definition:Line/Segment",
"File:Simson-line-2.png",
"Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles",
"Definition:Cyclic Quadrilateral",
"Angles in Same Segment of Circle are Equal",
"Definition:Cyclic Quadrilateral",
"Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles",
... |
proofwiki-18285 | Stone-Weierstrass Theorem/Compact Space | Let $T = \struct {X, \tau}$ be a compact topological space.
Let $\map C {X, \R}$ be the set of real-valued continuous functions on $T$.
Let $\times$ be the pointwise multiplication on $\map C {X, \R}$.
Let $\struct {\map C {X, \R}, \times}$ be the Banach algebra with respect to $\norm \cdot_\infty$.
{{ExtractTheorem|A ... | Let $\struct {C', \norm {\,\cdot\,}_{C'} }$ be the dual space of $\struct {\map C {X, \R}, \norm {\,\cdot\,}_\infty }$.
From Spanning Criterion of Normed Vector Space, it suffices to show that:
:$\forall \ell \in C' : \ell \restriction_\AA = 0 \implies \ell = 0$
Let $B' \subseteq C'$ be the closed unit ball, that is:
:... | Let $T = \struct {X, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $\map C {X, \R}$ be the [[Definition:Set|set]] of [[Definition:Real-Valued Function|real-valued]] [[Definition:Continuous Mapping|continuous functions]] on $T$.
Let $\times$ be the [[Definition:Pointwise Multiplic... | Let $\struct {C', \norm {\,\cdot\,}_{C'} }$ be the [[Definition:Normed Dual Space|dual space]] of $\struct {\map C {X, \R}, \norm {\,\cdot\,}_\infty }$.
From [[Spanning Criterion of Normed Vector Space]], it suffices to show that:
:$\forall \ell \in C' : \ell \restriction_\AA = 0 \implies \ell = 0$
Let $B' \subseteq ... | Stone-Weierstrass Theorem/Compact Space | https://proofwiki.org/wiki/Stone-Weierstrass_Theorem/Compact_Space | https://proofwiki.org/wiki/Stone-Weierstrass_Theorem/Compact_Space | [
"Stone-Weierstrass Theorem",
"Compact Topological Spaces",
"Banach Algebras",
"Hausdorff Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:Set",
"Definition:Real-Valued Function",
"Definition:Continuous Mapping",
"Definition:Pointwise Multiplication of Real-Valued Functions",
"Definition:Banach Algebra",
"Definition:Supremum Norm",
"Definition:Banach Algebra",
"Definition:Unital Subal... | [
"Definition:Normed Dual Space",
"Spanning Criterion of Normed Vector Space",
"Definition:Closed Unit Ball",
"Definition:Set",
"Definition:Extreme Point of Convex Set"
] |
proofwiki-18286 | Integers with Metric Induced by P-adic Valuation | Let $p \in \N$ be a prime.
Let $d: \Z^2 \to \R_{\ge 0}$ be the mapping defined as:
:$\forall x, y \in \Z: \map d {x, y} = \begin {cases} 0 & : x = y \\ \dfrac 1 r & : x - y = p^{r - 1} k: r \in \N_{>0}, k \in \Z, p \nmid k \end {cases}$
Then $d$ is a metric on $\Z$. | From Characterization of P-adic Valuation on Integers:
:$d$ is well-defined.
We prove the metric space axioms. | Let $p \in \N$ be a [[Definition:Prime Number|prime]].
Let $d: \Z^2 \to \R_{\ge 0}$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x, y \in \Z: \map d {x, y} = \begin {cases} 0 & : x = y \\ \dfrac 1 r & : x - y = p^{r - 1} k: r \in \N_{>0}, k \in \Z, p \nmid k \end {cases}$
Then $d$ is a [[Definition:D... | From [[Characterization of P-adic Valuation on Integers]]:
:$d$ is [[Definition:Well-Defined|well-defined]].
We prove the [[Axiom:Metric Space Axioms|metric space axioms]]. | Integers with Metric Induced by P-adic Valuation | https://proofwiki.org/wiki/Integers_with_Metric_Induced_by_P-adic_Valuation | https://proofwiki.org/wiki/Integers_with_Metric_Induced_by_P-adic_Valuation | [
"Definitions/Examples of Metric Spaces"
] | [
"Definition:Prime Number",
"Definition:Mapping",
"Definition:Distance Function"
] | [
"Characterization of P-adic Valuation on Integers",
"Definition:Well-Defined",
"Axiom:Metric Space Axioms",
"Characterization of P-adic Valuation on Integers",
"Characterization of P-adic Valuation on Integers"
] |
proofwiki-18287 | Equation of Chord of Contact on Ellipse in Reduced Form | Let $\EE$ be an ellipse embedded in a Cartesian plane in reduced form with the equation:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Let $P = \tuple {x_0, y_0}$ be a point which is outside the boundary of $\EE$.
Let $UV$ be the chord of contact on $\EE$ with respect to $P$.
Then $UV$ can be defined by the equation:
... | Let $\TT_1$ and $\TT_2$ be a tangents to $\EE$ passing through $P$.
Let:
:$\TT_1$ touch $\EE$ at $U = \tuple {x_1, y_1}$
:$\TT_2$ touch $\EE$ at $V = \tuple {x_2, y_2}$
Then the chord of contact on $\EE$ with respect to $P$ is defined as $UV$.
:480px
From Equation of Tangent to Ellipse in Reduced Form, $\TT_1$ is expre... | Let $\EE$ be an [[Definition:Ellipse|ellipse]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]] in [[Definition:Reduced Form of Ellipse|reduced form]] with the equation:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Let $P = \tuple {x_0, y_0}$ be a [[Definition:Point|point]] which is outside the [[Definit... | Let $\TT_1$ and $\TT_2$ be a [[Definition:Tangent Line|tangents]] to $\EE$ passing through $P$.
Let:
:$\TT_1$ [[Definition:Tangent to Circle|touch]] $\EE$ at $U = \tuple {x_1, y_1}$
:$\TT_2$ [[Definition:Tangent to Circle|touch]] $\EE$ at $V = \tuple {x_2, y_2}$
Then the [[Definition:Chord of Contact on Ellipse|chord... | Equation of Chord of Contact on Ellipse in Reduced Form | https://proofwiki.org/wiki/Equation_of_Chord_of_Contact_on_Ellipse_in_Reduced_Form | https://proofwiki.org/wiki/Equation_of_Chord_of_Contact_on_Ellipse_in_Reduced_Form | [
"Ellipses",
"Chords of Contact"
] | [
"Definition:Ellipse",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Definition:Point",
"Definition:Boundary (Geometry)",
"Definition:Chord of Contact/Ellipse",
"Definition:Equation"
] | [
"Definition:Tangent Line",
"Definition:Tangent Line/Circle",
"Definition:Tangent Line/Circle",
"Definition:Chord of Contact/Ellipse",
"File:Equation-of-polar-of-ellipse.png",
"Equation of Tangent to Ellipse in Reduced Form",
"Equation of Tangent to Circle Centered at Origin",
"Definition:Line/Straight... |
proofwiki-18288 | Polar is Locus of Harmonic Conjugates wrt Ellipse | Let $\EE$ be an ellipse embedded in the plane.
Let $P$ be an arbitrary point in the plane.
Let $\LL$ be the polar of $P$ {{WRT}} $\EE$.
Then $\LL$ is the locus of harmonic conjugates {{WRT}} $\EE$. | {{ProofWanted|Need to craft the definition of harmonic conjugates {{WRT}} $\EE$. See Harmonic Property of Pole and Polar wrt Circle and explore what Sommerville says about this in chapter $\text {III}$ section $5$.}} | Let $\EE$ be an [[Definition:Ellipse|ellipse]] embedded in [[Definition:The Plane|the plane]].
Let $P$ be an arbitrary [[Definition:Point|point]] in [[Definition:The Plane|the plane]].
Let $\LL$ be the [[Definition:Polar of Point wrt Ellipse|polar]] of $P$ {{WRT}} $\EE$.
Then $\LL$ is the [[Definition:Locus|locus]]... | {{ProofWanted|Need to craft the definition of [[Definition:Harmonic Conjugates wrt Curve|harmonic conjugates]] {{WRT}} $\EE$. See [[Harmonic Property of Pole and Polar wrt Circle]] and explore what Sommerville says about this in chapter $\text {III}$ section $5$.}} | Polar is Locus of Harmonic Conjugates wrt Ellipse | https://proofwiki.org/wiki/Polar_is_Locus_of_Harmonic_Conjugates_wrt_Ellipse | https://proofwiki.org/wiki/Polar_is_Locus_of_Harmonic_Conjugates_wrt_Ellipse | [
"Polars of Points",
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Plane Surface/The Plane",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Polar of Point/Ellipse",
"Definition:Locus",
"Definition:Harmonic Conjugates with respect to Curve"
] | [
"Definition:Harmonic Conjugates wrt Curve",
"Harmonic Property of Pole and Polar/Circle"
] |
proofwiki-18289 | Parabola has No Center | The parabola has no center. | By definition, the center of a conic section is the midpoint of the straight line segment whose endpoints are the foci.
From Parabola has One Focus, it is not possible to construct such a line segment.
Hence the result.
{{qed}}
Category:Parabolas
Category:Centers of Conic Sections
c9wr9ir2w5tdidbnpzdwbldzcao0n7u | The [[Definition:Parabola|parabola]] has no [[Definition:Center of Conic Section|center]]. | By definition, the [[Definition:Center of Conic Section|center]] of a [[Definition:Conic Section|conic section]] is the [[Definition:Midpoint of Line|midpoint]] of the [[Definition:Straight Line Segment|straight line segment]] whose [[Definition:Endpoint of Line|endpoints]] are the [[Definition:Focus of Conic Section|f... | Parabola has No Center | https://proofwiki.org/wiki/Parabola_has_No_Center | https://proofwiki.org/wiki/Parabola_has_No_Center | [
"Parabolas",
"Centers of Conic Sections"
] | [
"Definition:Parabola",
"Definition:Conic Section/Center"
] | [
"Definition:Conic Section/Center",
"Definition:Conic Section",
"Definition:Line/Midpoint",
"Definition:Line/Straight Line Segment",
"Definition:Line/Endpoint",
"Definition:Conic Section/Focus",
"Parabola has One Focus",
"Definition:Line/Straight Line Segment",
"Category:Parabolas",
"Category:Cente... |
proofwiki-18290 | Locus of Midpoints of Parallel Chords of Central Conic passes through Center | Let $\KK$ be a central conic.
Let $\DD$ be the locus of the midpoints of a system of parallel chords of $\KK$.
Then $\DD$ passes through the center of $\KK$. | We note from Circle is Ellipse with Equal Major and Minor Axes that a circle is a special case of the ellipse.
Hence there is no need to investigate the circle separately.
It remains to demonstrate the result for the ellipse and the hyperbola.
Let the central conic be expressed in reduced form.
By definition of reduced... | Let $\KK$ be a [[Definition:Central Conic|central conic]].
Let $\DD$ be the [[Definition:Locus|locus]] of the [[Definition:Midpoint of Line|midpoints]] of a system of [[Definition:Parallel Lines|parallel]] [[Definition:Chord of Conic Section|chords]] of $\KK$.
Then $\DD$ passes through the [[Definition:Center of Con... | We note from [[Circle is Ellipse with Equal Major and Minor Axes]] that a [[Definition:Circle|circle]] is a special case of the [[Definition:Ellipse|ellipse]].
Hence there is no need to investigate the [[Definition:Circle|circle]] separately.
It remains to demonstrate the result for the [[Definition:Ellipse|ellipse]]... | Locus of Midpoints of Parallel Chords of Central Conic passes through Center | https://proofwiki.org/wiki/Locus_of_Midpoints_of_Parallel_Chords_of_Central_Conic_passes_through_Center | https://proofwiki.org/wiki/Locus_of_Midpoints_of_Parallel_Chords_of_Central_Conic_passes_through_Center | [
"Diameters of Conic Sections",
"Central Conics"
] | [
"Definition:Central Conic",
"Definition:Locus",
"Definition:Line/Midpoint",
"Definition:Parallel (Geometry)/Lines",
"Definition:Chord of Conic Section",
"Definition:Conic Section/Center"
] | [
"Circle is Ellipse with Equal Major and Minor Axes",
"Definition:Circle",
"Definition:Ellipse",
"Definition:Circle",
"Definition:Ellipse",
"Definition:Hyperbola",
"Definition:Central Conic",
"Definition:Conic Section/Reduced Form",
"Definition:Conic Section/Reduced Form",
"Definition:Conic Section... |
proofwiki-18291 | Characterization of Derivative of Test Function | Let $\Phi, \phi \in \map \DD \R$ be test functions.
Let:
:$\ds X = \set {\Phi' : \Phi \in \map \DD \R} $
:$\ds Y = \set {\phi \in \map \DD \R : \int_{- \infty}^\infty \map \phi x \rd x = 0}$
Then $X = Y$.
Furthermore, for every $\phi \in Y$ there exists only one $\Phi \in \map \DD \R$ such that $\Phi' = \phi$. | === $X$ is a subset of $Y$ ===
By definition of test functions:
:$\paren {\Phi \in \map \DD \R} \implies \paren {\Phi' \in \map \DD \R}$
Moreover, a test function has compact support:
:$\ds \int_{-\infty}^\infty \map {\Phi'} x \rd x = \map \Phi \infty - \map \Phi {-\infty} = 0$
Hence:
:$X \subseteq Y$
{{qed|lemma}} | Let $\Phi, \phi \in \map \DD \R$ be [[Definition:Test Function|test functions]].
Let:
:$\ds X = \set {\Phi' : \Phi \in \map \DD \R} $
:$\ds Y = \set {\phi \in \map \DD \R : \int_{- \infty}^\infty \map \phi x \rd x = 0}$
Then $X = Y$.
Furthermore, for every $\phi \in Y$ there exists [[Definition:Unique|only one]] ... | === $X$ is a subset of $Y$ ===
By definition of [[Definition:Test Function|test functions]]:
:$\paren {\Phi \in \map \DD \R} \implies \paren {\Phi' \in \map \DD \R}$
Moreover, a [[Definition:Test Function|test function]] has [[Definition:Compact Set of Reals|compact]] [[Definition:Support of Continuous Real-Valued F... | Characterization of Derivative of Test Function | https://proofwiki.org/wiki/Characterization_of_Derivative_of_Test_Function | https://proofwiki.org/wiki/Characterization_of_Derivative_of_Test_Function | [
"Test Functions"
] | [
"Definition:Test Function",
"Definition:Unique"
] | [
"Definition:Test Function",
"Definition:Test Function",
"Definition:Compact Space/Real Analysis",
"Definition:Support of Continuous Mapping/Real-Valued",
"Definition:Test Function",
"Definition:Support of Continuous Mapping/Real-Valued",
"Definition:Support of Continuous Mapping/Real-Valued",
"Definit... |
proofwiki-18292 | Center of Conic is Center of Symmetry | Let $\KK$ be a central conic.
Let $C$ be the center of $\KK$.
Then $C$ is a center of symmetry for $\KK$. | First we note from Parabola has No Center that we do not need to consider the parabola.
Then we note from Circle is Ellipse with Equal Major and Minor Axes that a circle is a special case of the ellipse.
Hence there is no need to investigate the circle separately.
It remains to demonstrate the result for the ellipse an... | Let $\KK$ be a [[Definition:Central Conic|central conic]].
Let $C$ be the [[Definition:Center of Conic Section|center]] of $\KK$.
Then $C$ is a [[Definition:Center of Symmetry|center of symmetry]] for $\KK$. | First we note from [[Parabola has No Center]] that we do not need to consider the [[Definition:Parabola|parabola]].
Then we note from [[Circle is Ellipse with Equal Major and Minor Axes]] that a [[Definition:Circle|circle]] is a special case of the [[Definition:Ellipse|ellipse]].
Hence there is no need to investigate... | Center of Conic is Center of Symmetry | https://proofwiki.org/wiki/Center_of_Conic_is_Center_of_Symmetry | https://proofwiki.org/wiki/Center_of_Conic_is_Center_of_Symmetry | [
"Centers of Conic Sections"
] | [
"Definition:Central Conic",
"Definition:Conic Section/Center",
"Definition:Center of Symmetry"
] | [
"Parabola has No Center",
"Definition:Parabola",
"Circle is Ellipse with Equal Major and Minor Axes",
"Definition:Circle",
"Definition:Ellipse",
"Definition:Circle",
"Definition:Ellipse",
"Definition:Hyperbola",
"Definition:Central Conic",
"Definition:Conic Section/Reduced Form",
"Definition:Con... |
proofwiki-18293 | Equation of Hyperbola in Reduced Form/Polar Frame | Let $K$ be aligned in a polar plane in reduced form.
$K$ can be expressed by the equation:
:$\dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2} = \dfrac 1 {r^2}$ | Let the polar plane be aligned with its corresponding Cartesian plane in the conventional manner.
We have that
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
From Conversion between Cartesian and Polar Coordinates in Plane:
{{begin-eqn}}
{{eqn | l = x
| r = r \cos \theta
}}
{{eqn | l = y
| r = r \sin \theta... | Let $K$ be aligned in a [[Definition:Polar Plane|polar plane]] in [[Definition:Reduced Form of Hyperbola|reduced form]].
$K$ can be expressed by the equation:
:$\dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2} = \dfrac 1 {r^2}$ | Let the [[Definition:Polar Plane|polar plane]] be aligned with its corresponding [[Definition:Cartesian Plane|Cartesian plane]] in the conventional manner.
We have that
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
From [[Conversion between Cartesian and Polar Coordinates in Plane]]:
{{begin-eqn}}
{{eqn | l = x
... | Equation of Hyperbola in Reduced Form/Polar Frame | https://proofwiki.org/wiki/Equation_of_Hyperbola_in_Reduced_Form/Polar_Frame | https://proofwiki.org/wiki/Equation_of_Hyperbola_in_Reduced_Form/Polar_Frame | [
"Equation of Hyperbola in Reduced Form"
] | [
"Definition:Polar Coordinates/Polar Plane",
"Definition:Conic Section/Reduced Form/Hyperbola"
] | [
"Definition:Polar Coordinates/Polar Plane",
"Definition:Cartesian Plane",
"Conversion between Cartesian and Polar Coordinates in Plane"
] |
proofwiki-18294 | Asymptotes to Hyperbola in Reduced Form | Let $\KK$ be a hyperbola embedded in a cartesian plane in reduced form with the equation:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
$\KK$ has two asymptotes which can be described by the equation:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 0$
that is:
:$y = \pm \dfrac b a x$ | From Equation of Hyperbola in Reduced Form: Polar Frame, $\KK$ can be described in polar coordinates as:
:$\dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2} = \dfrac 1 {r^2}$
When $\theta = 0$ we have that $r = a$.
As $\theta$ increases, $\cos^2 \theta$ decreases and $\sin^2 \theta$ increases.
Hence $\dfrac 1... | Let $\KK$ be a [[Definition:Hyperbola|hyperbola]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] in [[Definition:Reduced Form of Hyperbola|reduced form]] with the equation:
:$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
$\KK$ has two [[Definition:Asymptote of Hyperbola|asymptotes]] which can be describe... | From [[Equation of Hyperbola in Reduced Form/Polar Frame|Equation of Hyperbola in Reduced Form: Polar Frame]], $\KK$ can be described in [[Definition:Polar Coordinates|polar coordinates]] as:
:$\dfrac {\cos^2 \theta} {a^2} - \dfrac {\sin^2 \theta} {b^2} = \dfrac 1 {r^2}$
When $\theta = 0$ we have that $r = a$.
As $\... | Asymptotes to Hyperbola in Reduced Form | https://proofwiki.org/wiki/Asymptotes_to_Hyperbola_in_Reduced_Form | https://proofwiki.org/wiki/Asymptotes_to_Hyperbola_in_Reduced_Form | [
"Hyperbolas"
] | [
"Definition:Hyperbola",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Hyperbola",
"Definition:Hyperbola/Asymptote"
] | [
"Equation of Hyperbola in Reduced Form/Polar Frame",
"Definition:Polar Coordinates",
"Definition:Homogeneous Quadratic Equation",
"Definition:Quadratic Equation in Two Variables",
"Homogeneous Quadratic Equation represents Two Straight Lines through Origin",
"Definition:Line/Straight Line",
"Definition:... |
proofwiki-18295 | Slope of Tangent to Curve at Point equals Value of Derivative | Let $\CC$ be a curve embedded in the Cartesian plane described using the equation:
:$y = \map f x$
where $f$ is a real function.
Let there exist a unique tangent $\TT$ to $\CC$ at a point $P = \tuple {x_0, y_0}$ on $\CC$.
Then the slope of $\CC$ at $P$ is equal to the derivative of $f$ at $P$. | We have been given that there exists a unique tangent $\TT$ to $\CC$ at $P$.
By definition of tangent, $\TT$ has a slope $M$ given by:
:$m = \ds \lim_{h \mathop \to 0} \frac {\map f {x_0 + h} - \map f {x_0} } h$
This is the definition of the derivative of $f$ at $P$.
{{qed}} | Let $\CC$ be a [[Definition:Curve|curve]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]] described using the equation:
:$y = \map f x$
where $f$ is a [[Definition:Real Function|real function]].
Let there exist a [[Definition:Unique|unique]] [[Definition:Tangent Line|tangent]] $\TT$ to $\CC$ at a [[Defi... | We have been given that there exists a [[Definition:Unique|unique]] [[Definition:Tangent Line|tangent]] $\TT$ to $\CC$ at $P$.
By definition of [[Definition:Tangent Line|tangent]], $\TT$ has a [[Definition:Slope of Straight Line|slope]] $M$ given by:
:$m = \ds \lim_{h \mathop \to 0} \frac {\map f {x_0 + h} - \map f {... | Slope of Tangent to Curve at Point equals Value of Derivative | https://proofwiki.org/wiki/Slope_of_Tangent_to_Curve_at_Point_equals_Value_of_Derivative | https://proofwiki.org/wiki/Slope_of_Tangent_to_Curve_at_Point_equals_Value_of_Derivative | [
"Slope of Tangent to Curve at Point equals Value of Derivative",
"Tangents",
"Differential Calculus",
"Analytic Geometry"
] | [
"Definition:Line/Curve",
"Definition:Cartesian Plane",
"Definition:Real Function",
"Definition:Unique",
"Definition:Tangent Line",
"Definition:Point",
"Definition:Slope/Curve",
"Definition:Derivative/Real Function/Derivative at Point"
] | [
"Definition:Unique",
"Definition:Tangent Line",
"Definition:Tangent Line",
"Definition:Slope/Straight Line",
"Definition:Derivative/Real Function/Derivative at Point"
] |
proofwiki-18296 | Conditions for Preservation of Covergence in Test Function Space under Differentiation | For all $n \in \N$ let $\Phi_n, \phi_n \in \map \DD \R$ be test functions.
Let $\mathbf 0 : \R \to 0$ be the zero mapping.
Let $\phi_n$ be such that:
:$\ds \int_{-\infty}^\infty \map {\phi_n} x \rd x = 0$
Let $\Phi_n$ be such that $\Phi_n' = \phi_n$.
Let $\sequence {\Phi_n}_{n \mathop \in \N}$ and $\sequence {\phi_n}_{... | By Characterization of Derivative of Test Function we have that for every $\phi_n$ there is a unique $\Phi_n$ such that:
:$\ds \map {\Phi_n} x = \int_{-\infty}^x \map {\phi_n} x \rd x$
Let $K = \closedint {-a} a$ be a closed real interval.
Suppose that $\sequence {\phi_n}_{n \mathop \in \N}$ is supported on $K$.
Hence:... | For all $n \in \N$ let $\Phi_n, \phi_n \in \map \DD \R$ be [[Definition:Test Function|test functions]].
Let $\mathbf 0 : \R \to 0$ be the [[Definition:Zero Mapping|zero mapping]].
Let $\phi_n$ be such that:
:$\ds \int_{-\infty}^\infty \map {\phi_n} x \rd x = 0$
Let $\Phi_n$ be such that $\Phi_n' = \phi_n$.
Let $\s... | By [[Characterization of Derivative of Test Function]] we have that for every $\phi_n$ there is a [[Definition:Unique|unique]] $\Phi_n$ such that:
:$\ds \map {\Phi_n} x = \int_{-\infty}^x \map {\phi_n} x \rd x$
Let $K = \closedint {-a} a$ be a [[Definition:Closed Real Interval|closed real interval]].
Suppose that $\... | Conditions for Preservation of Covergence in Test Function Space under Differentiation | https://proofwiki.org/wiki/Conditions_for_Preservation_of_Covergence_in_Test_Function_Space_under_Differentiation | https://proofwiki.org/wiki/Conditions_for_Preservation_of_Covergence_in_Test_Function_Space_under_Differentiation | [
"Test Functions",
"Convergence"
] | [
"Definition:Test Function",
"Definition:Zero Mapping",
"Definition:Sequence",
"Definition:Convergent Sequence/Test Function Space",
"Definition:Convergent Sequence/Test Function Space"
] | [
"Characterization of Derivative of Test Function",
"Definition:Unique",
"Definition:Real Interval/Closed",
"Definition:Support of Continuous Mapping/Real-Valued",
"Definition:Convergent Sequence/Test Function Space",
"Definition:Uniform Convergence/Real Sequence",
"Definition:Uniform Convergence/Real Se... |
proofwiki-18297 | Derivatives of Moment Generating Function of Gamma Distribution | The $n$th derivative of $M_X$ is given by:
:${M_X}^{\paren n} = \dfrac {\alpha^{\overline n} \beta^\alpha} {\paren {\beta - t}^{\alpha + n} }$
where $\alpha^{\overline n}$ denotes the $n$th rising factorial of $\alpha$. | The proof proceeds by induction on $n$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:${M_X}^{\paren n} = \dfrac {\alpha^{\overline n} \beta^\alpha} {\paren {\beta - t}^{\alpha + n} }$ | The [[Definition:Higher Derivative|$n$th derivative]] of $M_X$ is given by:
:${M_X}^{\paren n} = \dfrac {\alpha^{\overline n} \beta^\alpha} {\paren {\beta - t}^{\alpha + n} }$
where $\alpha^{\overline n}$ denotes the [[Definition:Rising Factorial|$n$th rising factorial]] of $\alpha$. | The proof proceeds by [[Principle of Mathematical Induction|induction]] on $n$.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:${M_X}^{\paren n} = \dfrac {\alpha^{\overline n} \beta^\alpha} {\paren {\beta - t}^{\alpha + n} }$ | Derivatives of Moment Generating Function of Gamma Distribution | https://proofwiki.org/wiki/Derivatives_of_Moment_Generating_Function_of_Gamma_Distribution | https://proofwiki.org/wiki/Derivatives_of_Moment_Generating_Function_of_Gamma_Distribution | [
"Moment Generating Function of Gamma Distribution"
] | [
"Definition:Derivative/Higher Derivatives/Higher Order",
"Definition:Rising Factorial"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-18298 | Expectation of Power of Gamma Distribution | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
Then:
:$\expect {X^n} = \dfrac {\alpha^{\overline n} } {\beta^n}$
where:
:$\expect {X^n}$ denotes the expectation of $X^n$
:$\alpha^{\overline n}$ denotes the $n$th rising factorial of $\alpha$. | From Moment in terms of Moment Generating Function:
:$\expect {X^n} = \map { {M_X}^{\paren n} } 0$
where ${M_X}^{\paren n}$ denotes the $n$th derivative of $M_X$.
Then:
{{begin-eqn}}
{{eqn | l = \expect {X^n}
| r = \map { {M_X}^{\paren n} } 0
| c =
}}
{{eqn | r = \valueat {\dfrac {\alpha^{\overline n} \bet... | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the [[Definition:Gamma Distribution|Gamma distribution]].
Then:
:$\expect {X^n} = \dfrac {\alpha^{\overline n} } {\beta^n}$
where:
:$\expect {X^n}$ denotes the [[Definition:Expectation|expectation]] of $X^n$
:$\alpha^{\overline ... | From [[Moment in terms of Moment Generating Function]]:
:$\expect {X^n} = \map { {M_X}^{\paren n} } 0$
where ${M_X}^{\paren n}$ denotes the [[Definition:Higher Derivative|$n$th derivative]] of $M_X$.
Then:
{{begin-eqn}}
{{eqn | l = \expect {X^n}
| r = \map { {M_X}^{\paren n} } 0
| c =
}}
{{eqn | r = \v... | Expectation of Power of Gamma Distribution | https://proofwiki.org/wiki/Expectation_of_Power_of_Gamma_Distribution | https://proofwiki.org/wiki/Expectation_of_Power_of_Gamma_Distribution | [
"Gamma Distribution"
] | [
"Definition:Gamma Distribution",
"Definition:Expectation",
"Definition:Rising Factorial"
] | [
"Moment in terms of Moment Generating Function",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Derivatives of Moment Generating Function of Gamma Distribution",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Category:Gamma Distribution"
] |
proofwiki-18299 | Dirac Comb is Distribution | Let $\phi \in \map \DD \R$ be a test function.
Suppose $\map {\operatorname {III} } 0$ is a Dirac comb such that:
:$\ds \map {\map {\operatorname {III} } 0} \phi := \sum_{n \mathop \in \Z} \map \phi n$
Then $\map {\operatorname {III} } 0$ is a Schwartz distribution. | By definition of test function, $\phi$ is supported on a compact subset of $\R$.
Hence:
:$\exists N \in \N : \forall x \in \R \setminus \closedint {-N} N : \map \phi x = 0$
Therefore:
:$\ds \sum_{n \mathop \in \Z} \map \phi n = \sum_{n \mathop = - N}^N \map \phi n$
This is a finite sequence of real numbers.
Thus, the s... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
Suppose $\map {\operatorname {III} } 0$ is a [[Definition:Dirac Comb|Dirac comb]] such that:
:$\ds \map {\map {\operatorname {III} } 0} \phi := \sum_{n \mathop \in \Z} \map \phi n$
Then $\map {\operatorname {III} } 0$ is a [[Definition:Schw... | By definition of [[Definition:Test Function|test function]], $\phi$ is [[Definition:Support of Schwartz Distribution|supported]] on a [[Definition:Compact Set of Reals|compact subset of $\R$]].
Hence:
:$\exists N \in \N : \forall x \in \R \setminus \closedint {-N} N : \map \phi x = 0$
Therefore:
:$\ds \sum_{n \math... | Dirac Comb is Distribution | https://proofwiki.org/wiki/Dirac_Comb_is_Distribution | https://proofwiki.org/wiki/Dirac_Comb_is_Distribution | [
"Examples of Schwartz Distributions"
] | [
"Definition:Test Function",
"Definition:Sampling Function",
"Definition:Schwartz Distribution"
] | [
"Definition:Test Function",
"Definition:Support of Schwartz Distribution",
"Definition:Compact Space/Real Analysis",
"Definition:Finite Sequence",
"Definition:Real Number",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Mapping",
"Definition:Support of Schwartz Distribution",
"Definition... |
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