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-18100 | Cosine of Integer Multiple of Argument/Formulation 6 | :$\map \cos {n \theta} = \paren {-2 \sin \theta} \map \sin {\paren {n - 1} \theta} + \map \cos {\paren {n - 2} \theta}$ | {{begin-eqn}}
{{eqn | l = \map \cos {n \theta}
| r = \map \cos {\paren {n - 1} \theta + \theta}
| c =
}}
{{eqn | r = -\sin \theta \map \sin {\paren {n - 1} \theta} + \cos \theta \map \cos {\paren {n - 1} \theta}
| c = Cosine of Sum
}}
{{eqn | r = - \sin \theta \map \sin {\paren {n - 1} \theta} + \co... | :$\map \cos {n \theta} = \paren {-2 \sin \theta} \map \sin {\paren {n - 1} \theta} + \map \cos {\paren {n - 2} \theta}$ | {{begin-eqn}}
{{eqn | l = \map \cos {n \theta}
| r = \map \cos {\paren {n - 1} \theta + \theta}
| c =
}}
{{eqn | r = -\sin \theta \map \sin {\paren {n - 1} \theta} + \cos \theta \map \cos {\paren {n - 1} \theta}
| c = [[Cosine of Sum]]
}}
{{eqn | r = - \sin \theta \map \sin {\paren {n - 1} \theta} +... | Cosine of Integer Multiple of Argument/Formulation 6 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_6 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_6 | [
"Cosine of Integer Multiple of Argument"
] | [] | [
"Cosine of Sum",
"Cosine of Sum",
"Sum of Squares of Sine and Cosine",
"Sine of Sum",
"Category:Cosine of Integer Multiple of Argument"
] |
proofwiki-18101 | Cross Product of Perpendicular Vectors | Let $\mathbf a$ and $\mathbf b$ be vector quantities which are perpendicular.
Let $\mathbf a \times \mathbf b$ denote the cross product of $\mathbf a$ with $\mathbf b$.
Then:
:$\mathbf a \times \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \mathbf {\hat n}$
where:
:$\norm {\mathbf a}$ denotes the length of $\mathbf a... | By definition of cross product:
:$\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$
where:
:$\norm {\mathbf a}$ denotes the length of $\mathbf a$
:$\theta$ denotes the angle from $\mathbf a$ to $\mathbf b$, measured in the positive direction
:$\hat {\mathbf n}$ is the unit vector perpendicular to bot... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector Quantity|vector quantities]] which are [[Definition:Perpendicular Lines|perpendicular]].
Let $\mathbf a \times \mathbf b$ denote the [[Definition:Vector Cross Product|cross product]] of $\mathbf a$ with $\mathbf b$.
Then:
:$\mathbf a \times \mathbf b = \norm {\m... | By definition of [[Definition:Vector Cross Product/Definition 2|cross product]]:
:$\norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$
where:
:$\norm {\mathbf a}$ denotes the [[Definition:Vector Length|length]] of $\mathbf a$
:$\theta$ denotes the [[Definition:Angle Between Vectors|angle]] from $\mat... | Cross Product of Perpendicular Vectors | https://proofwiki.org/wiki/Cross_Product_of_Perpendicular_Vectors | https://proofwiki.org/wiki/Cross_Product_of_Perpendicular_Vectors | [
"Vector Cross Product"
] | [
"Definition:Vector Quantity",
"Definition:Right Angle/Perpendicular",
"Definition:Vector Cross Product",
"Definition:Vector Length",
"Definition:Unit Vector",
"Definition:Right Angle/Perpendicular",
"Definition:Right-Hand Rule/Cross Product"
] | [
"Definition:Vector Cross Product/Definition 2",
"Definition:Vector Length",
"Definition:Angle between Vectors",
"Definition:Axis/Positive Direction",
"Definition:Unit Vector",
"Definition:Right Angle/Perpendicular",
"Definition:Right-Hand Rule/Cross Product",
"Definition:Right Angle/Perpendicular",
... |
proofwiki-18102 | Cross Product of Elements of Standard Ordered Basis | Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis of Cartesian $3$-space $S$.
Then:
:$\mathbf i \times \mathbf i = \mathbf j \times \mathbf j = \mathbf k \times \mathbf k = 0$
and:
{{begin-eqn}}
{{eqn | l = \mathbf i \times \mathbf j
| m = \mathbf k
| mo= =
| r = -\mathbf j ... | From Cross Product of Vector with Itself is Zero:
:$\mathbf i \times \mathbf i = \mathbf j \times \mathbf j = \mathbf k \times \mathbf k = 0$
Then we can take the definition of cross product:
:$\mathbf a \times \mathbf b = \begin {vmatrix}
\mathbf i & \mathbf j & \mathbf k \\
a_i & a_j & a_k \\
b_i & b_j & b_k \\
\end ... | Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the [[Definition:Standard Ordered Basis|standard ordered basis]] of [[Definition:Cartesian 3-Space|Cartesian $3$-space]] $S$.
Then:
:$\mathbf i \times \mathbf i = \mathbf j \times \mathbf j = \mathbf k \times \mathbf k = 0$
and:
{{begin-eqn}}
{{eqn | l = \mathbf i \t... | From [[Cross Product of Vector with Itself is Zero]]:
:$\mathbf i \times \mathbf i = \mathbf j \times \mathbf j = \mathbf k \times \mathbf k = 0$
Then we can take the definition of [[Definition:Vector Cross Product/Definition 1|cross product]]:
:$\mathbf a \times \mathbf b = \begin {vmatrix}
\mathbf i & \mathbf j & ... | Cross Product of Elements of Standard Ordered Basis | https://proofwiki.org/wiki/Cross_Product_of_Elements_of_Standard_Ordered_Basis | https://proofwiki.org/wiki/Cross_Product_of_Elements_of_Standard_Ordered_Basis | [
"Vector Cross Product",
"Standard Ordered Bases"
] | [
"Definition:Standard Ordered Basis",
"Definition:Cartesian 3-Space",
"Definition:Dot Product"
] | [
"Cross Product of Vector with Itself is Zero",
"Definition:Vector Cross Product/Definition 1",
"Vector Cross Product is Anticommutative"
] |
proofwiki-18103 | Total Vector Area of Tetrahedron is Zero | Let $T$ be a tetrahedron whose faces have vector area $\mathbf S_1$, $\mathbf S_2$, $\mathbf S_3$ and $\mathbf S_4$.
Let the positive direction be defined as outward.
:300px
Then the total vector area is zero:
:$\mathbf S_1 + \mathbf S_2 + \mathbf S_3 + \mathbf S_4 = \mathbf 0$ | Let the vector areas be resolved upon the faces of $T$.
Some of the projections will be positive and some negative.
Let $T$ be imagined in a fluid which is in equilibrium under pressure.
Each face experiences a force which is normal to its plane and proportional to its area.
Because the fluid inside is in equilibrium w... | Let $T$ be a [[Definition:Tetrahedron|tetrahedron]] whose [[Definition:Face of Polyhedron|faces]] have [[Definition:Vector Area|vector area]] $\mathbf S_1$, $\mathbf S_2$, $\mathbf S_3$ and $\mathbf S_4$.
Let the [[Definition:Positive Real Number|positive]] direction be defined as outward.
:[[File:Tetrahedron-vector-... | Let the [[Definition:Vector Area|vector areas]] be resolved upon the [[Definition:Face of Polyhedron|faces]] of $T$.
Some of the projections will be [[Definition:Positive Real Number|positive]] and some [[Definition:Negative Real Number|negative]].
Let $T$ be imagined in a [[Definition:Fluid|fluid]] which is in [[Def... | Total Vector Area of Tetrahedron is Zero | https://proofwiki.org/wiki/Total_Vector_Area_of_Tetrahedron_is_Zero | https://proofwiki.org/wiki/Total_Vector_Area_of_Tetrahedron_is_Zero | [
"Vector Area",
"Tetrahedra"
] | [
"Definition:Tetrahedron",
"Definition:Polyhedron/Face",
"Definition:Vector Area",
"Definition:Positive/Real Number",
"File:Tetrahedron-vector-area.png",
"Definition:Vector Area",
"Definition:Zero Vector"
] | [
"Definition:Vector Area",
"Definition:Polyhedron/Face",
"Definition:Positive/Real Number",
"Definition:Negative/Real Number",
"Definition:Fluid",
"Definition:Equilibrium (Mechanics)",
"Definition:Pressure",
"Definition:Polyhedron/Face",
"Definition:Force",
"Definition:Normal",
"Definition:Plane ... |
proofwiki-18104 | Total Vector Area of Polyhedron is Zero | Let $P$ be a polyhedron.
Let the positive direction be defined as outward.
Let $\mathbf T$ be the total vector area of all the faces of $P$.
Then:
:$\mathbf T = \mathbf 0$ | $P$ can be geometrically divided into a finite number of tetrahedra.
Every face of these tetrahedra which are internal to $P$ appears twice: once with a positive vector area, and once with a negative normal.
Hence for any polyhedron, the total vector area is the sum of the vector areas of all the tetrahedra.
The result... | Let $P$ be a [[Definition:Polyhedron|polyhedron]].
Let the [[Definition:Positive Real Number|positive]] direction be defined as outward.
Let $\mathbf T$ be the total [[Definition:Vector Area|vector area]] of all the [[Definition:Face of Polyhedron|faces]] of $P$.
Then:
:$\mathbf T = \mathbf 0$ | $P$ can be geometrically divided into a [[Definition:Finite Set|finite]] number of [[Definition:Tetrahedron|tetrahedra]].
Every [[Definition:Face of Polyhedron|face]] of these [[Definition:Tetrahedron|tetrahedra]] which are internal to $P$ appears twice: once with a [[Definition:Positive Real Number|positive]] [[Defin... | Total Vector Area of Polyhedron is Zero | https://proofwiki.org/wiki/Total_Vector_Area_of_Polyhedron_is_Zero | https://proofwiki.org/wiki/Total_Vector_Area_of_Polyhedron_is_Zero | [
"Vector Area",
"Polyhedra"
] | [
"Definition:Polyhedron",
"Definition:Positive/Real Number",
"Definition:Vector Area",
"Definition:Polyhedron/Face"
] | [
"Definition:Finite Set",
"Definition:Tetrahedron",
"Definition:Polyhedron/Face",
"Definition:Tetrahedron",
"Definition:Positive/Real Number",
"Definition:Vector Area",
"Definition:Negative/Real Number",
"Definition:Normal Vector",
"Definition:Polyhedron",
"Definition:Vector Area",
"Definition:Ve... |
proofwiki-18105 | Total Vector Area of Closed Surface is Zero | Let $S$ be a closed surface.
Let $\d \mathbf S$ be an infinitesimal vector area around some point $P$ of $S$.
Then the total surface area of $S$ is given by:
:$\ds \iint_S \d \mathbf S = \mathbf 0$ | {{ProofWanted|The book waffles on about slicing $S$ into vanishingly small tetrahedra, but the argument is flimsy.}} | Let $S$ be a [[Definition:Closed Surface|closed surface]].
Let $\d \mathbf S$ be an [[Definition:Infinitesimal|infinitesimal]] [[Definition:Vector Area|vector area]] around some [[Definition:Point|point]] $P$ of $S$.
Then the total [[Definition:Surface Area|surface area]] of $S$ is given by:
:$\ds \iint_S \d \mathbf... | {{ProofWanted|The book waffles on about slicing $S$ into vanishingly small tetrahedra, but the argument is flimsy.}} | Total Vector Area of Closed Surface is Zero | https://proofwiki.org/wiki/Total_Vector_Area_of_Closed_Surface_is_Zero | https://proofwiki.org/wiki/Total_Vector_Area_of_Closed_Surface_is_Zero | [
"Vector Area",
"Polyhedra"
] | [
"Definition:Closed Surface",
"Definition:Infinitesimal",
"Definition:Vector Area",
"Definition:Point",
"Definition:Surface Area"
] | [] |
proofwiki-18106 | Vectors are Coplanar iff Scalar Triple Product equals Zero | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a Cartesian $3$-space:
Let $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$ denote the scalar triple product of $\mathbf a$, $\mathbf b$ and $\mathbf c$.
Then:
:$\mathbf a \cdot \paren {\mathbf b \times \mathbf c} = 0$
{{iff}} $\mathbf a$, $\mathbf b$ and ... | From Magnitude of Scalar Triple Product equals Volume of Parallelepiped Contained by Vectors:
:$\size {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} }$ equals the volume of the parallelepiped contained by $\mathbf a, \mathbf b, \mathbf c$.
The result follows.
{{qed}} | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be [[Definition:Vector Quantity|vectors]] in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]]:
Let $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$ denote the [[Definition:Scalar Triple Product|scalar triple product]] of $\mathbf a$, $\mathbf b$ and $\mathbf c$.
... | From [[Magnitude of Scalar Triple Product equals Volume of Parallelepiped Contained by Vectors]]:
:$\size {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} }$ equals the [[Definition:Volume|volume]] of the [[Definition:Parallelepiped|parallelepiped]] contained by $\mathbf a, \mathbf b, \mathbf c$.
The result follow... | Vectors are Coplanar iff Scalar Triple Product equals Zero/Proof 1 | https://proofwiki.org/wiki/Vectors_are_Coplanar_iff_Scalar_Triple_Product_equals_Zero | https://proofwiki.org/wiki/Vectors_are_Coplanar_iff_Scalar_Triple_Product_equals_Zero/Proof_1 | [
"Vectors are Coplanar iff Scalar Triple Product equals Zero",
"Scalar Triple Product"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Scalar Triple Product",
"Definition:Coplanar Vectors"
] | [
"Magnitude of Scalar Triple Product equals Volume of Parallelepiped Contained by Vectors",
"Definition:Volume",
"Definition:Parallelepiped"
] |
proofwiki-18107 | Vectors are Coplanar iff Scalar Triple Product equals Zero | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a Cartesian $3$-space:
Let $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$ denote the scalar triple product of $\mathbf a$, $\mathbf b$ and $\mathbf c$.
Then:
:$\mathbf a \cdot \paren {\mathbf b \times \mathbf c} = 0$
{{iff}} $\mathbf a$, $\mathbf b$ and ... | If any of $\mathbf a$, $\mathbf b$, or $\mathbf c$ is $\mathbf 0$, then the vectors are trivially coplanar.
Also, the scalar triple product is zero from Scalar Triple Product as Product of Magnitudes.
So, let $\mathbf a$, $\mathbf b$, and $\mathbf c$ all be non-zero.
Suppose $\mathbf b$ and $\mathbf c$ are parallel.
Th... | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be [[Definition:Vector Quantity|vectors]] in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]]:
Let $\mathbf a \cdot \paren {\mathbf b \times \mathbf c}$ denote the [[Definition:Scalar Triple Product|scalar triple product]] of $\mathbf a$, $\mathbf b$ and $\mathbf c$.
... | If any of $\mathbf a$, $\mathbf b$, or $\mathbf c$ is $\mathbf 0$, then the [[Definition:Vector|vectors]] are trivially [[Definition:Coplanar Vectors|coplanar]].
Also, the [[Definition:Scalar Triple Product|scalar triple product]] is [[Definition:Zero (Number)|zero]] from [[Scalar Triple Product as Product of Magnitud... | Vectors are Coplanar iff Scalar Triple Product equals Zero/Proof 2 | https://proofwiki.org/wiki/Vectors_are_Coplanar_iff_Scalar_Triple_Product_equals_Zero | https://proofwiki.org/wiki/Vectors_are_Coplanar_iff_Scalar_Triple_Product_equals_Zero/Proof_2 | [
"Vectors are Coplanar iff Scalar Triple Product equals Zero",
"Scalar Triple Product"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Scalar Triple Product",
"Definition:Coplanar Vectors"
] | [
"Definition:Vector",
"Definition:Coplanar Vectors",
"Definition:Scalar Triple Product",
"Definition:Zero (Number)",
"Scalar Triple Product as Product of Magnitudes",
"Definition:Zero Vector",
"Definition:Parallel (Geometry)/Lines",
"Definition:Coplanar Vectors",
"Definition:Generator of Vector Space... |
proofwiki-18108 | Equivalence of Definitions of Scalar Triple Product | {{TFAE|def = Scalar Triple Product}}
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be vectors in a Cartesian $3$-space:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k
}}
{{eqn | l = \mathbf b
| r = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k
}}
{{eqn | l = \mathbf c... | {{begin-eqn}}
{{eqn | l = \mathbf a \cdot \paren {\mathbf b \times \mathbf c}
| r = \mathbf a \cdot \paren {\paren {b_j c_k - c_j b_k} \mathbf i + \paren {b_k c_i - c_k b_i} \mathbf j + \paren {b_i c_j - c_i b_j} \mathbf k}
| c = {{Defof|Vector Cross Product}}
}}
{{eqn | r = a_i \paren {b_j c_k - c_j b_k} +... | {{TFAE|def = Scalar Triple Product}}
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be [[Definition:Vector Quantity|vectors]] in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]]:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k
}}
{{eqn | l = \mathbf b
| r = b_i... | {{begin-eqn}}
{{eqn | l = \mathbf a \cdot \paren {\mathbf b \times \mathbf c}
| r = \mathbf a \cdot \paren {\paren {b_j c_k - c_j b_k} \mathbf i + \paren {b_k c_i - c_k b_i} \mathbf j + \paren {b_i c_j - c_i b_j} \mathbf k}
| c = {{Defof|Vector Cross Product}}
}}
{{eqn | r = a_i \paren {b_j c_k - c_j b_k} +... | Equivalence of Definitions of Scalar Triple Product | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Scalar_Triple_Product | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Scalar_Triple_Product | [
"Scalar Triple Product"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Standard Ordered Basis/Vector Space"
] | [
"Determinant/Examples/Order 3"
] |
proofwiki-18109 | Continuity of Linear Transformation/Normed Vector Space | Let $X, Y$ be a normed vector spaces over $\R$.
Let $T : X \to Y$ be a linear mapping.
{{TFAE}}:
:$(1): \quad T$ is continuous over $X$
:$(2): \quad T$ is continuous at $\mathbf 0$
:$(3): \quad \exists M > 0 : \forall x \in X : \norm {\map T x}_Y \le M \norm x_X$ | === $\paren 1 \implies \paren 2$ ===
Let $T$ be continuous on $X$.
$X$ is a vector space.
By definition, $\exists \mathbf 0 \in X$.
Hence, $T$ is continuous at $\mathbf 0$.
{{qed|lemma}} | Let $X, Y$ be a [[Definition:Normed Vector Space|normed vector spaces]] [[Definition:Vector Space over Division Ring|over]] $\R$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear mapping]].
{{TFAE}}:
:$(1): \quad T$ is [[Definition:Continuous Mapping (Normed Vector Space)/Space|continuous]] over $X$... | === $\paren 1 \implies \paren 2$ ===
Let $T$ be [[Definition:Continuous Mapping (Normed Vector Space)/Space|continuous]] on $X$.
$X$ is a [[Definition:Vector Space over Division Ring|vector space]].
By definition, $\exists \mathbf 0 \in X$.
Hence, $T$ is [[Definition:Continuous at Point of Normed Vector Space|conti... | Continuity of Linear Transformation/Normed Vector Space | https://proofwiki.org/wiki/Continuity_of_Linear_Transformation/Normed_Vector_Space | https://proofwiki.org/wiki/Continuity_of_Linear_Transformation/Normed_Vector_Space | [
"Linear Transformations",
"Normed Vector Spaces",
"Continuous Mappings",
"Continuous Mappings on Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Vector Space/Division Ring",
"Definition:Linear Transformation",
"Definition:Continuous Mapping (Normed Vector Space)/Space",
"Definition:Continuous Mapping (Normed Vector Space)/Point"
] | [
"Definition:Continuous Mapping (Normed Vector Space)/Space",
"Definition:Vector Space/Division Ring",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Continuous Mapping (Normed Vector Space)/Point",
"Definition:Vector Sp... |
proofwiki-18110 | Derivative of Sum of Vector-Valued Functions | Let $\mathbf a: \R \to \R^n$ and $\mathbf b: \R \to \R^n$ be differentiable vector-valued functions.
Then the derivative of $\map {\mathbf v} t = \map {\mathbf a} t + \map {\mathbf b} t$ is given by:
:$\dfrac {\d \mathbf v} {\d t} = \map {\dfrac \d {\d t} } {\mathbf a + \mathbf b} = \dfrac {\d \mathbf a} {\d t} + \dfra... | {{begin-eqn}}
{{eqn | l = \dfrac {\d \mathbf v} {\d t}
| r = \lim_{h \mathop \to 0} \dfrac {\map {\mathbf v} {t + h} - \map {\mathbf v} t} h
| c = {{Defof|Derivative of Vector-Valued Function}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \dfrac {\paren {\map {\mathbf a} {t + h} + \map {\mathbf b} {t + h} } - \par... | Let $\mathbf a: \R \to \R^n$ and $\mathbf b: \R \to \R^n$ be [[Definition:Differentiable Vector-Valued Function|differentiable]] [[Definition:Vector-Valued Function|vector-valued functions]].
Then the [[Definition:Derivative of Vector-Valued Function|derivative]] of $\map {\mathbf v} t = \map {\mathbf a} t + \map {\m... | {{begin-eqn}}
{{eqn | l = \dfrac {\d \mathbf v} {\d t}
| r = \lim_{h \mathop \to 0} \dfrac {\map {\mathbf v} {t + h} - \map {\mathbf v} t} h
| c = {{Defof|Derivative of Vector-Valued Function}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \dfrac {\paren {\map {\mathbf a} {t + h} + \map {\mathbf b} {t + h} } - \par... | Derivative of Sum of Vector-Valued Functions | https://proofwiki.org/wiki/Derivative_of_Sum_of_Vector-Valued_Functions | https://proofwiki.org/wiki/Derivative_of_Sum_of_Vector-Valued_Functions | [
"Vector Calculus"
] | [
"Definition:Differentiable Mapping/Vector-Valued Function",
"Definition:Vector-Valued Function",
"Definition:Derivative/Vector-Valued Function"
] | [] |
proofwiki-18111 | Derivative of Square of Vector-Valued Function | Let $\mathbf a: \R \to \R^n$ be a differentiable vector-valued function.
The derivative of its square is given by:
:$\map {\dfrac \d {\d x} } {\mathbf a^2} = 2 \mathbf a \cdot \dfrac {\d \mathbf a} {\d x} = 2 a \dfrac {\d a} {\d x}$
where $a = \norm {\mathbf a}$ is the magnitude of $\mathbf a$. | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\mathbf a \cdot \mathbf b}
| r = \dfrac {\d \mathbf a} {\d x} \cdot \mathbf b + \mathbf a \cdot \dfrac {\d \mathbf b} {\d x}
| c = Derivative of Dot Product of Vector-Valued Functions
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\mathbf a ... | Let $\mathbf a: \R \to \R^n$ be a [[Definition:Differentiable Vector-Valued Function|differentiable]] [[Definition:Vector-Valued Function|vector-valued function]].
The [[Definition:Derivative of Vector-Valued Function|derivative]] of its [[Definition:Square of Vector Quantity|square]] is given by:
:$\map {\dfrac \d ... | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\mathbf a \cdot \mathbf b}
| r = \dfrac {\d \mathbf a} {\d x} \cdot \mathbf b + \mathbf a \cdot \dfrac {\d \mathbf b} {\d x}
| c = [[Derivative of Dot Product of Vector-Valued Functions]]
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\mathb... | Derivative of Square of Vector-Valued Function | https://proofwiki.org/wiki/Derivative_of_Square_of_Vector-Valued_Function | https://proofwiki.org/wiki/Derivative_of_Square_of_Vector-Valued_Function | [
"Dot Product",
"Vector Calculus"
] | [
"Definition:Differentiable Mapping/Vector-Valued Function",
"Definition:Vector-Valued Function",
"Definition:Derivative/Vector-Valued Function",
"Definition:Square of Vector Quantity",
"Definition:Magnitude"
] | [
"Derivative of Dot Product of Vector-Valued Functions",
"Dot Product Operator is Commutative",
"Dot Product of Vector-Valued Function with its Derivative"
] |
proofwiki-18112 | Velocity of Point Moving on Surface of Sphere is Perpendicular to Radius | Let $P$ be a point moving on the surface of a sphere.
The velocity of $P$ is perpendicular to its radius at $P$. | Let $S$ be a sphere whose center is at $O$.
By definition of a sphere, all the points on the surface of $S$ are the same distance from its center.
Let $\map {\mathbf v} t$ denote the position vector of $P$ with respect to $O$ at time $t$.
Then the magnitude $\norm {\mathbf v}$ of $\mathbf v$ is contstant.
Hence from Do... | Let $P$ be a [[Definition:Point|point]] moving on the [[Definition:Surface|surface]] of a [[Definition:Sphere (Geometry)|sphere]].
The [[Definition:Velocity|velocity]] of $P$ is [[Definition:Perpendicular Lines|perpendicular]] to its [[Definition:Radius of Sphere|radius]] at $P$. | Let $S$ be a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is at $O$.
By definition of a [[Definition:Sphere (Geometry)|sphere]], all the [[Definition:Point|points]] on the [[Definition:Surface|surface]] of $S$ are the same [[Definition:Distance between Points|distance]] from its... | Velocity of Point Moving on Surface of Sphere is Perpendicular to Radius | https://proofwiki.org/wiki/Velocity_of_Point_Moving_on_Surface_of_Sphere_is_Perpendicular_to_Radius | https://proofwiki.org/wiki/Velocity_of_Point_Moving_on_Surface_of_Sphere_is_Perpendicular_to_Radius | [
"Mechanics"
] | [
"Definition:Point",
"Definition:Surface",
"Definition:Sphere/Geometry",
"Definition:Velocity",
"Definition:Right Angle/Perpendicular",
"Definition:Sphere/Geometry/Radius"
] | [
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Sphere/Geometry",
"Definition:Point",
"Definition:Surface",
"Definition:Distance between Points",
"Definition:Sphere/Geometry/Center",
"Definition:Position Vector",
"Definition:Time",
"Definition:Magnitude",
"Definiti... |
proofwiki-18113 | Derivative of Vector Triple Product of Vector-Valued Functions | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be differentiable vector-valued functions in Cartesian $3$-space.
The derivative of their vector triple product is given by:
:$\map {\dfrac \d {\d x} } {\mathbf a \times \paren {\mathbf b \times \mathbf c} } = \dfrac {\d \mathbf a} {\d x} \times \paren {\mathbf b \times \mat... | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\mathbf a \times \paren {\mathbf b \times \mathbf c} }
| r = \dfrac {\d \mathbf a} {\d x} \times \paren {\mathbf b \times \mathbf c} + \mathbf a \times \map {\dfrac \d {\d x} } {\mathbf b \times \mathbf c}
| c = Derivative of Vector Cross Product of Vector... | Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be [[Definition:Differentiable Vector-Valued Function|differentiable]] [[Definition:Vector-Valued Function|vector-valued functions]] in [[Definition:Cartesian 3-Space|Cartesian $3$-space]].
The [[Definition:Derivative of Vector-Valued Function|derivative]] of their [[Defin... | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\mathbf a \times \paren {\mathbf b \times \mathbf c} }
| r = \dfrac {\d \mathbf a} {\d x} \times \paren {\mathbf b \times \mathbf c} + \mathbf a \times \map {\dfrac \d {\d x} } {\mathbf b \times \mathbf c}
| c = [[Derivative of Vector Cross Product of Vect... | Derivative of Vector Triple Product of Vector-Valued Functions | https://proofwiki.org/wiki/Derivative_of_Vector_Triple_Product_of_Vector-Valued_Functions | https://proofwiki.org/wiki/Derivative_of_Vector_Triple_Product_of_Vector-Valued_Functions | [
"Differential Calculus",
"Vector Calculus",
"Vector Triple Product"
] | [
"Definition:Differentiable Mapping/Vector-Valued Function",
"Definition:Vector-Valued Function",
"Definition:Cartesian 3-Space",
"Definition:Derivative/Vector-Valued Function",
"Definition:Vector Triple Product"
] | [
"Derivative of Vector Cross Product of Vector-Valued Functions",
"Derivative of Vector Cross Product of Vector-Valued Functions",
"Vector Cross Product Distributes over Addition"
] |
proofwiki-18114 | Cosine of Integer Multiple of Argument/Formulation 7 | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - \paren {2 \sin \theta} \paren {\map \sin {\paren {n - 1} \theta} + \map \sin {\paren {n - 3} \theta} + \map \sin {\paren {n - 5} \theta} + \cdots}
| c =
}}
{{eqn | r = \sin^2 \f... | The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds \cos n \theta = \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$ | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - \paren {2 \sin \theta} \paren {\map \sin {\paren {n - 1} \theta} + \map \sin {\paren {n - 3} \theta} + \map \sin {\paren {n - 5} \theta} + \cdots}
| c =
}}
{{eqn | r = \sin^2 \f... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \cos n \theta = \sin^2 \frac {\paren {n + 1} \pi} 2 + \paren {\sin^2 \frac {n \pi} 2} \cos \theta - 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \... | Cosine of Integer Multiple of Argument/Formulation 7 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_7 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_7 | [
"Cosine of Integer Multiple of Argument"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-18115 | Derivative Operator is Linear Mapping | Let $I := \closedint a b$ be a closed real interval.
Let $C \closedint a b$ be the space of real-valued functions continuous on $I$.
Let $C^1 \closedint a b$ be the space of real-valued functions continuously differentiable on $I$.
Let $D$ be the derivative operator such that:
:$D : \map {C^1} I \to \map C I$
and $Dx :... | === Distributivity ===
{{begin-eqn}}
{{eqn | l = \map D {x + y}
| r = \paren {x + y}'
| c = Definition
}}
{{eqn | r = x' + y'
| c = Sum Rule for Derivatives
}}
{{eqn | r = Dx + Dy
| c = Definition
}}
{{end-eqn}}
{{qed|lemma}} | Let $I := \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $C \closedint a b$ be the [[Definition:Space of Real-Valued Functions Continuous on Closed Interval|space of real-valued functions continuous on $I$]].
Let $C^1 \closedint a b$ be the [[Definition:Space of Continuous Function... | === Distributivity ===
{{begin-eqn}}
{{eqn | l = \map D {x + y}
| r = \paren {x + y}'
| c = Definition
}}
{{eqn | r = x' + y'
| c = [[Sum Rule for Derivatives]]
}}
{{eqn | r = Dx + Dy
| c = Definition
}}
{{end-eqn}}
{{qed|lemma}} | Derivative Operator is Linear Mapping | https://proofwiki.org/wiki/Derivative_Operator_is_Linear_Mapping | https://proofwiki.org/wiki/Derivative_Operator_is_Linear_Mapping | [
"Linear Transformations",
"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:Derivative Operator",
"Definition:Linear Transformation"
] | [
"Sum Rule for Derivatives"
] |
proofwiki-18116 | Gradient Operator is Invariant under Coordinate Transformation | Let $R$ be a region of space in which there exists an scalar field $F$.
Let $\mathbf V$ denote the gradient of $F$.
Then $\mathbf V$ is invariant under a change of coordinate system on $R$. | The result follows directly from the geometrical representation of the gradient operator.
{{qed}} | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]] in which there exists an [[Definition:Scalar Field|scalar field]] $F$.
Let $\mathbf V$ denote the [[Definition:Gradient Operator|gradient]] of $F$.
Then $\mathbf V$ is invariant under a change of [[Definition:Coordinate System|coordinat... | The result follows directly from the [[Definition:Geometrical Representation of Gradient Operator|geometrical representation of the gradient operator]].
{{qed}} | Gradient Operator is Invariant under Coordinate Transformation | https://proofwiki.org/wiki/Gradient_Operator_is_Invariant_under_Coordinate_Transformation | https://proofwiki.org/wiki/Gradient_Operator_is_Invariant_under_Coordinate_Transformation | [
"Gradient Operator"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Scalar Field",
"Definition:Gradient Operator",
"Definition:Coordinate System"
] | [
"Definition:Gradient Operator/Geometrical Representation"
] |
proofwiki-18117 | Electric Force is Gradient of Electric Potential Field | Let $R$ be a region of space in which there exists an electric potential field $F$.
The electrostatic force experienced within $R$ is the negative of the gradient of $F$:
:$\mathbf V = -\grad F$ | The electrostatic force at a point of $R$ is in the direction of the greatest rate of decrease of electric potential.
That is, it is normal to the equipotential surfaces.
It also has a magnitude equal to the rate of decrease.
Hence the result.
{{qed}} | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]] in which there exists an [[Definition:Electric Potential|electric potential field]] $F$.
The [[Definition:Electrostatic Force|electrostatic force]] experienced within $R$ is the [[Definition:Negative|negative]] of the [[Definition:Gradie... | The [[Definition:Electrostatic Force|electrostatic force]] at a [[Definition:Point|point]] of $R$ is in the [[Definition:Direction|direction]] of the greatest [[Definition:Rate of Decrease|rate of decrease]] of [[Definition:Electric Potential|electric potential]].
That is, it is [[Definition:Normal Vector|normal]] to ... | Electric Force is Gradient of Electric Potential Field | https://proofwiki.org/wiki/Electric_Force_is_Gradient_of_Electric_Potential_Field | https://proofwiki.org/wiki/Electric_Force_is_Gradient_of_Electric_Potential_Field | [
"Gradient Operator",
"Electric Potential",
"Electric Fields"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Potential/Electric",
"Definition:Electrostatic Force",
"Definition:Negative",
"Definition:Gradient Operator"
] | [
"Definition:Electrostatic Force",
"Definition:Point",
"Definition:Direction",
"Definition:Rate of Change/Decrease",
"Definition:Potential/Electric",
"Definition:Normal Vector",
"Definition:Equipotential Surface",
"Definition:Magnitude"
] |
proofwiki-18118 | Cartesian Definition of Gradient defines Gradient Operator | Let $R$ be a region of Cartesian $3$-space $\R^3$.
Let $\map F {x, y, z}$ be a scalar field acting over $R$.
Let $\tuple {i, j, k}$ be the standard ordered basis on $\R^3$.
Let $\grad F$ be defined according to the Cartesian $3$-space definition of the gradient of $F$:
{{begin-eqn}}
{{eqn | l = \grad F
| o = :=
... | The vector rates of increase of $F$ in the directions of the $3$ axes are:
:$\dfrac {\partial F} {\partial x} \mathbf i$, $\dfrac {\partial F} {\partial y} \mathbf j$, $\dfrac {\partial F} {\partial z} \mathbf k$
Their sum will be a vector with the magnitude and direction of the most rapid rate of increase of $F$.
It r... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Cartesian 3-Space|Cartesian $3$-space]] $\R^3$.
Let $\map F {x, y, z}$ be a [[Definition:Scalar Field (Physics)|scalar field]] acting over $R$.
Let $\tuple {i, j, k}$ be the [[Definition:Standard Ordered Basis on Vector Space|standard ordered basis]] on $\R^3$... | The [[Definition:Vector Quantity|vector]] [[Definition:Rate of Increase|rates of increase]] of $F$ in the [[Definition:Direction|directions]] of the $3$ [[Definition:Coordinate Axis|axes]] are:
:$\dfrac {\partial F} {\partial x} \mathbf i$, $\dfrac {\partial F} {\partial y} \mathbf j$, $\dfrac {\partial F} {\partial z}... | Cartesian Definition of Gradient defines Gradient Operator | https://proofwiki.org/wiki/Cartesian_Definition_of_Gradient_defines_Gradient_Operator | https://proofwiki.org/wiki/Cartesian_Definition_of_Gradient_defines_Gradient_Operator | [
"Gradient Operator"
] | [
"Definition:Region",
"Definition:Cartesian 3-Space",
"Definition:Scalar Field (Physics)",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Gradient Operator/Cartesian 3-Space",
"Definition:Gradient Operator",
"Definition:Gradient Operator/Geometrical Representation",
"Definition:Unit Norm... | [
"Definition:Vector Quantity",
"Definition:Rate of Change/Increase",
"Definition:Direction",
"Definition:Axis/Coordinate Axes",
"Definition:Vector Sum",
"Definition:Vector Quantity",
"Definition:Magnitude",
"Definition:Direction",
"Definition:Rate of Change/Increase",
"Definition:Dot Product",
"D... |
proofwiki-18119 | Vector Field is Expressible as Gradient of Scalar Field iff Conservative | Let $R$ be a region of space.
Let $\mathbf V$ be a vector field acting over $R$.
Then $\mathbf V$ can be expressed as the gradient of some scalar field $F$ {{iff}} $\mathbf V$ is a conservative vector field. | Let $\mathbf V_F$ be a vector field which is the gradient of some scalar field $F$:
:$\mathbf V_F = \grad F = \nabla F$
:360px
Let $A$ and $B$ be two points in $R$.
Let $\text {Path $1$}$ be an arbitrary path from $A$ to $B$ lying entirely in $R$.
At the point $P$, let $\d \mathbf l$ be a small element of $\text {Path ... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]].
Let $\mathbf V$ be a [[Definition:Vector Field|vector field]] acting over $R$.
Then $\mathbf V$ can be expressed as the [[Definition:Gradient Operator|gradient]] of some [[Definition:Scalar Field|scalar field]] $F$ {{iff}} $\mathbf V$ ... | Let $\mathbf V_F$ be a [[Definition:Vector Field|vector field]] which is the [[Definition:Gradient Operator|gradient]] of some [[Definition:Scalar Field|scalar field]] $F$:
:$\mathbf V_F = \grad F = \nabla F$
:[[File:Line-Integrals-in-Lamellar-Field.png|360px]]
Let $A$ and $B$ be two [[Definition:Point|points]] in ... | Vector Field is Expressible as Gradient of Scalar Field iff Conservative | https://proofwiki.org/wiki/Vector_Field_is_Expressible_as_Gradient_of_Scalar_Field_iff_Conservative | https://proofwiki.org/wiki/Vector_Field_is_Expressible_as_Gradient_of_Scalar_Field_iff_Conservative | [
"Gradient Operator",
"Conservative Vector Fields"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Vector Field",
"Definition:Gradient Operator",
"Definition:Scalar Field",
"Definition:Conservative Vector Field"
] | [
"Definition:Vector Field",
"Definition:Gradient Operator",
"Definition:Scalar Field",
"File:Line-Integrals-in-Lamellar-Field.png",
"Definition:Point",
"Definition:Path (Topology)",
"Definition:Point",
"Definition:Magnitude",
"Definition:Position Vector",
"Definition:Point",
"Definition:Contour I... |
proofwiki-18120 | Divergence Operator on Vector Space is Dot Product of Del Operator | Let $R$ be a region of Cartesian $3$-space $\R^3$.
Let $\map {\mathbf V} {x, y, z}$ be a vector field acting over $R$.
Let $\tuple {i, j, k}$ be the standard ordered basis on $\R^3$.
Then
:$\operatorname {div} \mathbf V = \nabla \cdot \mathbf V$
where:
:$\operatorname {div} \mathbf V $ denotes the divergence of $\mathb... | We have by definition of divergence of $\mathbf V$:
:$\operatorname {div} \mathbf V = \dfrac {\partial V_x} {\partial x} + \dfrac {\partial V_y} {\partial y} + \dfrac {\partial V_z} {\partial z}$
Now:
{{begin-eqn}}
{{eqn | l = \nabla \cdot \mathbf V
| r = \paren {\mathbf i \dfrac \partial {\partial x} + \mathbf j... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Cartesian 3-Space|Cartesian $3$-space]] $\R^3$.
Let $\map {\mathbf V} {x, y, z}$ be a [[Definition:Vector Field|vector field]] acting over $R$.
Let $\tuple {i, j, k}$ be the [[Definition:Standard Ordered Basis on Vector Space|standard ordered basis]] on $\R^3$... | We have by definition of [[Definition:Divergence Operator on Cartesian 3-Space|divergence]] of $\mathbf V$:
:$\operatorname {div} \mathbf V = \dfrac {\partial V_x} {\partial x} + \dfrac {\partial V_y} {\partial y} + \dfrac {\partial V_z} {\partial z}$
Now:
{{begin-eqn}}
{{eqn | l = \nabla \cdot \mathbf V
| r =... | Divergence Operator on Vector Space is Dot Product of Del Operator | https://proofwiki.org/wiki/Divergence_Operator_on_Vector_Space_is_Dot_Product_of_Del_Operator | https://proofwiki.org/wiki/Divergence_Operator_on_Vector_Space_is_Dot_Product_of_Del_Operator | [
"Divergence Operator"
] | [
"Definition:Region",
"Definition:Cartesian 3-Space",
"Definition:Vector Field",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Divergence Operator/Cartesian 3-Space",
"Definition:Del Operator"
] | [
"Definition:Divergence Operator/Cartesian 3-Space"
] |
proofwiki-18121 | Divergence Operator is Invariant under Coordinate Transformation | Let $R$ be a region of space in which there exists an vector field $\mathbf V$.
Let $\nabla \cdot \mathbf V$ denote the divergence of $\mathbf V$.
Then $\nabla \cdot \mathbf V$ is invariant under a change of coordinate system on $R$. | The result follows directly from the physical interpretation of the divergence operator.
{{qed}} | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]] in which there exists an [[Definition:Vector Field|vector field]] $\mathbf V$.
Let $\nabla \cdot \mathbf V$ denote the [[Definition:Divergence Operator|divergence]] of $\mathbf V$.
Then $\nabla \cdot \mathbf V$ is invariant under a chan... | The result follows directly from the [[Definition:Physical Interpretation of Divergence|physical interpretation of the divergence operator]].
{{qed}} | Divergence Operator is Invariant under Coordinate Transformation | https://proofwiki.org/wiki/Divergence_Operator_is_Invariant_under_Coordinate_Transformation | https://proofwiki.org/wiki/Divergence_Operator_is_Invariant_under_Coordinate_Transformation | [
"Divergence Operator"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Vector Field",
"Definition:Divergence Operator",
"Definition:Coordinate System"
] | [
"Definition:Divergence Operator/Physical Interpretation"
] |
proofwiki-18122 | Curl Operator on Vector Space is Cross Product of Del Operator | Let $R$ be a region of Cartesian $3$-space $\R^3$.
Let $\map {\mathbf V} {x, y, z}$ be a vector field acting over $R$.
Then
:$\curl \mathbf V = \nabla \times \mathbf V$
where:
:$\curl \mathbf V $ denotes the curl of $\mathbf V$
:$\nabla$ denotes the del operator. | Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
We have by definition of curl of $\mathbf V$:
:$\curl \mathbf V = \paren {\dfrac {\partial V_z} {\partial y} - \dfrac {\partial V_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial V_x} {\partial z} - \dfrac {\partial V_z} {\par... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Cartesian 3-Space|Cartesian $3$-space]] $\R^3$.
Let $\map {\mathbf V} {x, y, z}$ be a [[Definition:Vector Field|vector field]] acting over $R$.
Then
:$\curl \mathbf V = \nabla \times \mathbf V$
where:
:$\curl \mathbf V $ denotes the [[Definition:Geometrical R... | Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the [[Definition:Standard Ordered Basis on Vector Space|standard ordered basis]] on $\R^3$.
We have by definition of [[Definition:Geometrical Representation of Curl Operator|curl]] of $\mathbf V$:
:$\curl \mathbf V = \paren {\dfrac {\partial V_z} {\partial y} - \dfrac... | Curl Operator on Vector Space is Cross Product of Del Operator | https://proofwiki.org/wiki/Curl_Operator_on_Vector_Space_is_Cross_Product_of_Del_Operator | https://proofwiki.org/wiki/Curl_Operator_on_Vector_Space_is_Cross_Product_of_Del_Operator | [
"Curl Operator"
] | [
"Definition:Region",
"Definition:Cartesian 3-Space",
"Definition:Vector Field",
"Definition:Curl Operator/Geometrical Representation",
"Definition:Del Operator"
] | [
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Curl Operator/Geometrical Representation",
"Determinant Form of Curl Operator"
] |
proofwiki-18123 | Left Shift Operator is Linear Mapping | Let $X = Y = \ell^2$ be 2-sequence spaces over real numbers.
Let $L : X \to Y$ be the left shift operator.
Then $L$ is a linear mapping. | Let $x = \tuple {x_1, x_2,x_3, \ldots}, y = \tuple {y_1, y_2, y_3, \ldots} \in \ell^2$
Let $\alpha \in \R$. | Let $X = Y = \ell^2$ be [[P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space|2-sequence spaces over real numbers]].
Let $L : X \to Y$ be the [[Definition:Left Shift Operator|left shift operator]].
Then $L$ is a [[Definition:Linear Mapping|linear mappi... | Let $x = \tuple {x_1, x_2,x_3, \ldots}, y = \tuple {y_1, y_2, y_3, \ldots} \in \ell^2$
Let $\alpha \in \R$. | Left Shift Operator is Linear Mapping | https://proofwiki.org/wiki/Left_Shift_Operator_is_Linear_Mapping | https://proofwiki.org/wiki/Left_Shift_Operator_is_Linear_Mapping | [
"Linear Transformations"
] | [
"P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space",
"Definition:Left Shift Operator",
"Definition:Linear Transformation"
] | [] |
proofwiki-18124 | Right Shift Operator is Linear Mapping | Let $X = Y = \ell^2$ be 2-sequence spaces over real numbers.
Let $R : X \to Y$ be the right shift operator.
Then $R$ is a linear mapping. | Let $x = \tuple {x_1, x_2,x_3, \ldots}, y = \tuple {y_1, y_2, y_3, \ldots} \in \ell^2$
Let $\alpha \in \R$. | Let $X = Y = \ell^2$ be [[P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space|2-sequence spaces over real numbers]].
Let $R : X \to Y$ be the [[Definition:Right Shift Operator|right shift operator]].
Then $R$ is a [[Definition:Linear Mapping|linear map... | Let $x = \tuple {x_1, x_2,x_3, \ldots}, y = \tuple {y_1, y_2, y_3, \ldots} \in \ell^2$
Let $\alpha \in \R$. | Right Shift Operator is Linear Mapping | https://proofwiki.org/wiki/Right_Shift_Operator_is_Linear_Mapping | https://proofwiki.org/wiki/Right_Shift_Operator_is_Linear_Mapping | [
"Linear Transformations"
] | [
"P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space",
"Definition:Right Shift Operator",
"Definition:Linear Transformation"
] | [] |
proofwiki-18125 | Angular Velocity is Vector Quantity | The physical quantity that is angular velocity can be correctly handled as a vector.
{{explain|The definition of angular velocity is currently being left open until a rigorous book-definition can be found for it which puts that definition firmly into context. The current works under analysis seem to take its definition... | In order to show that angular velocity is a vector, it is sufficient to demonstrate that it fulfils the vector space axioms.
Specifically, all we need to do is demonstrate the following.
Let $\bsomega_1$ be the angular velocity of a body about an axis which passes through a fixed point $O$.
Let $P$ be a point in $B$ wh... | The [[Definition:Physical Quantity|physical quantity]] that is [[Definition:Angular Velocity|angular velocity]] can be correctly handled as a [[Definition:Vector Quantity|vector]].
{{explain|The definition of [[Definition:Angular Velocity|angular velocity]] is currently being left open until a rigorous book-definition... | In order to show that [[Definition:Angular Velocity|angular velocity]] is a [[Definition:Vector Quantity|vector]], it is [[Definition:Sufficient Condition|sufficient]] to demonstrate that it fulfils the [[Axiom:Vector Space Axioms|vector space axioms]].
Specifically, all we need to do is demonstrate the following.
Le... | Angular Velocity is Vector Quantity | https://proofwiki.org/wiki/Angular_Velocity_is_Vector_Quantity | https://proofwiki.org/wiki/Angular_Velocity_is_Vector_Quantity | [
"Angular Velocity"
] | [
"Definition:Physical Quantity",
"Definition:Angular Velocity",
"Definition:Vector Quantity",
"Definition:Angular Velocity"
] | [
"Definition:Angular Velocity",
"Definition:Vector Quantity",
"Definition:Conditional/Sufficient Condition",
"Axiom:Vector Space Axioms",
"Definition:Angular Velocity",
"Definition:Body",
"Definition:Rotation (Geometry)/Axis",
"Definition:Fixed Point (Physics)",
"Definition:Point",
"Definition:Posi... |
proofwiki-18126 | Infinite Limit Operator is Linear Mapping | Let $c$ be the space of convergent sequences.
Let $\R$ be the set of real numbers.
Let $L : c \to \R$ be the infinite limit operator.
Then $L$ is a linear mapping. | Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N}, \mathbf y = \sequence {y_n}_{n \mathop \in \N} \in c$.
Suppose $\mathbf x$ and $\mathbf y$ converge to $x$ and $y$ respectively.
Let $\alpha \in \R$. | Let $c$ be the [[Definition:Space of Convergent Sequences|space of convergent sequences]].
Let $\R$ be the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]].
Let $L : c \to \R$ be the [[Definition:Infinite Limit Operator|infinite limit operator]].
Then $L$ is a [[Definition:Linear Mapping|linear map... | Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N}, \mathbf y = \sequence {y_n}_{n \mathop \in \N} \in c$.
Suppose $\mathbf x$ and $\mathbf y$ [[Definition:Convergent Real Sequence|converge]] to $x$ and $y$ respectively.
Let $\alpha \in \R$. | Infinite Limit Operator is Linear Mapping | https://proofwiki.org/wiki/Infinite_Limit_Operator_is_Linear_Mapping | https://proofwiki.org/wiki/Infinite_Limit_Operator_is_Linear_Mapping | [
"Linear Transformations",
"Operator Theory"
] | [
"Definition:Space of Convergent Sequences",
"Definition:Set",
"Definition:Real Number",
"Definition:Infinite Limit Operator",
"Definition:Linear Transformation"
] | [
"Definition:Convergent Sequence/Real Numbers"
] |
proofwiki-18127 | Gradient of Divergence is Conservative | Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.
Let $\mathbf V$ be a vector field on $\R^3$:
Then the gradient of the divergence of $\mathbf V$ is a conservative vector field. | The divergence of $\mathbf V$ is by definition a scalar field.
Then from Vector Field is Expressible as Gradient of Scalar Field iff Conservative it follows that $\grad \operatorname {div} \mathbf V$ is a conservative vector field.
{{qed}} | Let $\map {\R^3} {x, y, z}$ denote the [[Definition:Real Cartesian Space|real Cartesian space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]].
Let $\mathbf V$ be a [[Definition:Vector Field|vector field]] on $\R^3$:
Then the [[Definition:Gradient Operator|gradient]] of the [[Definition:Divergence Operat... | The [[Definition:Divergence Operator|divergence]] of $\mathbf V$ is by definition a [[Definition:Scalar Field (Physics)|scalar field]].
Then from [[Vector Field is Expressible as Gradient of Scalar Field iff Conservative]] it follows that $\grad \operatorname {div} \mathbf V$ is a [[Definition:Conservative Vector Fiel... | Gradient of Divergence is Conservative | https://proofwiki.org/wiki/Gradient_of_Divergence_is_Conservative | https://proofwiki.org/wiki/Gradient_of_Divergence_is_Conservative | [
"Gradient Operator",
"Divergence Operator",
"Conservative Vector Fields"
] | [
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Dimension of Vector Space",
"Definition:Vector Field",
"Definition:Gradient Operator",
"Definition:Divergence Operator",
"Definition:Conservative Vector Field"
] | [
"Definition:Divergence Operator",
"Definition:Scalar Field (Physics)",
"Vector Field is Expressible as Gradient of Scalar Field iff Conservative",
"Definition:Conservative Vector Field"
] |
proofwiki-18128 | Gradient of Divergence | Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Let $\mathbf V: \R^3 \to \R^3$ be a vector field on $\R^3$:
:$\mathbf V := \tuple {\map {V_x} {x, y, z}, \map {V_y} {x, y, z}, \map {V_z} {x, y, z} }$
Th... | From Divergence Operator on Vector Space is Dot Product of Del Operator and definition of the gradient operator:
{{begin-eqn}}
{{eqn | l = \operatorname {div} \mathbf V
| r = \nabla \cdot \mathbf V
}}
{{eqn | l = \grad \mathbf U
| r = \nabla U
}}
{{end-eqn}}
where $\nabla$ denotes the del operator.
Hence:
{... | Let $\map {\R^3} {x, y, z}$ denote the [[Definition:Real Cartesian Space|real Cartesian space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]].
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the [[Definition:Standard Ordered Basis on Vector Space|standard ordered basis on $\R^3$]].
Let $\mathbf V: \R^3... | From [[Divergence Operator on Vector Space is Dot Product of Del Operator]] and definition of the [[Definition:Gradient Operator|gradient operator]]:
{{begin-eqn}}
{{eqn | l = \operatorname {div} \mathbf V
| r = \nabla \cdot \mathbf V
}}
{{eqn | l = \grad \mathbf U
| r = \nabla U
}}
{{end-eqn}}
where $\na... | Gradient of Divergence | https://proofwiki.org/wiki/Gradient_of_Divergence | https://proofwiki.org/wiki/Gradient_of_Divergence | [
"Gradient Operator",
"Divergence Operator",
"Gradient of Divergence"
] | [
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Dimension of Vector Space",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Vector Field",
"Definition:Gradient Operator",
"Definition:Divergence Operator"
] | [
"Divergence Operator on Vector Space is Dot Product of Del Operator",
"Definition:Gradient Operator",
"Definition:Del Operator"
] |
proofwiki-18129 | Sine of Integer Multiple of Argument/Formulation 8 | :$\sin n \theta = \map \sin {\paren {n - 1 } \theta} \paren { a_0 - \cfrac 1 {a_1 - \cfrac 1 {a_2 - \cfrac 1 {\ddots \cfrac {} {a_{n-3} - \cfrac 1 {a_{n-2}}} }}} }$
where $a_0 = a_1 = a_2 = \ldots = a_{n-2} = 2 \cos \theta$. | From Sine of Integer Multiple of Argument Formulation 4 we have:
{{begin-eqn}}
{{eqn | l = \map \sin {n \theta}
| r = \paren {2 \cos \theta } \map \sin {\paren {n - 1 } \theta} - \map \sin {\paren {n - 2 } \theta}
| c =
}}
{{eqn | r = \map \sin {\paren {n - 1 } \theta} \paren {\paren {2 \cos \theta } - \... | :$\sin n \theta = \map \sin {\paren {n - 1 } \theta} \paren { a_0 - \cfrac 1 {a_1 - \cfrac 1 {a_2 - \cfrac 1 {\ddots \cfrac {} {a_{n-3} - \cfrac 1 {a_{n-2}}} }}} }$
where $a_0 = a_1 = a_2 = \ldots = a_{n-2} = 2 \cos \theta$. | From [[Sine of Integer Multiple of Argument/Formulation 4|Sine of Integer Multiple of Argument Formulation 4]] we have:
{{begin-eqn}}
{{eqn | l = \map \sin {n \theta}
| r = \paren {2 \cos \theta } \map \sin {\paren {n - 1 } \theta} - \map \sin {\paren {n - 2 } \theta}
| c =
}}
{{eqn | r = \map \sin {\par... | Sine of Integer Multiple of Argument/Formulation 8 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_8 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_8 | [
"Sine of Integer Multiple of Argument",
"Continued Fractions",
"Examples of Continued Fractions"
] | [] | [
"Sine of Integer Multiple of Argument/Formulation 4",
"Sine of Integer Multiple of Argument/Formulation 4",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Double Angle Formulas/Sine"
] |
proofwiki-18130 | Curl of Vector Field is Solenoidal | Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.
Let $\mathbf V$ be a vector field on $\R^3$:
Then the curl of $\mathbf V$ is a solenoidal vector field. | By definition, a solenoidal vector field is one whose divergence is zero.
The result follows from Divergence of Curl is Zero.
{{qed}} | Let $\map {\R^3} {x, y, z}$ denote the [[Definition:Real Cartesian Space|real Cartesian space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]].
Let $\mathbf V$ be a [[Definition:Vector Field|vector field]] on $\R^3$:
Then the [[Definition:Curl Operator|curl]] of $\mathbf V$ is a [[Definition:Solenoidal V... | By definition, a [[Definition:Solenoidal Vector Field|solenoidal vector field]] is one whose [[Definition:Divergence Operator|divergence]] is [[Definition:Zero (Number)|zero]].
The result follows from [[Divergence of Curl is Zero]].
{{qed}} | Curl of Vector Field is Solenoidal | https://proofwiki.org/wiki/Curl_of_Vector_Field_is_Solenoidal | https://proofwiki.org/wiki/Curl_of_Vector_Field_is_Solenoidal | [
"Solenoidal Vector Fields",
"Curl Operator"
] | [
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Dimension of Vector Space",
"Definition:Vector Field",
"Definition:Curl Operator",
"Definition:Solenoidal Vector Field"
] | [
"Definition:Solenoidal Vector Field",
"Definition:Divergence Operator",
"Definition:Zero (Number)",
"Divergence of Curl is Zero"
] |
proofwiki-18131 | Laplacian on Scalar Field is Divergence of Gradient | Let $\R^n$ denote the real Cartesian space of $n$ dimensions.
Let $U$ be a scalar field over $\R^n$.
Let $\nabla^2 U$ denote the laplacian on $U$.
Then:
:$\nabla^2 U = \operatorname {div} \grad U$
where:
:$\operatorname {div}$ denotes the divergence operator
:$\grad$ denotes the gradient operator. | From Divergence Operator on Vector Space is Dot Product of Del Operator and definition of the gradient operator:
{{begin-eqn}}
{{eqn | l = \operatorname {div} \mathbf V
| r = \nabla \cdot \mathbf V
}}
{{eqn | l = \grad \mathbf U
| r = \nabla U
}}
{{end-eqn}}
where $\nabla$ denotes the del operator.
Let $\tu... | Let $\R^n$ denote the [[Definition:Real Cartesian Space|real Cartesian space]] of [[Definition:Dimension of Vector Space|$n$ dimensions]].
Let $U$ be a [[Definition:Scalar Field (Physics)|scalar field]] over $\R^n$.
Let $\nabla^2 U$ denote the [[Definition:Laplacian on Scalar Field|laplacian]] on $U$.
Then:
:$\nabl... | From [[Divergence Operator on Vector Space is Dot Product of Del Operator]] and definition of the [[Definition:Gradient Operator|gradient operator]]:
{{begin-eqn}}
{{eqn | l = \operatorname {div} \mathbf V
| r = \nabla \cdot \mathbf V
}}
{{eqn | l = \grad \mathbf U
| r = \nabla U
}}
{{end-eqn}}
where $\na... | Laplacian on Scalar Field is Divergence of Gradient | https://proofwiki.org/wiki/Laplacian_on_Scalar_Field_is_Divergence_of_Gradient | https://proofwiki.org/wiki/Laplacian_on_Scalar_Field_is_Divergence_of_Gradient | [
"Laplacian",
"Divergence Operator",
"Gradient Operator"
] | [
"Definition:Cartesian Product/Cartesian Space/Real Cartesian Space",
"Definition:Dimension of Vector Space",
"Definition:Scalar Field (Physics)",
"Definition:Laplacian/Scalar Field",
"Definition:Divergence Operator",
"Definition:Gradient Operator"
] | [
"Divergence Operator on Vector Space is Dot Product of Del Operator",
"Definition:Gradient Operator",
"Definition:Del Operator",
"Definition:Standard Ordered Basis/Vector Space"
] |
proofwiki-18132 | Condition for Vector Field to satisfy Laplace's Equation | Let $\mathbf V$ be a vector field over a region of space $R$.
Then:
:$\mathbf V$ is both solenoidal and conservative
{{iff}}:
:$\mathbf V$ is the gradient of a scalar field $F$ over $R$ which satisfies Laplace's equation:
::$\nabla^2 F \equiv 0$ | === Sufficient Condition ===
Let $\mathbf V$ be both solenoidal and conservative.
Then from Vector Field is Expressible as Gradient of Scalar Field iff Conservative:
:$\mathbf V = \grad F$
for some scalar field $F$ over $R$.
Because $\mathbf V$ is solenoidal, we have:
:$\operatorname {div} \mathbf V = 0$
that is:
:$\op... | Let $\mathbf V$ be a [[Definition:Vector Field|vector field]] over a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]] $R$.
Then:
:$\mathbf V$ is both [[Definition:Solenoidal Vector Field|solenoidal]] and [[Definition:Conservative Vector Field|conservative]]
{{iff}}:
:$\mathbf V$ is the [[Definition:... | === Sufficient Condition ===
Let $\mathbf V$ be both [[Definition:Solenoidal Vector Field|solenoidal]] and [[Definition:Conservative Vector Field|conservative]].
Then from [[Vector Field is Expressible as Gradient of Scalar Field iff Conservative]]:
:$\mathbf V = \grad F$
for some [[Definition:Scalar Field (Physics)|... | Condition for Vector Field to satisfy Laplace's Equation | https://proofwiki.org/wiki/Condition_for_Vector_Field_to_satisfy_Laplace's_Equation | https://proofwiki.org/wiki/Condition_for_Vector_Field_to_satisfy_Laplace's_Equation | [
"Solenoidal Vector Fields",
"Conservative Vector Fields",
"Laplace's Equation"
] | [
"Definition:Vector Field",
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Solenoidal Vector Field",
"Definition:Conservative Vector Field",
"Definition:Gradient Operator",
"Definition:Scalar Field (Physics)",
"Definition:Laplace's Equation"
] | [
"Definition:Solenoidal Vector Field",
"Definition:Conservative Vector Field",
"Vector Field is Expressible as Gradient of Scalar Field iff Conservative",
"Definition:Scalar Field (Physics)",
"Definition:Solenoidal Vector Field",
"Laplacian on Scalar Field is Divergence of Gradient",
"Definition:Laplacia... |
proofwiki-18133 | Condition for Vector Field to satisfy Poisson's Equation | Let $\mathbf V$ be a vector field over a region of space $R$.
Then:
:$\mathbf V$ is conservative but not solenoidal
{{iff}}:
:$\mathbf V$ is the gradient of a scalar field $F$ over $R$ which satisfies Poisson's equation over $R$:
::$\nabla^2 F = \phi$
:where $\phi$ is a function which is not identically zero. | === Sufficient Condition ===
Let $\mathbf V$ be conservative but not solenoidal.
From Vector Field is Expressible as Gradient of Scalar Field iff Conservative:
:$\mathbf V = \grad F$
for some scalar field $F$ over $R$.
Because $\mathbf V$ is not solenoidal, we have:
:$\exists \mathbf v \in R: \operatorname {div} \mathb... | Let $\mathbf V$ be a [[Definition:Vector Field|vector field]] over a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]] $R$.
Then:
:$\mathbf V$ is [[Definition:Conservative Vector Field|conservative]] but not [[Definition:Solenoidal Vector Field|solenoidal]]
{{iff}}:
:$\mathbf V$ is the [[Definition:G... | === Sufficient Condition ===
Let $\mathbf V$ be [[Definition:Conservative Vector Field|conservative]] but not [[Definition:Solenoidal Vector Field|solenoidal]].
From [[Vector Field is Expressible as Gradient of Scalar Field iff Conservative]]:
:$\mathbf V = \grad F$
for some [[Definition:Scalar Field (Physics)|scalar... | Condition for Vector Field to satisfy Poisson's Equation | https://proofwiki.org/wiki/Condition_for_Vector_Field_to_satisfy_Poisson's_Equation | https://proofwiki.org/wiki/Condition_for_Vector_Field_to_satisfy_Poisson's_Equation | [
"Solenoidal Vector Fields",
"Conservative Vector Fields",
"Poisson's Differential Equation"
] | [
"Definition:Vector Field",
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Conservative Vector Field",
"Definition:Solenoidal Vector Field",
"Definition:Gradient Operator",
"Definition:Scalar Field (Physics)",
"Definition:Poisson's Differential Equation",
"Definition:Function",
"Defi... | [
"Definition:Conservative Vector Field",
"Definition:Solenoidal Vector Field",
"Vector Field is Expressible as Gradient of Scalar Field iff Conservative",
"Definition:Scalar Field (Physics)",
"Definition:Solenoidal Vector Field",
"Laplacian on Scalar Field is Divergence of Gradient",
"Definition:Zero (Nu... |
proofwiki-18134 | Sine of Integer Multiple of Argument/Formulation 9 | :$\sin n \theta = \map \cos {\paren {n - 1} \theta} \paren {a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n - r} } } } } }$
where:
:$r = \begin {cases} 2 & : \text {$n$ is even} \\ 1 & : \text {$n$ is odd} \end {cases}$
:$a_k = \begin {cases} 2 \sin \theta & : \text {$k$ is even} \\ -2 \sin \thet... | {{begin-eqn}}
{{eqn | l = \map \sin {n \theta}
| r = \paren {2 \sin \theta } \map \cos {\paren {n - 1 } \theta} + \map \sin {\paren {n - 2 } \theta}
| c = Sine of Integer Multiple of Argument/Formulation 6
}}
{{eqn | r = \map \cos {\paren {n - 1 } \theta} \paren { \paren {2 \sin \theta } + \frac {\map \s... | :$\sin n \theta = \map \cos {\paren {n - 1} \theta} \paren {a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n - r} } } } } }$
where:
:$r = \begin {cases} 2 & : \text {$n$ is even} \\ 1 & : \text {$n$ is odd} \end {cases}$
:$a_k = \begin {cases} 2 \sin \theta & : \text {$k$ is even} \\ -2 \sin \the... | {{begin-eqn}}
{{eqn | l = \map \sin {n \theta}
| r = \paren {2 \sin \theta } \map \cos {\paren {n - 1 } \theta} + \map \sin {\paren {n - 2 } \theta}
| c = [[Sine of Integer Multiple of Argument/Formulation 6]]
}}
{{eqn | r = \map \cos {\paren {n - 1 } \theta} \paren { \paren {2 \sin \theta } + \frac {\ma... | Sine of Integer Multiple of Argument/Formulation 9 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_9 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_9 | [
"Sine of Integer Multiple of Argument",
"Continued Fractions",
"Examples of Continued Fractions"
] | [] | [
"Sine of Integer Multiple of Argument/Formulation 6",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Cosine of Integer Multiple of Argument/Formulation 6",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Terminal Object",
"Definition:Fraction/Denom... |
proofwiki-18135 | Cosine of Integer Multiple of Argument/Formulation 8 | :$\cos n \theta = \map \cos {\paren {n - 1 } \theta} \paren { a_0 - \cfrac 1 {a_1 - \cfrac 1 {a_2 - \cfrac 1 {\ddots \cfrac {} {a_{n-2} - \cfrac 1 {a_{n-1}}} }}} }$
where $a_0 = a_1 = a_2 = \ldots = a_{n-2} = 2 \cos \theta$ and
::A terminal $a_{n-1} = \cos \theta$ term. | From Cosine of Integer Multiple of Argument Formulation 4 we have:
{{begin-eqn}}
{{eqn | l = \map \cos {n \theta}
| r = \paren {2 \cos \theta } \map \cos {\paren {n - 1 } \theta} - \map \cos {\paren {n - 2 } \theta}
| c =
}}
{{eqn | r = \map \cos {\paren {n - 1 } \theta} \paren {\paren {2 \cos \theta } -... | :$\cos n \theta = \map \cos {\paren {n - 1 } \theta} \paren { a_0 - \cfrac 1 {a_1 - \cfrac 1 {a_2 - \cfrac 1 {\ddots \cfrac {} {a_{n-2} - \cfrac 1 {a_{n-1}}} }}} }$
where $a_0 = a_1 = a_2 = \ldots = a_{n-2} = 2 \cos \theta$ and
::A [[Definition:Terminal Object|terminal]] $a_{n-1} = \cos \theta$ [[Definition:Term of Seq... | From [[Cosine of Integer Multiple of Argument/Formulation 4|Cosine of Integer Multiple of Argument Formulation 4]] we have:
{{begin-eqn}}
{{eqn | l = \map \cos {n \theta}
| r = \paren {2 \cos \theta } \map \cos {\paren {n - 1 } \theta} - \map \cos {\paren {n - 2 } \theta}
| c =
}}
{{eqn | r = \map \cos {... | Cosine of Integer Multiple of Argument/Formulation 8 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_8 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_8 | [
"Cosine of Integer Multiple of Argument",
"Continued Fractions",
"Examples of Continued Fractions"
] | [
"Definition:Terminal Object",
"Definition:Term of Sequence"
] | [
"Cosine of Integer Multiple of Argument/Formulation 4",
"Cosine of Integer Multiple of Argument/Formulation 4",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Terminal Object",
"Definition:Term of Sequence"
] |
proofwiki-18136 | Poisson's Differential Equation for Rotational and Solenoidal Field | Let $R$ be a region of ordinary space.
Let $\mathbf V$ be a vector field over $R$.
Let $\mathbf V$ be both rotational and solenoidal.
Let $\mathbf A$ be a vector field such that $\mathbf V = \curl \mathbf A$.
Then $\mathbf V$ satisfies this version of Poisson's differential equation:
:$\curl \mathbf V = -\nabla^2 \math... | As $\mathbf V$ is rotational it is not conservative.
Hence from Vector Field is Expressible as Gradient of Scalar Field iff Conservative $\mathbf V$ cannot be the gradient of some scalar field.
However, by definition of rotational vector field:
:$\curl \mathbf V \ne \bszero$
As $\mathbf V$ is solenoidal:
:$\operatornam... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|ordinary space]].
Let $\mathbf V$ be a [[Definition:Vector Field|vector field]] over $R$.
Let $\mathbf V$ be both [[Definition:Rotational Vector Field|rotational]] and [[Definition:Solenoidal Vector Field|solenoidal]].
Let $\mathbf A$ be a [[D... | As $\mathbf V$ is [[Definition:Rotational Vector Field|rotational]] it is not [[Definition:Conservative Vector Field|conservative]].
Hence from [[Vector Field is Expressible as Gradient of Scalar Field iff Conservative]] $\mathbf V$ cannot be the [[Definition:Gradient Operator|gradient]] of some [[Definition:Scalar Fi... | Poisson's Differential Equation for Rotational and Solenoidal Field | https://proofwiki.org/wiki/Poisson's_Differential_Equation_for_Rotational_and_Solenoidal_Field | https://proofwiki.org/wiki/Poisson's_Differential_Equation_for_Rotational_and_Solenoidal_Field | [
"Poisson's Differential Equation",
"Rotational Vector Fields",
"Solenoidal Vector Fields",
"Poisson's Differential Equation for Rotational and Solenoidal Field"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Vector Field",
"Definition:Rotational Vector Field",
"Definition:Solenoidal Vector Field",
"Definition:Vector Field",
"Definition:Poisson's Differential Equation"
] | [
"Definition:Rotational Vector Field",
"Definition:Conservative Vector Field",
"Vector Field is Expressible as Gradient of Scalar Field iff Conservative",
"Definition:Gradient Operator",
"Definition:Scalar Field (Physics)",
"Definition:Rotational Vector Field",
"Definition:Solenoidal Vector Field",
"Di... |
proofwiki-18137 | Helmholtz's Theorem | Let $R$ be a region of ordinary space.
Let $\mathbf V$ be a vector field over $R$.
Let $\mathbf V$ be both non-conservative and non-solenoidal.
Then $\mathbf V$ can be decomposed into the sum of $2$ vector fields:
:one being conservative, with scalar potential $S$, but not solenoidal
:one being solenoidal, with vector ... | Let us write:
:$\mathbf V = \grad S + \curl \mathbf A$
where:
:$S$ is a scalar field
:$\mathbf A$ is a vector field chosen to be solenoidal
{{explain|It is not clear why this is always possible}}
Then:
{{begin-eqn}}
{{eqn | l = \operatorname {div} \mathbf V
| r = \operatorname {div} \grad S + \operatorname {div} ... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|ordinary space]].
Let $\mathbf V$ be a [[Definition:Vector Field|vector field]] over $R$.
Let $\mathbf V$ be both non-[[Definition:Conservative Vector Field|conservative]] and non-[[Definition:Solenoidal Vector Field|solenoidal]].
Then $\mathbf... | Let us write:
:$\mathbf V = \grad S + \curl \mathbf A$
where:
:$S$ is a [[Definition:Scalar Field (Physics)|scalar field]]
:$\mathbf A$ is a [[Definition:Vector Field|vector field]] chosen to be [[Definition:Solenoidal Vector Field|solenoidal]]
{{explain|It is not clear why this is always possible}}
Then:
{{begin-eq... | Helmholtz's Theorem | https://proofwiki.org/wiki/Helmholtz's_Theorem | https://proofwiki.org/wiki/Helmholtz's_Theorem | [
"Vector Calculus"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Vector Field",
"Definition:Conservative Vector Field",
"Definition:Solenoidal Vector Field",
"Definition:Vector Sum",
"Definition:Vector Field",
"Definition:Conservative Vector Field",
"Definition:Scalar Potential",
"Definition:Solenoid... | [
"Definition:Scalar Field (Physics)",
"Definition:Vector Field",
"Definition:Solenoidal Vector Field",
"Divergence of Curl is Zero",
"Laplacian on Scalar Field is Divergence of Gradient",
"Definition:Solenoidal Vector Field",
"Curl of Gradient is Zero",
"Curl of Curl is Gradient of Divergence minus Lap... |
proofwiki-18138 | Divergence of Product of Scalar Field with Gradient of Scalar Field | Let $R$ be a region of space.
Let $U$ and $W$ be scalar fields over $R$.
Then:
:$\map {\operatorname {div} } {U \grad W} = U \nabla^2 W + \paren {\grad U} \cdot \paren {\grad W}$
where:
:$\operatorname {div}$ denotes the divergence operator
:$\grad$ denotes the gradient operator
:$\nabla^2$ denotes the Laplacian. | {{begin-eqn}}
{{eqn | l = \map {\operatorname {div} } {U \mathbf A}
| r = \map U {\operatorname {div} \mathbf A} + \mathbf A \cdot \grad U
| c = Product Rule for Divergence
}}
{{eqn | ll= \leadsto
| l = \map {\operatorname {div} } {U \grad W}
| r = \map U {\operatorname {div} \grad W} + \paren {... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]].
Let $U$ and $W$ be [[Definition:Scalar Field (Physics)|scalar fields]] over $R$.
Then:
:$\map {\operatorname {div} } {U \grad W} = U \nabla^2 W + \paren {\grad U} \cdot \paren {\grad W}$
where:
:$\operatorname {div}$ denotes the [[Def... | {{begin-eqn}}
{{eqn | l = \map {\operatorname {div} } {U \mathbf A}
| r = \map U {\operatorname {div} \mathbf A} + \mathbf A \cdot \grad U
| c = [[Product Rule for Divergence]]
}}
{{eqn | ll= \leadsto
| l = \map {\operatorname {div} } {U \grad W}
| r = \map U {\operatorname {div} \grad W} + \par... | Divergence of Product of Scalar Field with Gradient of Scalar Field | https://proofwiki.org/wiki/Divergence_of_Product_of_Scalar_Field_with_Gradient_of_Scalar_Field | https://proofwiki.org/wiki/Divergence_of_Product_of_Scalar_Field_with_Gradient_of_Scalar_Field | [
"Gradient Operator",
"Divergence Operator"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Scalar Field (Physics)",
"Definition:Divergence Operator",
"Definition:Gradient Operator",
"Definition:Laplacian/Scalar Field"
] | [
"Product Rule for Divergence",
"Laplacian on Scalar Field is Divergence of Gradient"
] |
proofwiki-18139 | Divergence of Product of Scalar Field with Curl of Vector Field | Let $R$ be a region of space.
Let $U$ be a scalar field over $R$.
Let $\mathbf A = \curl \mathbf B$ be a vector field over $R$ whose vector potential is $\mathbf B$.
Then:
:$\map {\operatorname {div} } {U \curl \mathbf B} = \paren {\curl \mathbf B} \cdot \paren {\grad U}$
where:
:$\operatorname {div}$ denotes the diver... | {{begin-eqn}}
{{eqn | l = \map {\operatorname {div} } {U \mathbf A}
| r = \map U {\operatorname {div} \mathbf A} + \mathbf A \cdot \grad U
| c = Product Rule for Divergence
}}
{{eqn | ll= \leadsto
| l = \map {\operatorname {div} } {U \curl \mathbf B}
| r = \map U {\operatorname {div} \curl \math... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]].
Let $U$ be a [[Definition:Scalar Field (Physics)|scalar field]] over $R$.
Let $\mathbf A = \curl \mathbf B$ be a [[Definition:Vector Field|vector field]] over $R$ whose [[Definition:Vector Potential|vector potential]] is $\mathbf B$.
... | {{begin-eqn}}
{{eqn | l = \map {\operatorname {div} } {U \mathbf A}
| r = \map U {\operatorname {div} \mathbf A} + \mathbf A \cdot \grad U
| c = [[Product Rule for Divergence]]
}}
{{eqn | ll= \leadsto
| l = \map {\operatorname {div} } {U \curl \mathbf B}
| r = \map U {\operatorname {div} \curl \... | Divergence of Product of Scalar Field with Curl of Vector Field | https://proofwiki.org/wiki/Divergence_of_Product_of_Scalar_Field_with_Curl_of_Vector_Field | https://proofwiki.org/wiki/Divergence_of_Product_of_Scalar_Field_with_Curl_of_Vector_Field | [
"Gradient Operator",
"Divergence Operator",
"Curl Operator"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Scalar Field (Physics)",
"Definition:Vector Field",
"Definition:Vector Potential",
"Definition:Divergence Operator",
"Definition:Gradient Operator",
"Definition:Curl Operator"
] | [
"Product Rule for Divergence",
"Divergence of Curl is Zero"
] |
proofwiki-18140 | Gradient of Newtonian Potential | Let $R$ be a region of space.
Let $S$ be a Newtonian potential over $R$ defined as:
:$\forall \mathbf r = x \mathbf i + y \mathbf j + z \mathbf k \in R: \map S {\mathbf r} = \dfrac k r$
where:
:$\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis on $R$
:$\mathbf r = x \mathbf i + y \mathbf j + z \m... | From the geometry of the sphere, the equal surfaces of $S$ are concentric spheres whose centers are at the origin.
As the origin is approached, the scalar potential is unbounded above.
We have:
{{begin-eqn}}
{{eqn | l = \grad S
| r = \map \grad {\dfrac k r}
| c = Definition of $S$
}}
{{eqn | r = \map \nabla... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]].
Let $S$ be a [[Definition:Newtonian Potential|Newtonian potential]] over $R$ defined as:
:$\forall \mathbf r = x \mathbf i + y \mathbf j + z \mathbf k \in R: \map S {\mathbf r} = \dfrac k r$
where:
:$\tuple {\mathbf i, \mathbf j, \mathb... | From the [[Definition:Sphere (Geometry)|geometry of the sphere]], the [[Definition:Equal Surface|equal surfaces]] of $S$ are [[Definition:Concentric Spheres|concentric spheres]] whose [[Definition:Center of Sphere|centers]] are at the [[Definition:Origin|origin]].
As the [[Definition:Origin|origin]] is approached, the... | Gradient of Newtonian Potential | https://proofwiki.org/wiki/Gradient_of_Newtonian_Potential | https://proofwiki.org/wiki/Gradient_of_Newtonian_Potential | [
"Gradient Operator",
"Newtonian Potentials"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Newtonian Potential",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Position Vector",
"Definition:Point",
"Definition:Coordinate System/Origin",
"Definition:Magnitude",
"Definition:Constant",
"Definition:Gradient Operato... | [
"Definition:Sphere/Geometry",
"Definition:Equal Surface",
"Definition:Concentric Spheres",
"Definition:Sphere/Geometry/Center",
"Definition:Coordinate System/Origin",
"Definition:Coordinate System/Origin",
"Definition:Scalar Potential",
"Definition:Bounded Above Mapping/Real-Valued/Unbounded",
"Powe... |
proofwiki-18141 | Newtonian Potential satisfies Laplace's Equation | Let $R$ be a region of space.
Let $S$ be a Newtonian potential over $R$ defined as:
:$\forall \mathbf r = x \mathbf i + y \mathbf j + z \mathbf k \in R: \map S {\mathbf r} = \dfrac k r$
where:
:$\tuple {\mathbf i, \mathbf j, \mathbf k}$ is the standard ordered basis on $R$
:$\mathbf r = x \mathbf i + y \mathbf j + z \m... | From Gradient of Newtonian Potential:
:$\grad S = -\dfrac {k \mathbf r} {r^3}$
where $\grad$ denotes the gradient operator.
Then:
{{begin-eqn}}
{{eqn | l = \operatorname {div} \grad S
| r = \map {\operatorname {div} } {-\dfrac {k \mathbf r} {r^3} }
| c = where $\operatorname {div}$ denotes divergence
}}
{{e... | Let $R$ be a [[Definition:Region|region]] of [[Definition:Ordinary Space|space]].
Let $S$ be a [[Definition:Newtonian Potential|Newtonian potential]] over $R$ defined as:
:$\forall \mathbf r = x \mathbf i + y \mathbf j + z \mathbf k \in R: \map S {\mathbf r} = \dfrac k r$
where:
:$\tuple {\mathbf i, \mathbf j, \mathb... | From [[Gradient of Newtonian Potential]]:
:$\grad S = -\dfrac {k \mathbf r} {r^3}$
where $\grad$ denotes the [[Definition:Gradient Operator|gradient operator]].
Then:
{{begin-eqn}}
{{eqn | l = \operatorname {div} \grad S
| r = \map {\operatorname {div} } {-\dfrac {k \mathbf r} {r^3} }
| c = where $\ope... | Newtonian Potential satisfies Laplace's Equation | https://proofwiki.org/wiki/Newtonian_Potential_satisfies_Laplace's_Equation | https://proofwiki.org/wiki/Newtonian_Potential_satisfies_Laplace's_Equation | [
"Laplace's Equation",
"Newtonian Potentials"
] | [
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Newtonian Potential",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Position Vector",
"Definition:Point",
"Definition:Coordinate System/Origin",
"Definition:Magnitude",
"Definition:Constant",
"Definition:Laplace's Equati... | [
"Gradient of Newtonian Potential",
"Definition:Gradient Operator",
"Definition:Divergence Operator",
"Divergence Operator on Vector Space is Dot Product of Del Operator",
"Power Rule for Derivatives"
] |
proofwiki-18142 | Riemann Integral Operator is Linear Mapping | Let $C \closedint a b$ be the space of continuous Riemann integrable functions.
Let $\R$ be the set of real numbers.
Let $I : C \closedint a b \to \R$ be the Riemann integral operator.
Then $I$ is a linear mapping. | Let $x, y \in C \closedint a b$ be Riemann integrable.
Let $\alpha \in \R$. | Let $C \closedint a b$ be the [[Definition:Continuous Real-Valued Function Space|space of continuous]] [[Definition:Riemann Integrable Function|Riemann integrable functions]].
Let $\R$ be the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]].
Let $I : C \closedint a b \to \R$ be the [[Definition:Riema... | Let $x, y \in C \closedint a b$ be [[Definition:Riemann Integrable Function|Riemann integrable]].
Let $\alpha \in \R$. | Riemann Integral Operator is Linear Mapping | https://proofwiki.org/wiki/Riemann_Integral_Operator_is_Linear_Mapping | https://proofwiki.org/wiki/Riemann_Integral_Operator_is_Linear_Mapping | [
"Linear Transformations",
"Operator Theory"
] | [
"Definition:Continuous Real-Valued Function Space",
"Definition:Definite Integral/Riemann",
"Definition:Set",
"Definition:Real Number",
"Definition:Definite Integral/Riemann/Integral Operator",
"Definition:Linear Transformation"
] | [
"Definition:Definite Integral/Riemann"
] |
proofwiki-18143 | Total Solid Angle Subtended by Spherical Surface | The total solid angle subtended by a spherical surface is $4 \pi$. | :400px
Let $\d S$ be an element of a surface $S$.
Let $\mathbf n$ be a unit normal on $\d S$ positive outwards.
From a point $O$, let a conical pencil touch the boundary of $S$.
Let $\mathbf r_1$ be a unit vector in the direction of the position vector $\mathbf r = r \mathbf r_1$ with respect to $O$.
Let spheres be dra... | The total [[Definition:Solid Angle|solid angle]] [[Definition:Solid Angle Subtended|subtended]] by a [[Definition:Sphere (Geometry)|spherical surface]] is $4 \pi$. | :[[File:Angle-subtended-by-spherical-surface.png|400px]]
Let $\d S$ be an element of a [[Definition:Surface|surface]] $S$.
Let $\mathbf n$ be a [[Definition:Unit Normal|unit normal]] on $\d S$ [[Definition:Positive Real Number|positive]] outwards.
From a [[Definition:Point|point]] $O$, let a [[Definition:Conical Pen... | Total Solid Angle Subtended by Spherical Surface | https://proofwiki.org/wiki/Total_Solid_Angle_Subtended_by_Spherical_Surface | https://proofwiki.org/wiki/Total_Solid_Angle_Subtended_by_Spherical_Surface | [
"Spheres",
"Solid Angles"
] | [
"Definition:Solid Angle",
"Definition:Solid Angle/Subtend",
"Definition:Sphere/Geometry"
] | [
"File:Angle-subtended-by-spherical-surface.png",
"Definition:Surface",
"Definition:Unit Normal",
"Definition:Positive/Real Number",
"Definition:Point",
"Definition:Conical Pencil",
"Definition:Boundary",
"Definition:Unit Vector",
"Definition:Direction",
"Definition:Position Vector",
"Definition:... |
proofwiki-18144 | Operator of Integrated Weighted Derivatives is Linear Mapping | Let $n \in \N$.
Let $I := \closedint a b$ be a closed real interval.
Let $\map {a_i} x : I \to \R$ be Riemann integrable functions.
Let $f, g \in \map {C^n} I$ be Riemann integrable real-valued functions of differentiability class $k$.
Let $L : \map {C^n} I \to \R$ be the operator of integrated weighted derivatives.
Th... | === Distributivity ===
{{begin-eqn}}
{{eqn | l = \map L {f + g}
| r = \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \dfrac {\d^i}{ {\d x}^i} \paren{\map f x + \map g x} \rd x
| c = {{Defof|Operator of Integrated Weighted Derivatives}}
}}
{{eqn | r = \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \paren{\map ... | Let $n \in \N$.
Let $I := \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\map {a_i} x : I \to \R$ be [[Definition:Riemann Integrable Function|Riemann integrable functions]].
Let $f, g \in \map {C^n} I$ be [[Definition:Riemann Integrable Function|Riemann integrable]] [[Definition:... | === Distributivity ===
{{begin-eqn}}
{{eqn | l = \map L {f + g}
| r = \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \dfrac {\d^i}{ {\d x}^i} \paren{\map f x + \map g x} \rd x
| c = {{Defof|Operator of Integrated Weighted Derivatives}}
}}
{{eqn | r = \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \paren{\map... | Operator of Integrated Weighted Derivatives is Linear Mapping | https://proofwiki.org/wiki/Operator_of_Integrated_Weighted_Derivatives_is_Linear_Mapping | https://proofwiki.org/wiki/Operator_of_Integrated_Weighted_Derivatives_is_Linear_Mapping | [
"Linear Transformations",
"Operator Theory"
] | [
"Definition:Real Interval/Closed",
"Definition:Definite Integral/Riemann",
"Definition:Definite Integral/Riemann",
"Definition:Real-Valued Function",
"Definition:Space of Continuous Functions of Differentiability Class k",
"Definition:Operator of Integrated Weighted Derivatives",
"Definition:Linear Tran... | [
"Linear Combination of Derivatives",
"Linear Combination of Integrals/Definite",
"Linear Combination of Derivatives"
] |
proofwiki-18145 | Cosine of Integer Multiple of Argument/Formulation 9 | :$\cos n \theta = \map \sin {\paren {n - 1} \theta} \paren {a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n - r} } } } } }$
where:
:$r = \begin {cases} 1 & : \text {$n$ is even} \\ 2 & : \text {$n$ is odd} \end {cases}$
:$a_k = \begin {cases} -2 \sin \theta & : \text {$k$ is even and $k < n - 1$... | {{begin-eqn}}
{{eqn | l = \map \cos {n \theta}
| r = \paren {-2 \sin \theta } \map \sin {\paren {n - 1 } \theta} + \map \cos {\paren {n - 2 } \theta}
| c = Cosine of Integer Multiple of Argument/Formulation 6
}}
{{eqn | r = \map \sin {\paren {n - 1 } \theta} \paren { -2 \sin \theta + \frac {\map \cos {\p... | :$\cos n \theta = \map \sin {\paren {n - 1} \theta} \paren {a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n - r} } } } } }$
where:
:$r = \begin {cases} 1 & : \text {$n$ is even} \\ 2 & : \text {$n$ is odd} \end {cases}$
:$a_k = \begin {cases} -2 \sin \theta & : \text {$k$ is even and $k < n - 1... | {{begin-eqn}}
{{eqn | l = \map \cos {n \theta}
| r = \paren {-2 \sin \theta } \map \sin {\paren {n - 1 } \theta} + \map \cos {\paren {n - 2 } \theta}
| c = [[Cosine of Integer Multiple of Argument/Formulation 6]]
}}
{{eqn | r = \map \sin {\paren {n - 1 } \theta} \paren { -2 \sin \theta + \frac {\map \cos... | Cosine of Integer Multiple of Argument/Formulation 9 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_9 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_9 | [
"Cosine of Integer Multiple of Argument",
"Continued Fractions",
"Examples of Continued Fractions"
] | [] | [
"Cosine of Integer Multiple of Argument/Formulation 6",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Sine of Integer Multiple of Argument/Formulation 6",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Terminal Object",
"Definition:Fraction/Denom... |
proofwiki-18146 | Component of Vector is Scalar Projection on Standard Ordered Basis Element | Let $\tuple {\mathbf e_1, \mathbf e_2, \mathbf e_3}$ be the standard ordered basis of Cartesian $3$-space $S$.
{{explain|It needs to be emphasised that the the standard ordered basis of Cartesian $3$-space is actually an orthonormal basis.}}
Let $\mathbf a = a_1 \mathbf e_1 + a_2 \mathbf e_2 + a_3 \mathbf e_3$ be a vec... | Using the Einstein summation convention
{{begin-eqn}}
{{eqn | l = \mathbf a \cdot \mathbf e_i
| r = a_j \cdot \mathbf e_j \cdot \mathbf e_i
| c =
}}
{{eqn | r = a_j \delta_{i j}
| c = Dot Product of Orthonormal Basis Vectors
}}
{{eqn | r = a_i
| c =
}}
{{end-eqn}}
{{qed}} | Let $\tuple {\mathbf e_1, \mathbf e_2, \mathbf e_3}$ be the [[Definition:Standard Ordered Basis|standard ordered basis]] of [[Definition:Cartesian 3-Space|Cartesian $3$-space]] $S$.
{{explain|It needs to be emphasised that the the [[Definition:Standard Ordered Basis|standard ordered basis]] of [[Definition:Cartesian 3... | Using the [[Definition:Einstein Summation Convention|Einstein summation convention]]
{{begin-eqn}}
{{eqn | l = \mathbf a \cdot \mathbf e_i
| r = a_j \cdot \mathbf e_j \cdot \mathbf e_i
| c =
}}
{{eqn | r = a_j \delta_{i j}
| c = [[Dot Product of Orthonormal Basis Vectors]]
}}
{{eqn | r = a_i
|... | Component of Vector is Scalar Projection on Standard Ordered Basis Element | https://proofwiki.org/wiki/Component_of_Vector_is_Scalar_Projection_on_Standard_Ordered_Basis_Element | https://proofwiki.org/wiki/Component_of_Vector_is_Scalar_Projection_on_Standard_Ordered_Basis_Element | [
"Scalar Projections"
] | [
"Definition:Standard Ordered Basis",
"Definition:Cartesian 3-Space",
"Definition:Standard Ordered Basis",
"Definition:Cartesian 3-Space",
"Definition:Vector Quantity"
] | [
"Definition:Einstein Summation Convention",
"Dot Product of Orthonormal Basis Vectors"
] |
proofwiki-18147 | Sextuple Angle Formulas/Sine/Corollary | :$\sin 6 \theta = 6 \sin \theta \cos \theta - 32 \sin^3 \theta \cos \theta + 32 \sin^5 \theta \cos \theta$ | {{begin-eqn}}
{{eqn | l = \sin 6 \theta
| r = \paren {2 \cos \theta } \sin 5 \theta - \sin 4 \theta
| c = Sine of Integer Multiple of Argument/Formulation 4
}}
{{eqn | r = \paren {2 \cos \theta } \paren { 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta } - \paren {4 \sin \theta \cos \theta - 8 \sin^3 \t... | :$\sin 6 \theta = 6 \sin \theta \cos \theta - 32 \sin^3 \theta \cos \theta + 32 \sin^5 \theta \cos \theta$ | {{begin-eqn}}
{{eqn | l = \sin 6 \theta
| r = \paren {2 \cos \theta } \sin 5 \theta - \sin 4 \theta
| c = [[Sine of Integer Multiple of Argument/Formulation 4]]
}}
{{eqn | r = \paren {2 \cos \theta } \paren { 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta } - \paren {4 \sin \theta \cos \theta - 8 \sin^... | Sextuple Angle Formulas/Sine/Corollary | https://proofwiki.org/wiki/Sextuple_Angle_Formulas/Sine/Corollary | https://proofwiki.org/wiki/Sextuple_Angle_Formulas/Sine/Corollary | [
"Sextuple Angle Formula for Sine"
] | [] | [
"Sine of Integer Multiple of Argument/Formulation 4",
"Category:Sextuple Angle Formula for Sine"
] |
proofwiki-18148 | Complex Conjugation is not Linear Mapping | Let $\overline \cdot: \C \to \C: z \mapsto \overline z$ be the complex conjugation over the field of complex numbers.
Then complex conjugation is not a linear mapping. | {{begin-eqn}}
{{eqn | l = \overline {i \cdot 1}
| r = \overline i \cdot \overline 1
| c = Product of Complex Conjugates
}}
{{eqn | r = - i \cdot \overline 1
| c = {{defof|Complex Conjugate}}
}}
{{eqn | o = \ne
| r = i \cdot \overline 1
}}
{{end-eqn}}
By definition, it is not a linear mapping.
{{... | Let $\overline \cdot: \C \to \C: z \mapsto \overline z$ be the [[Definition:Complex Conjugation|complex conjugation]] over the [[Definition:Field of Complex Numbers|field of complex numbers]].
Then [[Definition:Complex Conjugation|complex conjugation]] is not a [[Definition:Linear Mapping|linear mapping]]. | {{begin-eqn}}
{{eqn | l = \overline {i \cdot 1}
| r = \overline i \cdot \overline 1
| c = [[Product of Complex Conjugates]]
}}
{{eqn | r = - i \cdot \overline 1
| c = {{defof|Complex Conjugate}}
}}
{{eqn | o = \ne
| r = i \cdot \overline 1
}}
{{end-eqn}}
By definition, it is not a [[Definition:... | Complex Conjugation is not Linear Mapping | https://proofwiki.org/wiki/Complex_Conjugation_is_not_Linear_Mapping | https://proofwiki.org/wiki/Complex_Conjugation_is_not_Linear_Mapping | [
"Complex Conjugates",
"Linear Transformations"
] | [
"Definition:Complex Conjugate/Complex Conjugation",
"Definition:Field of Complex Numbers",
"Definition:Complex Conjugate/Complex Conjugation",
"Definition:Linear Transformation"
] | [
"Product of Complex Conjugates",
"Definition:Linear Transformation"
] |
proofwiki-18149 | Space of Piecewise Linear Functions on Closed Interval is Dense in Space of Continuous Functions on Closed Interval | Let $I = \closedint a b$.
Let $\map \CC I$ be the set of continuous functions on $I$.
Let $\map {\mathrm {PL} } I$ be the set of piecewise linear functions on $I$.
Let $d$ be the metric induced by the supremum norm.
Then $\map {\mathrm {PL} } I$ is dense in $\struct {\map \CC I, d}$. | Let $f \in \map \CC I$.
Let $\epsilon \in \R_{>0}$ be a real number.
From Open Ball Characterization of Denseness:
:it suffices to find a $p \in \map {\mathrm {PL} } I$ such that $p$ is contained in the open ball $\map {B_\epsilon} f$.
From Continuous Function on Closed Real Interval is Uniformly Continuous:
:$f$ is u... | Let $I = \closedint a b$.
Let $\map \CC I$ be the set of [[Definition:Continuous Function|continuous functions]] on $I$.
Let $\map {\mathrm {PL} } I$ be the set of [[Definition:Piecewise Linear Function|piecewise linear functions]] on $I$.
Let $d$ be the [[Definition:Metric Induced by Norm|metric induced]] by the... | Let $f \in \map \CC I$.
Let $\epsilon \in \R_{>0}$ be a [[Definition:Real Number|real number]].
From [[Open Ball Characterization of Denseness]]:
:it suffices to find a $p \in \map {\mathrm {PL} } I$ such that $p$ is contained in the [[Definition:Open Ball|open ball]] $\map {B_\epsilon} f$.
From [[Continuous Funct... | Space of Piecewise Linear Functions on Closed Interval is Dense in Space of Continuous Functions on Closed Interval | https://proofwiki.org/wiki/Space_of_Piecewise_Linear_Functions_on_Closed_Interval_is_Dense_in_Space_of_Continuous_Functions_on_Closed_Interval | https://proofwiki.org/wiki/Space_of_Piecewise_Linear_Functions_on_Closed_Interval_is_Dense_in_Space_of_Continuous_Functions_on_Closed_Interval | [
"Functional Analysis"
] | [
"Definition:Continuous Function",
"Definition:Piecewise Linear Function",
"Definition:Metric Induced by Norm",
"Definition:Supremum Norm",
"Definition:Everywhere Dense"
] | [
"Definition:Real Number",
"Open Set Characterization of Denseness/Open Ball",
"Definition:Open Ball",
"Continuous Function on Closed Real Interval is Uniformly Continuous",
"Definition:Uniform Continuity/Real Function",
"Definition:Subdivision of Interval/Finite",
"Definition:Continuous Real Function",
... |
proofwiki-18150 | Legendre Symbol is Multiplicative | Let $p$ be a odd prime.
Let $a, b \in \Z$.
Then:
:$\paren {\dfrac {a b} p} = \paren {\dfrac a p} \paren {\dfrac b p}$
where $\paren {\dfrac a p}$ is the Legendre symbol. | We have:
{{begin-eqn}}
{{eqn | l = \paren {\frac {a b} p}
| r = \paren {a b}^{\frac {p - 1} 2} \bmod p
| c = {{Defof|Legendre Symbol|index = 2}}
}}
{{eqn | r = a^{\frac {p - 1} 2} b^{\frac {p - 1} 2} \bmod p
| c = Power of Product
}}
{{eqn | r = \paren {\frac a p} \paren {\frac b p}
| c = {{Defof|Legendre ... | Let $p$ be a [[Definition:Odd Prime|odd prime]].
Let $a, b \in \Z$.
Then:
:$\paren {\dfrac {a b} p} = \paren {\dfrac a p} \paren {\dfrac b p}$
where $\paren {\dfrac a p}$ is the [[Definition:Legendre Symbol|Legendre symbol]]. | We have:
{{begin-eqn}}
{{eqn | l = \paren {\frac {a b} p}
| r = \paren {a b}^{\frac {p - 1} 2} \bmod p
| c = {{Defof|Legendre Symbol|index = 2}}
}}
{{eqn | r = a^{\frac {p - 1} 2} b^{\frac {p - 1} 2} \bmod p
| c = [[Power of Product]]
}}
{{eqn | r = \paren {\frac a p} \paren {\frac b p}
| c = {{Defof|Lege... | Legendre Symbol is Multiplicative | https://proofwiki.org/wiki/Legendre_Symbol_is_Multiplicative | https://proofwiki.org/wiki/Legendre_Symbol_is_Multiplicative | [
"Legendre Symbol"
] | [
"Definition:Odd Prime",
"Definition:Legendre Symbol"
] | [
"Exponent Combination Laws/Power of Product",
"Congruence of Product",
"Category:Legendre Symbol"
] |
proofwiki-18151 | Legendre Symbol of Congruent Integers | Let $p$ be a odd prime.
Let $a, b \in \Z$ be such that $a \equiv b \pmod p$.
Then:
:$\paren {\dfrac a p} = \paren {\dfrac b p}$
where $\paren {\dfrac a p}$ is the Legendre symbol. | {{begin-eqn}}
{{eqn | l = \paren {\frac a p}
| r = a^{\frac {p - 1} 2} \bmod p
| c = {{Defof|Legendre Symbol|index = 2}}
}}
{{eqn | r = b^{\frac {p - 1} 2} \bmod p
| c = Congruence of Powers
}}
{{eqn | r = \paren {\frac b p}
| c = {{Defof|Legendre Symbol|index = 2}}
}}
{{end-eqn}}
{{qed}}
Category:Legendre Symbol
i... | Let $p$ be a [[Definition:Odd Prime|odd prime]].
Let $a, b \in \Z$ be such that $a \equiv b \pmod p$.
Then:
:$\paren {\dfrac a p} = \paren {\dfrac b p}$
where $\paren {\dfrac a p}$ is the [[Definition:Legendre Symbol|Legendre symbol]]. | {{begin-eqn}}
{{eqn | l = \paren {\frac a p}
| r = a^{\frac {p - 1} 2} \bmod p
| c = {{Defof|Legendre Symbol|index = 2}}
}}
{{eqn | r = b^{\frac {p - 1} 2} \bmod p
| c = [[Congruence of Powers]]
}}
{{eqn | r = \paren {\frac b p}
| c = {{Defof|Legendre Symbol|index = 2}}
}}
{{end-eqn}}
{{qed}}
[[Category:Legendre S... | Legendre Symbol of Congruent Integers | https://proofwiki.org/wiki/Legendre_Symbol_of_Congruent_Integers | https://proofwiki.org/wiki/Legendre_Symbol_of_Congruent_Integers | [
"Legendre Symbol"
] | [
"Definition:Odd Prime",
"Definition:Legendre Symbol"
] | [
"Congruence of Powers",
"Category:Legendre Symbol"
] |
proofwiki-18152 | Derivative of Reciprocal | Let $f: \R_{\ne 0} \to \R$ be the reciprocal function defined as:
:$\map f x = \dfrac 1 x$
Then its derivative is given by:
:$\map {f'} x = -\dfrac 1 {x^2}$ | We have:
{{begin-eqn}}
{{eqn | l = \dfrac 1 x
| r = x^{-1}
| c =
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\dfrac 1 x}
| r = \paren {-1} x^{-2}
| c = Power Rule for Derivatives
}}
{{eqn | r = -\dfrac 1 {x^2}
| c =
}}
{{end-eqn}}
{{qed}} | Let $f: \R_{\ne 0} \to \R$ be the [[Definition:Reciprocal|reciprocal function]] defined as:
:$\map f x = \dfrac 1 x$
Then its [[Definition:Derivative of Real Function|derivative]] is given by:
:$\map {f'} x = -\dfrac 1 {x^2}$ | We have:
{{begin-eqn}}
{{eqn | l = \dfrac 1 x
| r = x^{-1}
| c =
}}
{{eqn | ll= \leadsto
| l = \map {\dfrac \d {\d x} } {\dfrac 1 x}
| r = \paren {-1} x^{-2}
| c = [[Power Rule for Derivatives]]
}}
{{eqn | r = -\dfrac 1 {x^2}
| c =
}}
{{end-eqn}}
{{qed}} | Derivative of Reciprocal | https://proofwiki.org/wiki/Derivative_of_Reciprocal | https://proofwiki.org/wiki/Derivative_of_Reciprocal | [
"Power Rule for Derivatives",
"Reciprocals"
] | [
"Definition:Reciprocal",
"Definition:Derivative/Real Function"
] | [
"Power Rule for Derivatives"
] |
proofwiki-18153 | Distance between Two Points in Plane in Polar Coordinates | Let $A = \polar {r_1, \theta_1}$ and $B = \polar {r_2, \theta_2}$ be two points in a polar coordinate plane
The distance $d$ between $A$ and $B$ is given by:
:$d = \sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \map \cos {\theta_1 - \theta_2} }$ | Let $A$ and $B$ be embedded as suggested in a polar coordinate plane whose pole is at $O$.
:320px
The distance $d$ is the side $AB$ of the triangle $AOB$.
We have that:
:$OA = r_1$
:$OB = r_2$
and:
:$\theta_2 - \theta_1$ is the opposite angle to $AB$.
Hence we can use the Cosine Rule:
:$AB^2 = r_1^2 + r_2^2 - 2 r_1 r_2... | Let $A = \polar {r_1, \theta_1}$ and $B = \polar {r_2, \theta_2}$ be two [[Definition:Point|points]] in a [[Definition:Polar Coordinate Plane|polar coordinate plane]]
The [[Definition:Distance between Points|distance]] $d$ between $A$ and $B$ is given by:
:$d = \sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \map \cos {\theta_1 - \t... | Let $A$ and $B$ be embedded as suggested in a [[Definition:Polar Coordinate Plane|polar coordinate plane]] whose [[Definition:Pole (Polar Coordinates)|pole]] is at $O$.
:[[File:Distance-polar-form.png|320px]]
The [[Definition:Distance between Points|distance]] $d$ is the [[Definition:Side of Polygon|side]] $AB$ of ... | Distance between Two Points in Plane in Polar Coordinates/Proof 1 | https://proofwiki.org/wiki/Distance_between_Two_Points_in_Plane_in_Polar_Coordinates | https://proofwiki.org/wiki/Distance_between_Two_Points_in_Plane_in_Polar_Coordinates/Proof_1 | [
"Distance between Two Points in Plane in Polar Coordinates",
"Distance Formula",
"Analytic Geometry"
] | [
"Definition:Point",
"Definition:Polar Coordinates/Polar Plane",
"Definition:Distance between Points"
] | [
"Definition:Polar Coordinates/Polar Plane",
"Definition:Polar Coordinates/Pole",
"File:Distance-polar-form.png",
"Definition:Distance between Points",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Opposite",
"Law of Cosines",
"Cosine Function is Even"
] |
proofwiki-18154 | Distance between Two Points in Plane in Polar Coordinates | Let $A = \polar {r_1, \theta_1}$ and $B = \polar {r_2, \theta_2}$ be two points in a polar coordinate plane
The distance $d$ between $A$ and $B$ is given by:
:$d = \sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \map \cos {\theta_1 - \theta_2} }$ | From Conversion between Cartesian and Polar Coordinates in Plane, $A$ and $B$ can be expressed in Cartesian coordinates as follows:
{{begin-eqn}}
{{eqn | l = x_1
| r = r_1 \cos \theta_1
}}
{{eqn | l = y_1
| r = r_1 \sin \theta_1
}}
{{eqn | l = x_2
| r = r_2 \cos \theta_2
}}
{{eqn | l = y_2
| r =... | Let $A = \polar {r_1, \theta_1}$ and $B = \polar {r_2, \theta_2}$ be two [[Definition:Point|points]] in a [[Definition:Polar Coordinate Plane|polar coordinate plane]]
The [[Definition:Distance between Points|distance]] $d$ between $A$ and $B$ is given by:
:$d = \sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \map \cos {\theta_1 - \t... | From [[Conversion between Cartesian and Polar Coordinates in Plane]], $A$ and $B$ can be expressed in [[Definition:Cartesian Coordinates|Cartesian coordinates]] as follows:
{{begin-eqn}}
{{eqn | l = x_1
| r = r_1 \cos \theta_1
}}
{{eqn | l = y_1
| r = r_1 \sin \theta_1
}}
{{eqn | l = x_2
| r = r_2 \c... | Distance between Two Points in Plane in Polar Coordinates/Proof 2 | https://proofwiki.org/wiki/Distance_between_Two_Points_in_Plane_in_Polar_Coordinates | https://proofwiki.org/wiki/Distance_between_Two_Points_in_Plane_in_Polar_Coordinates/Proof_2 | [
"Distance between Two Points in Plane in Polar Coordinates",
"Distance Formula",
"Analytic Geometry"
] | [
"Definition:Point",
"Definition:Polar Coordinates/Polar Plane",
"Definition:Distance between Points"
] | [
"Conversion between Cartesian and Polar Coordinates in Plane",
"Definition:Cartesian Coordinate System",
"Definition:Distance between Points",
"Distance Formula",
"Sum of Squares of Sine and Cosine",
"Cosine of Difference"
] |
proofwiki-18155 | Linear Mappings between Vector Spaces form Vector Space | Let $\struct {F, +, \times}$ be a field whose unity is $1_F$.
Let $X, Y$ be vector spaces over the same field $\struct {F, +, \times}$.
Let $\map \LL {X, Y}$ be the set of linear mappings.
Let $x \in X$.
Define pointwise addition $T + S \in \map \LL {X, Y}$ such that:
:$\forall x \in X: \map {\paren {T + S} } x := \map... | Let $T, S, P \in \map \LL {X, Y}$ such that:
:$T, S, P: X \to Y$
Let $\lambda, \mu \in F$. | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Unity of Field|unity]] is $1_F$.
Let $X, Y$ be [[Definition:Vector Space|vector spaces]] over the same [[Definition:Field (Abstract Algebra)|field]] $\struct {F, +, \times}$.
Let $\map \LL {X, Y}$ be the [[Definition:Se... | Let $T, S, P \in \map \LL {X, Y}$ such that:
:$T, S, P: X \to Y$
Let $\lambda, \mu \in F$. | Linear Mappings between Vector Spaces form Vector Space | https://proofwiki.org/wiki/Linear_Mappings_between_Vector_Spaces_form_Vector_Space | https://proofwiki.org/wiki/Linear_Mappings_between_Vector_Spaces_form_Vector_Space | [
"Examples of Vector Spaces",
"Linear Transformations"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Multiplicative Identity",
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Set of All Linear Transformations/Vector Space",
"Definition:Pointwise Addition of Linear Transformations",
"Definition:Pointwise Scalar Multiplicati... | [] |
proofwiki-18156 | Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x are Linearly Independent | Let $x, a, b \in \R$.
Then $e^{a x} \map \cos {b x}$ and $e^{b x} \map \sin {b x}$ are linearly independent. | Let $\alpha_1, \alpha_2 \in \R$ be such that:
:$\alpha_1 e^{a x} \map \cos {b x} + \alpha_2 e^{a x} \map \sin {b x} = 0$
Suppose $x = 0$.
Then $\alpha_1 = 0$.
Suppose $\ds x = \frac \pi {2 b}$.
Then:
:$\ds \alpha_2 \map \exp {\frac {\pi a} {2 b}} = 0$
Hence, $\alpha_2 = 0$.
The conclusion follows from the definition of... | Let $x, a, b \in \R$.
Then $e^{a x} \map \cos {b x}$ and $e^{b x} \map \sin {b x}$ are [[Definition:Linearly Independent Real Functions|linearly independent]]. | Let $\alpha_1, \alpha_2 \in \R$ be such that:
:$\alpha_1 e^{a x} \map \cos {b x} + \alpha_2 e^{a x} \map \sin {b x} = 0$
Suppose $x = 0$.
Then $\alpha_1 = 0$.
Suppose $\ds x = \frac \pi {2 b}$.
Then:
:$\ds \alpha_2 \map \exp {\frac {\pi a} {2 b}} = 0$
Hence, $\alpha_2 = 0$.
The conclusion follows from the defin... | Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x are Linearly Independent | https://proofwiki.org/wiki/Exponential_of_a_x_by_Cosine_of_b_x_and_Exponential_of_a_x_by_Sine_of_b_x_are_Linearly_Independent | https://proofwiki.org/wiki/Exponential_of_a_x_by_Cosine_of_b_x_and_Exponential_of_a_x_by_Sine_of_b_x_are_Linearly_Independent | [
"Derivatives involving Exponential Function",
"Derivatives involving Sine Function",
"Derivatives involving Cosine Function",
"Linear Independence"
] | [
"Definition:Linearly Independent Real Functions"
] | [
"Definition:Linearly Independent Real Functions"
] |
proofwiki-18157 | Difference of Squares of Sines | :$\sin^2 A - \sin^2 B = \map \sin {A + B} \map \sin {A - B}$ | {{begin-eqn}}
{{eqn | l = \sin^2 A - \sin^2 B
| r = \sin^2 A - \sin^2 B + 0
| c =
}}
{{eqn | r = \sin^2 A - \sin^2 B + \paren {\sin^2 A \sin^2 B - \sin^2 A \sin^2 B}
| c =
}}
{{eqn | r = \sin^2 A \paren {1 - \sin^2 B} - \sin^2 B \paren {1 - \sin^2 A}
| c =
}}
{{eqn | r = \sin^2 A \paren {\cos^2... | :$\sin^2 A - \sin^2 B = \map \sin {A + B} \map \sin {A - B}$ | {{begin-eqn}}
{{eqn | l = \sin^2 A - \sin^2 B
| r = \sin^2 A - \sin^2 B + 0
| c =
}}
{{eqn | r = \sin^2 A - \sin^2 B + \paren {\sin^2 A \sin^2 B - \sin^2 A \sin^2 B}
| c =
}}
{{eqn | r = \sin^2 A \paren {1 - \sin^2 B} - \sin^2 B \paren {1 - \sin^2 A}
| c =
}}
{{eqn | r = \sin^2 A \paren {\cos^2... | Difference of Squares of Sines | https://proofwiki.org/wiki/Difference_of_Squares_of_Sines | https://proofwiki.org/wiki/Difference_of_Squares_of_Sines | [
"Sine Function",
"Difference of Two Powers"
] | [] | [
"Sum of Squares of Sine and Cosine",
"Category:Sine Function",
"Category:Difference of Two Powers"
] |
proofwiki-18158 | Reversal of Order of Vertices of Triangle causes Reversal of Sign of Area | Let $\triangle ABC$ be a triangle embedded in the plane.
Let $\Area \triangle ABC = \AA$.
Then:
:$\Area \triangle CBA = -\AA$. | $\triangle CBA$ is the same as $\triangle ABC$ but with its vertices in the reverse order.
We have that:
:if $\triangle ABC$ is traversed anticlockwise going $AB \to BC \to CA$, then $\triangle CBA$ is traversed clockwise going $CB \to BA \to AC$
:if $\triangle ABC$ is traversed clockwise going $AB \to BC \to CA$, then... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] embedded in [[Definition:The Plane|the plane]].
Let $\Area \triangle ABC = \AA$.
Then:
:$\Area \triangle CBA = -\AA$. | $\triangle CBA$ is the same as $\triangle ABC$ but with its [[Definition:Vertex of Polygon|vertices]] in the reverse order.
We have that:
:if $\triangle ABC$ is traversed [[Definition:Anticlockwise|anticlockwise]] going $AB \to BC \to CA$, then $\triangle CBA$ is traversed [[Definition:Clockwise|clockwise]] going $CB ... | Reversal of Order of Vertices of Triangle causes Reversal of Sign of Area | https://proofwiki.org/wiki/Reversal_of_Order_of_Vertices_of_Triangle_causes_Reversal_of_Sign_of_Area | https://proofwiki.org/wiki/Reversal_of_Order_of_Vertices_of_Triangle_causes_Reversal_of_Sign_of_Area | [
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Plane Surface/The Plane"
] | [
"Definition:Polygon/Vertex",
"Definition:Anticlockwise",
"Definition:Clockwise",
"Definition:Clockwise",
"Definition:Anticlockwise",
"Definition:Sign of Area of Triangle"
] |
proofwiki-18159 | Condition for 3 Points in Plane to be Collinear | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be points in the Cartesian plane.
Then:
:$A$, $B$ and $C$ are collinear
{{iff}} the determinant:
:<nowiki>$\begin {vmatrix}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1 \\
\end {vmatrix}$</nowiki>
equals zero. | We have that:
:$A$, $B$ and $C$ are collinear
{{iff}}:
:the area of $\triangle ABC = 0$
{{iff}}:
:$\dfrac 1 2 \size {\paren {\begin {vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end {vmatrix} } } = 0$ (from Area of Triangle in Determinant Form)
{{iff}}:
:$\begin {vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 ... | Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be [[Definition:Point|points]] in the [[Definition:Cartesian Plane|Cartesian plane]].
Then:
:$A$, $B$ and $C$ are [[Definition:Collinear Points|collinear]]
{{iff}} the [[Definition:Determinant|determinant]]:
:<nowiki>$\begin {vmatrix}
x_1 & y_1 ... | We have that:
:$A$, $B$ and $C$ are [[Definition:Collinear Points|collinear]]
{{iff}}:
:the [[Definition:Area|area]] of $\triangle ABC = 0$
{{iff}}:
:$\dfrac 1 2 \size {\paren {\begin {vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end {vmatrix} } } = 0$ (from [[Area of Triangle in Determinant Form]])
{{... | Condition for 3 Points in Plane to be Collinear | https://proofwiki.org/wiki/Condition_for_3_Points_in_Plane_to_be_Collinear | https://proofwiki.org/wiki/Condition_for_3_Points_in_Plane_to_be_Collinear | [
"Collinear Points"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Collinear/Points",
"Definition:Determinant",
"Definition:Zero (Number)"
] | [
"Definition:Collinear/Points",
"Definition:Area",
"Area of Triangle in Determinant Form"
] |
proofwiki-18160 | Joachimsthal's Section-Formulae | Let $P = \tuple {x_1, y_1}$ and $Q = \tuple {x_2, y_2}$ be points in the Cartesian plane.
Let $R = \tuple {x, y}$ be a point on $PQ$ dividing $PQ$ in the ratio:
:$PR : RQ = l : m$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \dfrac {l x_2 + m x_1} {l + m}
}}
{{eqn | l = y
| r = \dfrac {l y_2 + m y_1} {l + m}
}}
... | :600px
Let the ordinates $PL$, $QM$ and $RN$ be constructed for $P$, $Q$ and $R$ respectively.
Then we have:
{{begin-eqn}}
{{eqn | l = OL
| r = x_1
}}
{{eqn | l = OM
| r = x_2
}}
{{eqn | l = ON
| r = x
}}
{{eqn | l = LP
| r = y_1
}}
{{eqn | l = MQ
| r = y_2
}}
{{eqn | l = NR
| r = y
... | Let $P = \tuple {x_1, y_1}$ and $Q = \tuple {x_2, y_2}$ be [[Definition:Point|points]] in the [[Definition:Cartesian Plane|Cartesian plane]].
Let $R = \tuple {x, y}$ be a [[Definition:Point|point]] on $PQ$ dividing $PQ$ in the [[Definition:Ratio|ratio]]:
:$PR : RQ = l : m$
Then:
{{begin-eqn}}
{{eqn | l = x
|... | :[[File:Joachimsthals-section-formulae.png|600px]]
Let the [[Definition:Ordinate|ordinates]] $PL$, $QM$ and $RN$ be constructed for $P$, $Q$ and $R$ respectively.
Then we have:
{{begin-eqn}}
{{eqn | l = OL
| r = x_1
}}
{{eqn | l = OM
| r = x_2
}}
{{eqn | l = ON
| r = x
}}
{{eqn | l = LP
| r =... | Joachimsthal's Section-Formulae | https://proofwiki.org/wiki/Joachimsthal's_Section-Formulae | https://proofwiki.org/wiki/Joachimsthal's_Section-Formulae | [
"Analytic Geometry"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Ratio"
] | [
"File:Joachimsthals-section-formulae.png",
"Definition:Ordinate"
] |
proofwiki-18161 | Value of Position-Ratio | Let $k$ denote the position-ratio of $R$.
Then:
:$k = \dfrac {PQ} {RQ} - 1$ | {{begin-eqn}}
{{eqn | l = k
| r = \dfrac {PR} {RQ}
| c =
}}
{{eqn | r = \dfrac {PQ + QR} {RQ}
| c =
}}
{{eqn | r = \dfrac {PQ} {RQ} - 1
| c =
}}
{{end-eqn}}
{{qed}} | Let $k$ denote the [[Definition:Position-Ratio of Point|position-ratio]] of $R$.
Then:
:$k = \dfrac {PQ} {RQ} - 1$ | {{begin-eqn}}
{{eqn | l = k
| r = \dfrac {PR} {RQ}
| c =
}}
{{eqn | r = \dfrac {PQ + QR} {RQ}
| c =
}}
{{eqn | r = \dfrac {PQ} {RQ} - 1
| c =
}}
{{end-eqn}}
{{qed}} | Value of Position-Ratio | https://proofwiki.org/wiki/Value_of_Position-Ratio | https://proofwiki.org/wiki/Value_of_Position-Ratio | [
"Position-Ratios"
] | [
"Definition:Position-Ratio of Point"
] | [] |
proofwiki-18162 | Span of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x is preserved under Differentiation wrt x | Let $a, b, x, \alpha_1, \alpha_2 \in \R$ be real numbers.
Let $f_1$ and $f_2$ be the real functions defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
| r = \map \exp {a x} \map \sin {b x}
}}
{{end-eqn}}
Let $\map... | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\alpha_1 f_1 + \alpha_2 f_2}
| r = \map {\dfrac \d {\d x} } {\alpha_1 \map \exp {a x} \map \cos {b x} + \alpha_2 \map \exp {a x} \map \sin {b x} }
}}
{{eqn | r = \alpha_1 a \map \exp {a x} \map \cos {b x} - \alpha_1 b \map \exp {a x} \map \sin {b x} + \alpha_2 a... | Let $a, b, x, \alpha_1, \alpha_2 \in \R$ be [[Definition:Real Number|real numbers]].
Let $f_1$ and $f_2$ be the [[Definition:Real Function|real functions]] defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
| r... | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\alpha_1 f_1 + \alpha_2 f_2}
| r = \map {\dfrac \d {\d x} } {\alpha_1 \map \exp {a x} \map \cos {b x} + \alpha_2 \map \exp {a x} \map \sin {b x} }
}}
{{eqn | r = \alpha_1 a \map \exp {a x} \map \cos {b x} - \alpha_1 b \map \exp {a x} \map \sin {b x} + \alpha_2 a... | Span of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x is preserved under Differentiation wrt x | https://proofwiki.org/wiki/Span_of_Exponential_of_a_x_by_Cosine_of_b_x_and_Exponential_of_a_x_by_Sine_of_b_x_is_preserved_under_Differentiation_wrt_x | https://proofwiki.org/wiki/Span_of_Exponential_of_a_x_by_Cosine_of_b_x_and_Exponential_of_a_x_by_Sine_of_b_x_is_preserved_under_Differentiation_wrt_x | [
"Real Analysis"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Continuous Real-Valued Function Space",
"Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Vector Space",
"Definition:G... | [] |
proofwiki-18163 | Position of Centroid of Triangle on Median | Let $\triangle ABC$ be a triangle.
Let $AL$, $BM$ and $CN$ be the medians of $\triangle ABC$ meeting at the centroid $G$ of $\triangle ABC$.
Then $G$ is $\dfrac 1 3$ of the way along $AL$ from $L$, and similarly for the other medians. | :520px
Let $\triangle ABC$ be embedded in a Cartesian plane such that $A = \tuple {x_1, y_1}$, $B = \tuple {x_2, y_2}$ and $C = \tuple {x_3, y_3}$.
The coordinates of $L$ are $\tuple {\dfrac {x_2 + x_3} 2, \dfrac {y_2 + y_3} 2}$.
Let $G$ be the point dividing $AL$ in the ratio $2 : 1$.
The coordinates of $G$ are $\tupl... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $AL$, $BM$ and $CN$ be the [[Definition:Median of Triangle|medians]] of $\triangle ABC$ meeting at the [[Definition:Centroid of Triangle|centroid]] $G$ of $\triangle ABC$.
Then $G$ is $\dfrac 1 3$ of the way along $AL$ from $L$, and similarly ... | :[[File:CentroidOfTriangle.png|520px]]
Let $\triangle ABC$ be embedded in a [[Definition:Cartesian Plane|Cartesian plane]] such that $A = \tuple {x_1, y_1}$, $B = \tuple {x_2, y_2}$ and $C = \tuple {x_3, y_3}$.
The [[Definition:Cartesian Coordinates|coordinates]] of $L$ are $\tuple {\dfrac {x_2 + x_3} 2, \dfrac {y_2... | Position of Centroid of Triangle on Median/Proof 1 | https://proofwiki.org/wiki/Position_of_Centroid_of_Triangle_on_Median | https://proofwiki.org/wiki/Position_of_Centroid_of_Triangle_on_Median/Proof_1 | [
"Position of Centroid of Triangle on Median",
"Centroids of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Median of Triangle",
"Definition:Centroid/Triangle",
"Definition:Median of Triangle"
] | [
"File:CentroidOfTriangle.png",
"Definition:Cartesian Plane",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Ratio",
"Definition:Cartesian Coordinate System",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Ratio",
"Definition:Point",
"Definiti... |
proofwiki-18164 | Position of Centroid of Triangle on Median | Let $\triangle ABC$ be a triangle.
Let $AL$, $BM$ and $CN$ be the medians of $\triangle ABC$ meeting at the centroid $G$ of $\triangle ABC$.
Then $G$ is $\dfrac 1 3$ of the way along $AL$ from $L$, and similarly for the other medians. | :520px
Let $\triangle ABC$ be a triangle.
Let $AL$, $BM$ and $CN$ be the medians of $\triangle ABC$.
Let the medians be concurrent at the centroid, $G$.
By the definition of median, the sides of $\triangle ABC$ are bisected.
{{begin-eqn}}
{{eqn | l = BL
| r = LC
| c = {{hypothesis}}
}}
{{eqn | l = BC
... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $AL$, $BM$ and $CN$ be the [[Definition:Median of Triangle|medians]] of $\triangle ABC$ meeting at the [[Definition:Centroid of Triangle|centroid]] $G$ of $\triangle ABC$.
Then $G$ is $\dfrac 1 3$ of the way along $AL$ from $L$, and similarly ... | :[[File:CentroidOfTriangle.png|520px]]
Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $AL$, $BM$ and $CN$ be the [[Definition:Median of Triangle|medians]] of $\triangle ABC$.
Let the [[Definition:Median of Triangle|medians]] be [[Definition:Concurrent|concurrent]] at the [[Definition:Centr... | Position of Centroid of Triangle on Median/Proof 2 | https://proofwiki.org/wiki/Position_of_Centroid_of_Triangle_on_Median | https://proofwiki.org/wiki/Position_of_Centroid_of_Triangle_on_Median/Proof_2 | [
"Position of Centroid of Triangle on Median",
"Centroids of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Median of Triangle",
"Definition:Centroid/Triangle",
"Definition:Median of Triangle"
] | [
"File:CentroidOfTriangle.png",
"Definition:Triangle (Geometry)",
"Definition:Median of Triangle",
"Definition:Median of Triangle",
"Definition:Concurrent",
"Definition:Centroid",
"Definition:Median of Triangle",
"Definition:Side",
"Definition:Bisection",
"Menelaus's Theorem",
"Definition:Distanc... |
proofwiki-18165 | Position of Centroid of Triangle on Median | Let $\triangle ABC$ be a triangle.
Let $AL$, $BM$ and $CN$ be the medians of $\triangle ABC$ meeting at the centroid $G$ of $\triangle ABC$.
Then $G$ is $\dfrac 1 3$ of the way along $AL$ from $L$, and similarly for the other medians. | :400px
By Medians of Triangle Form Six Triangles of Equal Area:
:these six triangles formed by the medians of $\triangle ABC$ have equal area:
* $\triangle AGN$
* $\triangle BGN$
* $\triangle BGL$
* $\triangle CGL$
* $\triangle CGM$
* $\triangle AGM$
{{WLOG}} consider one of the three medians of $\triangle ABC$, $AGL$.... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $AL$, $BM$ and $CN$ be the [[Definition:Median of Triangle|medians]] of $\triangle ABC$ meeting at the [[Definition:Centroid of Triangle|centroid]] $G$ of $\triangle ABC$.
Then $G$ is $\dfrac 1 3$ of the way along $AL$ from $L$, and similarly ... | :[[File:CentroidOfTriangle.png|400px]]
By [[Medians of Triangle Form Six Triangles of Equal Area]]:
:these six [[Definition:Triangle (Geometry)|triangles]] formed by the [[Definition:Median of Triangle|medians]] of $\triangle ABC$ have equal [[Definition:Area|area]]:
* $\triangle AGN$
* $\triangle BGN$
* $\triangle BG... | Position of Centroid of Triangle on Median/Proof 3 | https://proofwiki.org/wiki/Position_of_Centroid_of_Triangle_on_Median | https://proofwiki.org/wiki/Position_of_Centroid_of_Triangle_on_Median/Proof_3 | [
"Position of Centroid of Triangle on Median",
"Centroids of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Median of Triangle",
"Definition:Centroid/Triangle",
"Definition:Median of Triangle"
] | [
"File:CentroidOfTriangle.png",
"Medians of Triangle Form Six Triangles of Equal Area",
"Definition:Triangle (Geometry)",
"Definition:Median of Triangle",
"Definition:Area",
"Definition:Median of Triangle",
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Base",
"Definition:Polygon/V... |
proofwiki-18166 | Equation of Straight Line in Plane/Point-Slope Form/Parametric Form | Let $\LL$ be a straight line embedded in a cartesian plane, given in point-slope form as:
::$y - y_0 = \paren {x - x_0} \tan \psi$
where $\psi$ is the angle between $\LL$ and the $x$-axis.
Then $\LL$ can be expressed by the parametric equations:
:$\begin {cases} x = x_0 + t \cos \psi \\ y = y_0 + t \sin \psi \end {case... | Let $P_0$ be the point $\tuple {x_0, y_0}$.
Let $P$ be an arbitrary point on $\LL$.
Let $t$ be the distance from $P_0$ to $P$ measured as positive when in the positive $x$ direction.
The equation for $P$ is then:
{{begin-eqn}}
{{eqn | l = y - y_0
| r = \paren {x - x_0} \tan \psi
| c =
}}
{{eqn | ll= \leads... | Let $\LL$ be a [[Definition:Straight Line|straight line]] embedded in a [[Definition:Cartesian Plane|cartesian plane]], given in [[Equation of Straight Line in Plane/Point-Slope Form|point-slope form]] as:
::$y - y_0 = \paren {x - x_0} \tan \psi$
where $\psi$ is the [[Definition:Angle|angle]] between $\LL$ and the [[De... | Let $P_0$ be the [[Definition:Point|point]] $\tuple {x_0, y_0}$.
Let $P$ be an arbitrary [[Definition:Point|point]] on $\LL$.
Let $t$ be the [[Definition:Distance between Points|distance]] from $P_0$ to $P$ measured as [[Definition:Positive Real Number|positive]] when in the [[Definition:Positive Real Number|positive... | Equation of Straight Line in Plane/Point-Slope Form/Parametric Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Point-Slope_Form/Parametric_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Point-Slope_Form/Parametric_Form | [
"Equation of Straight Line in Plane/Point-Slope Form"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/Point-Slope Form",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Parametric Equation"
] | [
"Definition:Point",
"Definition:Point",
"Definition:Distance between Points",
"Definition:Positive/Real Number",
"Definition:Positive/Real Number",
"Definition:Axis/X-Axis"
] |
proofwiki-18167 | Equation of Straight Line in Plane/Two-Point Form/Parametric Form | Let $\LL$ be a straight line embedded in a cartesian plane, given in two-point form as:
::$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$
Then $\LL$ can be expressed by the parametric equations:
:$\begin {cases} x = x_1 + t \paren {x_2 - x_1} \\ y = y_1 + t \paren {y_2 - y_1} \end {cases}$ | Let $P = \tuple {x, y}$ be an arbitrary point on $\LL$.
Let $t = \dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$.
We then have:
{{begin-eqn}}
{{eqn | l = t
| r = \dfrac {x - x_1} {x_2 - x_1}
| c =
}}
{{eqn | ll= \leadsto
| l = x - x_1
| r = t \paren {x_2 - x_1}
| c =
}}
{{eq... | Let $\LL$ be a [[Definition:Straight Line|straight line]] embedded in a [[Definition:Cartesian Plane|cartesian plane]], given in [[Two-Point Form of Equation of Straight Line in Plane|two-point form]] as:
::$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$
Then $\LL$ can be expressed by the [[Definition:P... | Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] on $\LL$.
Let $t = \dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$.
We then have:
{{begin-eqn}}
{{eqn | l = t
| r = \dfrac {x - x_1} {x_2 - x_1}
| c =
}}
{{eqn | ll= \leadsto
| l = x - x_1
| r = t \paren {x_2 - ... | Equation of Straight Line in Plane/Two-Point Form/Parametric Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form/Parametric_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form/Parametric_Form | [
"Two-Point Form of Equation of Straight Line in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/Two-Point Form",
"Definition:Parametric Equation"
] | [
"Definition:Point"
] |
proofwiki-18168 | Equation of Straight Line in Plane/Two-Point Form/Determinant Form | Let $\LL$ be a straight line embedded in a Cartesian plane, given in two-point form as:
:$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$
Then $\LL$ can be expressed in the form:
:$\begin {vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end {vmatrix} = 0$ | {{begin-eqn}}
{{eqn | l = \frac {x - x_1} {x_2 - x_1}
| r = \frac {y - y_1} {y_2 - y_1}
}}
{{eqn | ll= \leadsto
| l = \paren {x - x_1} \paren {y_2 - y_1}
| r = \paren {x_2 - x_1} \paren {y - y_1}
}}
{{eqn | ll= \leadsto
| l = \paren {x - x_1} \paren {y_2 - y_1} - \paren {x_2 - x_1} \paren {y - y... | Let $\LL$ be a [[Definition:Straight Line|straight line]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]], given in [[Two-Point Form of Equation of Straight Line in Plane|two-point form]] as:
:$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$
Then $\LL$ can be expressed in the form:
:$\begin... | {{begin-eqn}}
{{eqn | l = \frac {x - x_1} {x_2 - x_1}
| r = \frac {y - y_1} {y_2 - y_1}
}}
{{eqn | ll= \leadsto
| l = \paren {x - x_1} \paren {y_2 - y_1}
| r = \paren {x_2 - x_1} \paren {y - y_1}
}}
{{eqn | ll= \leadsto
| l = \paren {x - x_1} \paren {y_2 - y_1} - \paren {x_2 - x_1} \paren {y - y... | Equation of Straight Line in Plane/Two-Point Form/Determinant Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form/Determinant_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form/Determinant_Form | [
"Two-Point Form of Equation of Straight Line in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/Two-Point Form"
] | [
"Determinant/Examples/Order 2",
"Determinant with Unit Element in Otherwise Zero Column",
"Multiple of Row Added to Row of Determinant",
"Determinant with Row Multiplied by Constant",
"Multiple of Row Added to Row of Determinant"
] |
proofwiki-18169 | Equation of Straight Line in Plane/Normal Form/Polar Form | Let $\LL$ be defined in normal form:
:$x \cos \alpha + y \sin \alpha = p$
Then $\LL$ can be presented in polar coordinates as:
:$r \map \cos {\theta - \alpha} = p$ | Let $O$ be the origin of the Cartesian plane and the pole of the corresponding polar frame.
Let $OX$ denote the polar axis, coincident with the $x$-axis.
Let $P$ be an arbitrary point on $\LL$, expressed in polar coordinates as $\polar {r, \theta}$.
Let $N$ be the point on $\LL$ where the normal to $\LL$ intersects $\L... | Let $\LL$ be defined in [[Equation of Straight Line in Plane/Normal Form|normal form]]:
:$x \cos \alpha + y \sin \alpha = p$
Then $\LL$ can be presented in [[Definition:Polar Coordinates|polar coordinates]] as:
:$r \map \cos {\theta - \alpha} = p$ | Let $O$ be the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian plane]] and the [[Definition:Pole (Polar Coordinates)|pole]] of the corresponding [[Definition:Polar Coordinates|polar frame]].
Let $OX$ denote the [[Definition:Polar Axis (Polar Coordinates)|polar axis]], coincident with the [[D... | Equation of Straight Line in Plane/Normal Form/Polar Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Normal_Form/Polar_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Normal_Form/Polar_Form | [
"Equations of Straight Lines in Plane"
] | [
"Equation of Straight Line in Plane/Normal Form",
"Definition:Polar Coordinates"
] | [
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Polar Coordinates/Pole",
"Definition:Polar Coordinates",
"Definition:Polar Coordinates/Polar Axis",
"Definition:Axis/X-Axis",
"Definition:Point",
"Definition:Polar Coordinates",
"Definition:Point",
"Definition:Normal"... |
proofwiki-18170 | Differentiation of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x wrt x as Nonsingular Matrix | Let $a, b, x \in \R$ be real numbers.
Suppose $a \ne 0 \ne b$.
Let $f_1$ and $f_2$ be the real functions defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
| r = \map \exp {a x} \map \sin {b x}
}}
{{end-eqn}}
Let ... | {{begin-eqn}}
{{eqn | l = \map D {f_1}
| r = \dfrac \d {\d x} \paren {\map \exp {a x} \map \cos {b x} }
}}
{{eqn | r = a \map \exp {a x} \map \cos {b x} - b \map \exp {a x} \map \sin {b x}
}}
{{eqn | r = a f_1 - b f_2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map D {f_2}
| r = \dfrac \d {\d x} \paren {\map ... | Let $a, b, x \in \R$ be [[Definition:Real Number|real numbers]].
Suppose $a \ne 0 \ne b$.
Let $f_1$ and $f_2$ be the [[Definition:Real Function|real functions]] defined as:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \map {f_1} x
| r = \map \exp {a x} \map \cos {b x}
}}
{{eqn | l = \map {f_2} x
... | {{begin-eqn}}
{{eqn | l = \map D {f_1}
| r = \dfrac \d {\d x} \paren {\map \exp {a x} \map \cos {b x} }
}}
{{eqn | r = a \map \exp {a x} \map \cos {b x} - b \map \exp {a x} \map \sin {b x}
}}
{{eqn | r = a f_1 - b f_2
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = \map D {f_2}
| r = \dfrac \d {\d x} \paren {\map... | Differentiation of Exponential of a x by Cosine of b x and Exponential of a x by Sine of b x wrt x as Nonsingular Matrix | https://proofwiki.org/wiki/Differentiation_of_Exponential_of_a_x_by_Cosine_of_b_x_and_Exponential_of_a_x_by_Sine_of_b_x_wrt_x_as_Nonsingular_Matrix | https://proofwiki.org/wiki/Differentiation_of_Exponential_of_a_x_by_Cosine_of_b_x_and_Exponential_of_a_x_by_Sine_of_b_x_wrt_x_as_Nonsingular_Matrix | [
"Derivatives involving Exponential Function",
"Derivatives involving Sine Function",
"Derivatives involving Cosine Function",
"Inverse Matrices"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Continuous Real-Valued Function Space",
"Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Vector Space",
"Definition:D... | [
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse",
"Definition:Nonsingular Matrix"
] |
proofwiki-18171 | Angle between Straight Lines in Plane/General Form | Let $L_1$ and $L_2$ be straight lines embedded in a cartesian plane, given in general form:
{{begin-eqn}}
{{eqn | q = L_1
| l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | q = L_2
| l = l_2 x + m_2 y + n_2
| r = 0
}}
{{end-eqn}}
Then the angle $\psi$ between $L_1$ and $L_2$ is given by:
:$\tan \psi =... | From the general equation for the straight line:
{{begin-eqn}}
{{eqn | q = L_1
| l = y
| r = -\dfrac {l_1} {m_1} x + \dfrac {n_1} {m_1}
}}
{{eqn | q = L_2
| l = y
| r = -\dfrac {l_2} {m_2} x + \dfrac {n_2} {m_2}
}}
{{end-eqn}}
Hence the slope of $L_1$ and $L_2$ are $-\dfrac {l_1} {m_1}$ and $-\d... | Let $L_1$ and $L_2$ be [[Definition:Straight Line|straight lines]] embedded in a [[Definition:Cartesian Plane|cartesian plane]], given in [[Equation of Straight Line in Plane/General Equation|general form]]:
{{begin-eqn}}
{{eqn | q = L_1
| l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | q = L_2
| l = l_2 ... | From the [[Equation of Straight Line in Plane/General Equation|general equation for the straight line]]:
{{begin-eqn}}
{{eqn | q = L_1
| l = y
| r = -\dfrac {l_1} {m_1} x + \dfrac {n_1} {m_1}
}}
{{eqn | q = L_2
| l = y
| r = -\dfrac {l_2} {m_2} x + \dfrac {n_2} {m_2}
}}
{{end-eqn}}
Hence the [... | Angle between Straight Lines in Plane/General Form | https://proofwiki.org/wiki/Angle_between_Straight_Lines_in_Plane/General_Form | https://proofwiki.org/wiki/Angle_between_Straight_Lines_in_Plane/General_Form | [
"Angle between Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Angle"
] | [
"Equation of Straight Line in Plane/General Equation",
"Definition:Slope/Straight Line",
"Definition:Angle",
"Angle between Straight Lines in Plane",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-18172 | Condition for Straight Lines in Plane to be Parallel/Slope Form | Let $L_1$ and $L_2$ be straight lines in the Cartesian plane.
Let the slope of $L_1$ and $L_2$ be $\mu_1$ and $\mu_2$ respectively.
Then $L_1$ is parallel to $L_2$ {{iff}}:
:$\mu_1 = \mu_2$ | Let $\psi$ be the angle between $L_1$ and $L_2$
When $L_1$ and $L_2$ are parallel:
:$\psi = 0$
by definition.
From Angle between Straight Lines in Plane:
:$\psi = \arctan \dfrac {m_1 - m_2} {1 + m_1 m_2}$
The result follows immediately.
{{qed}} | Let $L_1$ and $L_2$ be [[Definition:Straight Line|straight lines]] in the [[Definition:Cartesian Plane|Cartesian plane]].
Let the [[Definition:Slope of Straight Line|slope]] of $L_1$ and $L_2$ be $\mu_1$ and $\mu_2$ respectively.
Then $L_1$ is [[Definition:Parallel Lines|parallel]] to $L_2$ {{iff}}:
:$\mu_1 = \mu_2... | Let $\psi$ be the [[Definition:Angle|angle]] between $L_1$ and $L_2$
When $L_1$ and $L_2$ are [[Definition:Parallel Lines|parallel]]:
:$\psi = 0$
by definition.
From [[Angle between Straight Lines in Plane]]:
:$\psi = \arctan \dfrac {m_1 - m_2} {1 + m_1 m_2}$
The result follows immediately.
{{qed}} | Condition for Straight Lines in Plane to be Parallel/Slope Form/Proof 3 | https://proofwiki.org/wiki/Condition_for_Straight_Lines_in_Plane_to_be_Parallel/Slope_Form | https://proofwiki.org/wiki/Condition_for_Straight_Lines_in_Plane_to_be_Parallel/Slope_Form/Proof_3 | [
"Condition for Straight Lines in Plane to be Parallel"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Slope/Straight Line",
"Definition:Parallel (Geometry)/Lines"
] | [
"Definition:Angle",
"Definition:Parallel (Geometry)/Lines",
"Angle between Straight Lines in Plane"
] |
proofwiki-18173 | Condition for Straight Lines in Plane to be Perpendicular/General Equation | Let $L_1$ and $L_2$ be straight lines embedded in a cartesian plane, given in general form:
{{begin-eqn}}
{{eqn | q = L_1
| l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | q = L_2
| l = l_2 x + m_2 y + n_2
| r = 0
}}
{{end-eqn}}
Then $L_1$ is perpendicular to $L_2$ {{iff}}:
:$l_1 l_2 + m_1 m_2 = 0$ | From the general equation for the straight line:
{{begin-eqn}}
{{eqn | q = L_1
| l = y
| r = -\dfrac {l_1} {m_1} x + \dfrac {n_1} {m_1}
}}
{{eqn | q = L_2
| l = y
| r = -\dfrac {l_2} {m_2} x + \dfrac {n_2} {m_2}
}}
{{end-eqn}}
Hence the slope of $L_1$ and $L_2$ are $-\dfrac {l_1} {m_1}$ and $-\d... | Let $L_1$ and $L_2$ be [[Definition:Straight Line|straight lines]] embedded in a [[Definition:Cartesian Plane|cartesian plane]], given in [[Equation of Straight Line in Plane/General Equation|general form]]:
{{begin-eqn}}
{{eqn | q = L_1
| l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | q = L_2
| l = l_2 ... | From the [[Equation of Straight Line in Plane/General Equation|general equation for the straight line]]:
{{begin-eqn}}
{{eqn | q = L_1
| l = y
| r = -\dfrac {l_1} {m_1} x + \dfrac {n_1} {m_1}
}}
{{eqn | q = L_2
| l = y
| r = -\dfrac {l_2} {m_2} x + \dfrac {n_2} {m_2}
}}
{{end-eqn}}
Hence the [... | Condition for Straight Lines in Plane to be Perpendicular/General Equation | https://proofwiki.org/wiki/Condition_for_Straight_Lines_in_Plane_to_be_Perpendicular/General_Equation | https://proofwiki.org/wiki/Condition_for_Straight_Lines_in_Plane_to_be_Perpendicular/General_Equation | [
"Condition for Straight Lines in Plane to be Perpendicular"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Right Angle/Perpendicular"
] | [
"Equation of Straight Line in Plane/General Equation",
"Definition:Slope/Straight Line",
"Condition for Straight Lines in Plane to be Perpendicular/Slope Form"
] |
proofwiki-18174 | Condition for Straight Lines in Plane to be Perpendicular/General Equation/Corollary | Let $L$ be a straight line in the Cartesian plane.
Let $L$ be described by the general equation for the straight line:
:$l x + m y + n = 0$
Then the straight line $L'$ is perpendicular to $L$ {{iff}} $L'$ can be expressed in the form:
:$m x - l y = k$ | From the general equation for the straight line, $L$ can be expressed as:
:$y = -\dfrac l m x + \dfrac n m$
Hence the slope of $L$ is $-\dfrac l m$.
Let $L'$ be perpendicular to $L$.
From Condition for Straight Lines in Plane to be Perpendicular, the slope of $L'$ is $\dfrac m l$.
Hence $L'$ has the equation:
:$y = \df... | Let $L$ be a [[Definition:Straight Line|straight line]] in the [[Definition:Cartesian Plane|Cartesian plane]].
Let $L$ be described by the [[Equation of Straight Line in Plane/General Equation|general equation for the straight line]]:
:$l x + m y + n = 0$
Then the [[Definition:Straight Line|straight line]] $L'$ is [[... | From the [[Equation of Straight Line in Plane/General Equation|general equation for the straight line]], $L$ can be expressed as:
:$y = -\dfrac l m x + \dfrac n m$
Hence the [[Definition:Slope of Straight Line|slope]] of $L$ is $-\dfrac l m$.
Let $L'$ be [[Definition:Perpendicular|perpendicular]] to $L$.
From [[Con... | Condition for Straight Lines in Plane to be Perpendicular/General Equation/Corollary | https://proofwiki.org/wiki/Condition_for_Straight_Lines_in_Plane_to_be_Perpendicular/General_Equation/Corollary | https://proofwiki.org/wiki/Condition_for_Straight_Lines_in_Plane_to_be_Perpendicular/General_Equation/Corollary | [
"Condition for Straight Lines in Plane to be Perpendicular"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Line/Straight Line",
"Definition:Right Angle/Perpendicular"
] | [
"Equation of Straight Line in Plane/General Equation",
"Definition:Slope/Straight Line",
"Definition:Right Angle/Perpendicular",
"Condition for Straight Lines in Plane to be Perpendicular",
"Definition:Slope/Straight Line"
] |
proofwiki-18175 | Perpendicular Distance from Straight Line in Plane to Point/Normal Form | Let $\LL$ be a straight line in the Cartesian plane.
Let $\LL$ be expressed in normal form:
:$x \cos \alpha + y \sin \alpha = p$
Let $P$ be a point in the cartesian plane whose coordinates are given by:
:$P = \tuple {x_0, y_0}$
Then the perpendicular distance $d$ from $P$ to $\LL$ is given by:
:$\pm d = x_0 \cos \alpha... | First suppose that $P$ is on the opposite side of $\LL$ from the origin $O$.
Let $MP$ be the ordinate of $P$.
Let $N$ be the point of intersection between $\LL$ and the perpendicular through $O$.
Let $ON$ be produced to $N'$ where $PN'$ is the straight line through $P$ parallel to $\LL$.
:600px
We have that:
:$d = NN'$... | Let $\LL$ be a [[Definition:Straight Line|straight line]] in the [[Definition:Cartesian Plane|Cartesian plane]].
Let $\LL$ be expressed in [[Equation of Straight Line in Plane/Normal Form|normal form]]:
:$x \cos \alpha + y \sin \alpha = p$
Let $P$ be a [[Definition:Point|point]] in the [[Definition:Cartesian Plane|ca... | First suppose that $P$ is on the opposite side of $\LL$ from the [[Definition:Origin|origin]] $O$.
Let $MP$ be the [[Definition:Ordinate|ordinate]] of $P$.
Let $N$ be the [[Definition:Point|point]] of [[Definition:Intersection (Geometry)|intersection]] between $\LL$ and the [[Definition:Perpendicular|perpendicular]] ... | Perpendicular Distance from Straight Line in Plane to Point/Normal Form | https://proofwiki.org/wiki/Perpendicular_Distance_from_Straight_Line_in_Plane_to_Point/Normal_Form | https://proofwiki.org/wiki/Perpendicular_Distance_from_Straight_Line_in_Plane_to_Point/Normal_Form | [
"Perpendicular Distance from Straight Line in Plane to Point"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/Normal Form",
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Cartesian Coordinate System",
"Definition:Perpendicular Distance between Point and Straight Line",
"Definition:Coordinate Syste... | [
"Definition:Coordinate System/Origin",
"Definition:Ordinate",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Right Angle/Perpendicular",
"Definition:Production",
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"File:Distance-from-straight-line-normal-f... |
proofwiki-18176 | Sign of Half-Plane is Well-Defined | Let $\LL$ be a straight line embedded in a cartesian plane $\CC$, given by the equation:
:$l x + m y + n = 0$
Let $\HH_1$ and $\HH_2$ be the half-planes into which $\LL$ divides $\CC$.
Let the sign of a point $P = \tuple {x_1, y_1}$ in $\CC$ be defined as the sign of the expression $l x_1 + m y_1 + n$.
Then the sign of... | By definition of $\LL$, if $P$ is on $\LL$ then $l x_1 + m y_1 + n = 0$.
Similarly, if $P$ is not on $\LL$ then $l x_1 + m y_1 + n \ne 0$.
Let $P = \tuple {x_1, y_1}$ and $Q = \tuple {x_2, y_2}$ be two points not on $\LL$ such that the line $PQ$ intersects $\LL$ at $R = \tuple {x, y}$.
Let $PR : RQ = k$.
Then from Joac... | Let $\LL$ be a [[Definition:Straight Line|straight line]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] $\CC$, given by the [[Equation of Straight Line in Plane/General Equation|equation]]:
:$l x + m y + n = 0$
Let $\HH_1$ and $\HH_2$ be the [[Definition:Half-Plane|half-planes]] into which $\LL$ divides... | By definition of $\LL$, if $P$ is on $\LL$ then $l x_1 + m y_1 + n = 0$.
Similarly, if $P$ is not on $\LL$ then $l x_1 + m y_1 + n \ne 0$.
Let $P = \tuple {x_1, y_1}$ and $Q = \tuple {x_2, y_2}$ be two [[Definition:Point|points]] not on $\LL$ such that the [[Definition:Straight Line|line]] $PQ$ [[Definition:Intersec... | Sign of Half-Plane is Well-Defined | https://proofwiki.org/wiki/Sign_of_Half-Plane_is_Well-Defined | https://proofwiki.org/wiki/Sign_of_Half-Plane_is_Well-Defined | [
"Half-Planes"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Half-Plane",
"Definition:Sign of Number",
"Definition:Point",
"Definition:Sign of Number",
"Definition:Half-Plane/Sign",
"Definition:Well-Defined/Mapping",
"Definition:... | [
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Intersection (Geometry)",
"Joachimsthal's Section-Formulae",
"Definition:Sign of Number",
"Definition:Negative/Real Number",
"Definition:Position-Ratio of Point",
"Definition:Line/Segment",
"Definition:Half-Plane",
"Definition:Sign o... |
proofwiki-18177 | Autocovariance is Autocorrelation by Variance | Let $\map S z$ be a stochastic process giving rise to a time series $T$.
Let $S$ be weakly stationary of order $2$ or greater.
Let $\gamma_k$ denote the autocovariance coefficient of $S$ at lag $k$.
Let $\rho_k$ denote the autocorrelation coefficient of $S$ at lag $k$.
Then:
:$\gamma_k = \rho_k \sigma_z^2$
where $\sigm... | {{begin-eqn}}
{{eqn | l = \rho_k
| r = \dfrac {\expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} } } {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }
| c = {{Defof|Autocorrelation}}
}}
{{eqn | l = \gamma_k
| r = \expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} }
... | Let $\map S z$ be a [[Definition:Stochastic Process|stochastic process]] giving rise to a [[Definition:Time Series|time series]] $T$.
Let $S$ be [[Definition:Weakly Stationary Stochastic Process|weakly stationary of order $2$]] or greater.
Let $\gamma_k$ denote the [[Definition:Autocovariance Coefficient|autocovarian... | {{begin-eqn}}
{{eqn | l = \rho_k
| r = \dfrac {\expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} } } {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }
| c = {{Defof|Autocorrelation}}
}}
{{eqn | l = \gamma_k
| r = \expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} }
... | Autocovariance is Autocorrelation by Variance | https://proofwiki.org/wiki/Autocovariance_is_Autocorrelation_by_Variance | https://proofwiki.org/wiki/Autocovariance_is_Autocorrelation_by_Variance | [
"Autocovariance",
"Autocorrelation"
] | [
"Definition:Stochastic Process",
"Definition:Time Series",
"Definition:Weakly Stationary Stochastic Process",
"Definition:Autocovariance/Coefficient",
"Definition:Lag",
"Definition:Autocorrelation/Coefficient",
"Definition:Lag",
"Definition:Variance of Stochastic Process"
] | [
"Definition:Weakly Stationary Stochastic Process"
] |
proofwiki-18178 | Properties of Fourier Transform/Linearity | Let $a$ and $b$ be complex numbers.
Let $\map h x$ be a Lebesgue integrable function such that:
:$\map h x = a \map f x + b \map g x$
Then:
:$\map {\hat h} s = a \map {\hat f} s + b \map {\hat g} s$ | {{begin-eqn}}
{{eqn | l = \map {\hat h} \zeta
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} \paren {a \map f x + b \map g x} \rd x
| c =
}}
{{eqn | r = a \int_{-\infty}^\infty e^{-2 \pi i x s} \map f x \rd x + b \int_{-\infty}... | Let $a$ and $b$ be [[Definition:Complex Number|complex numbers]].
Let $\map h x$ be a [[Definition:Lebesgue Integrable Function|Lebesgue integrable function]] such that:
:$\map h x = a \map f x + b \map g x$
Then:
:$\map {\hat h} s = a \map {\hat f} s + b \map {\hat g} s$ | {{begin-eqn}}
{{eqn | l = \map {\hat h} \zeta
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} \paren {a \map f x + b \map g x} \rd x
| c =
}}
{{eqn | r = a \int_{-\infty}^\infty e^{-2 \pi i x s} \map f x \rd x + b \int_{-\infty}... | Properties of Fourier Transform/Linearity | https://proofwiki.org/wiki/Properties_of_Fourier_Transform/Linearity | https://proofwiki.org/wiki/Properties_of_Fourier_Transform/Linearity | [
"Fourier Transforms"
] | [
"Definition:Complex Number",
"Definition:Integrable Function/Lebesgue"
] | [
"Linear Combination of Integrals/Definite"
] |
proofwiki-18179 | Properties of Fourier Transform/Translation | Let $x_0$ be a real number.
Let $\map h x$ be a Lebesgue integrable function such that:
:$\map h x = \map f {x - x_0}$
Then:
:$\map {\hat h} s = e^{-2 \pi i x_0 s} \map {\hat f} s$ | {{begin-eqn}}
{{eqn | l = \map {\hat h} s
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map f {x - x_0 } \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty \paren {e^{-2 \pi i x_0 s} } e^{-2 \pi i \paren {x - x_0 } s} \map f {x... | Let $x_0$ be a [[Definition:Real Number|real number]].
Let $\map h x$ be a [[Definition:Lebesgue Integrable Function|Lebesgue integrable function]] such that:
:$\map h x = \map f {x - x_0}$
Then:
:$\map {\hat h} s = e^{-2 \pi i x_0 s} \map {\hat f} s$ | {{begin-eqn}}
{{eqn | l = \map {\hat h} s
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map f {x - x_0 } \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty \paren {e^{-2 \pi i x_0 s} } e^{-2 \pi i \paren {x - x_0 } s} \map f {x... | Properties of Fourier Transform/Translation | https://proofwiki.org/wiki/Properties_of_Fourier_Transform/Translation | https://proofwiki.org/wiki/Properties_of_Fourier_Transform/Translation | [
"Fourier Transforms"
] | [
"Definition:Real Number",
"Definition:Integrable Function/Lebesgue"
] | [
"Linear Combination of Integrals/Definite"
] |
proofwiki-18180 | Properties of Fourier Transform/Modulation | Let $s_0$ be a real number.
Let $\map h x$ be a Lebesgue integrable function such that:
:$\map h x = e^{2 \pi i x s_0} \map f x$
Then:
:$\map {\hat h} s = \map {\hat f} {s - s_0}$ | {{begin-eqn}}
{{eqn | l = \map {\hat h} s
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} e^{2 \pi i x s_0} \map f x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x \paren {s - s_0} } \map f x \rd x
| c = Pr... | Let $s_0$ be a [[Definition:Real Number|real number]].
Let $\map h x$ be a [[Definition:Lebesgue Integrable Function|Lebesgue integrable function]] such that:
:$\map h x = e^{2 \pi i x s_0} \map f x$
Then:
:$\map {\hat h} s = \map {\hat f} {s - s_0}$ | {{begin-eqn}}
{{eqn | l = \map {\hat h} s
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} e^{2 \pi i x s_0} \map f x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x \paren {s - s_0} } \map f x \rd x
| c = [[... | Properties of Fourier Transform/Modulation | https://proofwiki.org/wiki/Properties_of_Fourier_Transform/Modulation | https://proofwiki.org/wiki/Properties_of_Fourier_Transform/Modulation | [
"Fourier Transforms"
] | [
"Definition:Real Number",
"Definition:Integrable Function/Lebesgue"
] | [
"Exponent Combination Laws/Product of Powers"
] |
proofwiki-18181 | Properties of Fourier Transform/Scaling | Let $a$ be a non-zero real number.
Let $\map h x$ be a Lebesgue integrable function such that:
:$\map h x = \map f {a x}$
Then:
:$\map {\hat h} s = \dfrac 1 {\size a} \map {\hat f} {\dfrac s a}$ | {{begin-eqn}}
{{eqn | l = \map {\hat h} s
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map f {a x} \rd x
| c =
}}
{{eqn | r = \dfrac 1 {\size a} \int_{-\infty}^\infty e^{-2 \pi i \paren {a x} \paren {s / a} } \map f {a x} \m... | Let $a$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Real Number|real number]].
Let $\map h x$ be a [[Definition:Lebesgue Integrable Function|Lebesgue integrable function]] such that:
:$\map h x = \map f {a x}$
Then:
:$\map {\hat h} s = \dfrac 1 {\size a} \map {\hat f} {\dfrac s a}$ | {{begin-eqn}}
{{eqn | l = \map {\hat h} s
| r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map h x \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty e^{-2 \pi i x s} \map f {a x} \rd x
| c =
}}
{{eqn | r = \dfrac 1 {\size a} \int_{-\infty}^\infty e^{-2 \pi i \paren {a x} \paren {s / a} } \map f {a x} \m... | Properties of Fourier Transform/Scaling | https://proofwiki.org/wiki/Properties_of_Fourier_Transform/Scaling | https://proofwiki.org/wiki/Properties_of_Fourier_Transform/Scaling | [
"Fourier Transforms"
] | [
"Definition:Zero (Number)",
"Definition:Real Number",
"Definition:Integrable Function/Lebesgue"
] | [
"Exponent Combination Laws/Power of Power"
] |
proofwiki-18182 | Intersection of Straight Lines in General Form | Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, given by the equations:
{{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 point of intersection of $\LL_1$ and $\LL_2$ ha... | First note that by the parallel postulate $\LL_1$ and $\LL_2$ have a unique point of intersection {{iff}} they are not parallel.
From Condition for Straight Lines in Plane to be Parallel, $\LL_1$ and $\LL_2$ are parallel {{iff}} $l_1 m_2 = l_2 m_1$.
{{qed|lemma}}
Let the equations for $\LL_1$ and $\LL_2$ be given.
Let ... | Let $\LL_1$ and $\LL_2$ be [[Definition:Straight Line|straight lines]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] $\CC$, given by the [[Equation of Straight Line in Plane/General Equation|equations]]:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | lll = \L... | First note that by the [[Axiom:Parallel Postulate|parallel postulate]] $\LL_1$ and $\LL_2$ have a [[Definition:Unique|unique]] [[Definition:Point|point]] of [[Definition:Intersection (Geometry)|intersection]] {{iff}} they are not [[Definition:Parallel Lines|parallel]].
From [[Condition for Straight Lines in Plane to b... | Intersection of Straight Lines in General Form | https://proofwiki.org/wiki/Intersection_of_Straight_Lines_in_General_Form | https://proofwiki.org/wiki/Intersection_of_Straight_Lines_in_General_Form | [
"Straight Lines"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Cartesian Coordinate System",
"Definition:Point",
"Definition:Unique"
] | [
"Axiom:Parallel Postulate",
"Definition:Unique",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Condition for Straight Lines in Plane to be Parallel",
"Definition:Parallel (Geometry)/Lines",
"Definition:Point"
] |
proofwiki-18183 | Intersection of Straight Lines in General Form/Determinant Form | :$\dfrac x {\begin {vmatrix} m_1 & n_1 \\ m_2 & n_2 \end {vmatrix} } = \dfrac y {\begin {vmatrix} n_1 & l_1 \\ n_2 & l_2 \end {vmatrix} } = \dfrac 1 {\begin {vmatrix} l_1 & m_1 \\ l_2 & m_2 \end {vmatrix} }$
where $\begin {vmatrix} \cdot \end {vmatrix}$ denotes a determinant. | From Intersection of Straight Lines in General Form, the point of intersection of $\LL_1$ and $\LL_2$ has coordinates given by:
:$\dfrac x {m_1 n_2 - m_2 n_1} = \dfrac y {n_1 l_2 - n_2 l_1} = \dfrac 1 {l_1 m_2 - l_2 m_1}$
which exists and is unique {{iff}} $l_1 m_2 \ne l_2 m_1$.
The result follows by Determinant of Ord... | :$\dfrac x {\begin {vmatrix} m_1 & n_1 \\ m_2 & n_2 \end {vmatrix} } = \dfrac y {\begin {vmatrix} n_1 & l_1 \\ n_2 & l_2 \end {vmatrix} } = \dfrac 1 {\begin {vmatrix} l_1 & m_1 \\ l_2 & m_2 \end {vmatrix} }$
where $\begin {vmatrix} \cdot \end {vmatrix}$ denotes a [[Definition:Determinant|determinant]]. | From [[Intersection of Straight Lines in General Form]], the [[Definition:Point|point]] of [[Definition:Intersection (Geometry)|intersection]] of $\LL_1$ and $\LL_2$ has [[Definition:Cartesian Coordinates|coordinates]] given by:
:$\dfrac x {m_1 n_2 - m_2 n_1} = \dfrac y {n_1 l_2 - n_2 l_1} = \dfrac 1 {l_1 m_2 - l_2 m_... | Intersection of Straight Lines in General Form/Determinant Form | https://proofwiki.org/wiki/Intersection_of_Straight_Lines_in_General_Form/Determinant_Form | https://proofwiki.org/wiki/Intersection_of_Straight_Lines_in_General_Form/Determinant_Form | [
"Straight Lines"
] | [
"Definition:Determinant"
] | [
"Intersection of Straight Lines in General Form",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Cartesian Coordinate System",
"Definition:Unique",
"Determinant/Examples/Order 2",
"Category:Straight Lines"
] |
proofwiki-18184 | Fourier Transform of Normal Function | Let $\map f x$ be defined as $\sqrt \pi$ times the normal probability density function where $\mu = 0$ and $\sigma = \dfrac {\sqrt 2} 2$:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \dfrac {\sqrt {\pi} } {\dfrac {\sqrt 2} 2 \sqrt {2 \pi} } \map \exp {-\dfrac {\paren {x - 0}^2} {2 \paren {\dfrac {\sqrt 2} 2 }^2} }
... | 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^{-x^2} \rd x
| c =
}}
{{end-eqn}}
Taking the derivative with respect to $s$, we have:
{{begin-... | Let $\map f x$ be defined as $\sqrt \pi$ times the [[Definition:Normal Distribution|normal probability density function]] where $\mu = 0$ and $\sigma = \dfrac {\sqrt 2} 2$:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \dfrac {\sqrt {\pi} } {\dfrac {\sqrt 2} 2 \sqrt {2 \pi} } \map \exp {-\dfrac {\paren {x - 0}^2} {2... | 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^{-x^2} \rd x
| c =
}}
{{end-eqn}}
Taking ... | Fourier Transform of Normal Function | https://proofwiki.org/wiki/Fourier_Transform_of_Normal_Function | https://proofwiki.org/wiki/Fourier_Transform_of_Normal_Function | [
"Examples of Fourier Transforms"
] | [
"Definition:Normal Distribution",
"Definition:Fourier Transform/Real Function"
] | [
"Definition:Fourier Transform/Real Function",
"Definition:Derivative",
"Integration by Parts/Primitive",
"Solution to Separable Differential Equation",
"Definition:Primitive (Calculus)/Integration",
"Primitive of Function under its Derivative",
"Primitive of Power",
"Exponential of Natural Logarithm",... |
proofwiki-18185 | Multiples of Homogeneous Cartesian Coordinates represent Same Point | Let $\CC$ denote the Cartesian plane.
Let $P$ be an arbitrary point in $\CC$.
Let $P$ be expressed in homogeneous Cartesian coordinates as:
:$P = \tuple {X, Y, Z}$
Then $P$ can also be expressed as:
:$P = \tuple {\rho X, \rho Y, \rho Z}$
where $\rho \in \R$ is an arbitrary real number such that $\rho \ne 0$. | By definition of homogeneous Cartesian coordinates, $P$ can be expressed in conventional Cartesian coordinates as:
:$P = \tuple {x, y}$
where:
{{begin-eqn}}
{{eqn | l = x
| r = \dfrac X Z
}}
{{eqn | l = y
| r = \dfrac Y Z
}}
{{end-eqn}}
for arbitrary $Z$.
We have that:
{{begin-eqn}}
{{eqn | l = \dfrac X Z
... | Let $\CC$ denote the [[Definition:Cartesian Plane|Cartesian plane]].
Let $P$ be an arbitrary [[Definition:Point|point]] in $\CC$.
Let $P$ be expressed in [[Definition:Homogeneous Cartesian Coordinates|homogeneous Cartesian coordinates]] as:
:$P = \tuple {X, Y, Z}$
Then $P$ can also be expressed as:
:$P = \tuple {\r... | By definition of [[Definition:Homogeneous Cartesian Coordinates|homogeneous Cartesian coordinates]], $P$ can be expressed in conventional [[Definition:Cartesian Coordinates|Cartesian coordinates]] as:
:$P = \tuple {x, y}$
where:
{{begin-eqn}}
{{eqn | l = x
| r = \dfrac X Z
}}
{{eqn | l = y
| r = \dfrac Y ... | Multiples of Homogeneous Cartesian Coordinates represent Same Point | https://proofwiki.org/wiki/Multiples_of_Homogeneous_Cartesian_Coordinates_represent_Same_Point | https://proofwiki.org/wiki/Multiples_of_Homogeneous_Cartesian_Coordinates_represent_Same_Point | [
"Homogeneous Cartesian Coordinates"
] | [
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Real Number"
] | [
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Cartesian Coordinate System"
] |
proofwiki-18186 | Homogeneous Cartesian Coordinates represent Unique Finite Point | Let $\CC$ denote the Cartesian plane.
Let $P$ be an arbitrary point in $\CC$ which is not the point at infinity.
Let $Z \in \R_{\ne 0}$ be fixed.
Then $P$ can be expressed uniquely in homogeneous Cartesian coordinates as:
:$P = \tuple {X, Y, Z}$ | Let $P$ be expressed in (conventional) Cartesian coordinates as $\tuple {x, y}$.
As $Z$ is fixed and non-zero, there exists a unique $X \in \R$ and a unique $Y \in \R$ such that:
{{begin-eqn}}
{{eqn | l = x
| r = \dfrac X Z
}}
{{eqn | l = y
| r = \dfrac Y Z
}}
{{end-eqn}}
Then by definition of homogeneous C... | Let $\CC$ denote the [[Definition:Cartesian Plane|Cartesian plane]].
Let $P$ be an arbitrary [[Definition:Point|point]] in $\CC$ which is not the [[Definition:Point at Infinity|point at infinity]].
Let $Z \in \R_{\ne 0}$ be fixed.
Then $P$ can be expressed [[Definition:Unique|uniquely]] in [[Definition:Homogeneous C... | Let $P$ be expressed in (conventional) [[Definition:Cartesian Coordinates|Cartesian coordinates]] as $\tuple {x, y}$.
As $Z$ is fixed and non-[[Definition:Zero (Number)|zero]], there exists a [[Definition:Unique|unique]] $X \in \R$ and a [[Definition:Unique|unique]] $Y \in \R$ such that:
{{begin-eqn}}
{{eqn | l = x
... | Homogeneous Cartesian Coordinates represent Unique Finite Point | https://proofwiki.org/wiki/Homogeneous_Cartesian_Coordinates_represent_Unique_Finite_Point | https://proofwiki.org/wiki/Homogeneous_Cartesian_Coordinates_represent_Unique_Finite_Point | [
"Homogeneous Cartesian Coordinates"
] | [
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Point at Infinity",
"Definition:Unique",
"Definition:Homogeneous Cartesian Coordinates"
] | [
"Definition:Cartesian Coordinate System",
"Definition:Zero (Number)",
"Definition:Unique",
"Definition:Unique",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Unique"
] |
proofwiki-18187 | Equation of Straight Line in Plane/Homogeneous Cartesian Coordinates | A straight line $\LL$ is the set of all points $P$ in $\R^2$, where $P$ is described in homogeneous Cartesian coordinates as:
:$l X + m Y + n Z = 0$
where $l, m, n \in \R$ are given, and not both $l$ and $m$ are zero. | Let $P = \tuple {X, Y, Z}$ be a point on $L$ defined in homogeneous Cartesian coordinates.
Then by definition:
{{begin-eqn}}
{{eqn | l = x
| r = \dfrac X Z
}}
{{eqn | l = y
| r = \dfrac Y Z
}}
{{end-eqn}}
where $P = \tuple {x, y}$ described in conventional Cartesian coordinates.
Hence:
{{begin-eqn}}
{{eqn |... | A [[Definition:Straight Line|straight line]] $\LL$ is the [[Definition:Set|set]] of all [[Definition:Point|points]] $P$ in $\R^2$, where $P$ is described in [[Definition:Homogeneous Cartesian Coordinates|homogeneous Cartesian coordinates]] as:
:$l X + m Y + n Z = 0$
where $l, m, n \in \R$ are given, and not both $l$ an... | Let $P = \tuple {X, Y, Z}$ be a [[Definition:Point|point]] on $L$ defined in [[Definition:Homogeneous Cartesian Coordinates|homogeneous Cartesian coordinates]].
Then by definition:
{{begin-eqn}}
{{eqn | l = x
| r = \dfrac X Z
}}
{{eqn | l = y
| r = \dfrac Y Z
}}
{{end-eqn}}
where $P = \tuple {x, y}$ desc... | Equation of Straight Line in Plane/Homogeneous Cartesian Coordinates | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Homogeneous_Cartesian_Coordinates | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Homogeneous_Cartesian_Coordinates | [
"Equations of Straight Lines in Plane",
"Homogeneous Cartesian Coordinates"
] | [
"Definition:Line/Straight Line",
"Definition:Set",
"Definition:Point",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Zero (Number)"
] | [
"Definition:Point",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Cartesian Coordinate System"
] |
proofwiki-18188 | Intersection of Straight Lines in Homogeneous Cartesian Coordinate Form | Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$.
Let $\LL_1$ and $\LL_2$ be given in homogeneous Cartesian coordinates by the equations:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = l_1 X + m_1 Y + n_1 Z
| r = 0
}}
{{eqn | lll = \LL_2:
| l = l_2 X + m_2 Y + n_2 Z
| r = ... | First note that by the parallel postulate $\LL_1$ and $\LL_2$ have a unique point of intersection {{iff}} they are not parallel.
So, first let it be the case that $\LL_1$ and $\LL_2$ are not parallel.
Let the equations for $\LL_1$ and $\LL_2$ be given.
Let $P = \tuple {X, Y, Z}$ be the point on both $\LL_1$ and $\LL_2$... | Let $\LL_1$ and $\LL_2$ be [[Definition:Straight Line|straight lines]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] $\CC$.
Let $\LL_1$ and $\LL_2$ be given in [[Definition:Homogeneous Cartesian Coordinates|homogeneous Cartesian coordinates]] by the [[Equation of Straight Line in Plane/Homogeneous Carte... | First note that by the [[Axiom:Parallel Postulate|parallel postulate]] $\LL_1$ and $\LL_2$ have a [[Definition:Unique|unique]] [[Definition:Point|point]] of [[Definition:Intersection (Geometry)|intersection]] {{iff}} they are not [[Definition:Parallel Lines|parallel]].
So, first let it be the case that $\LL_1$ and $\L... | Intersection of Straight Lines in Homogeneous Cartesian Coordinate Form | https://proofwiki.org/wiki/Intersection_of_Straight_Lines_in_Homogeneous_Cartesian_Coordinate_Form | https://proofwiki.org/wiki/Intersection_of_Straight_Lines_in_Homogeneous_Cartesian_Coordinate_Form | [
"Straight Lines",
"Homogeneous Cartesian Coordinates"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Homogeneous Cartesian Coordinates",
"Equation of Straight Line in Plane/Homogeneous Cartesian Coordinates",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Unique",
"Definition:Homogeneous Cartesian Coordi... | [
"Axiom:Parallel Postulate",
"Definition:Unique",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Definition:Parallel (Geometry)/Lines",
"Definition:Point",
"Definition:Homogeneous Cartesian Coordinates",
"Definition:Cartesian Coordinate System",
"... |
proofwiki-18189 | Left Shift Operator on 2-Sequence Space is Continuous | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the $2$-sequence space with $2$-norm.
Let $L : \ell^2 \to \ell^2$ be the left shift operator.
Then $L$ is continuous on $\struct {\ell^2, \norm {\, \cdot \,}_2}$. | Let $\sequence {a_n}_{n \mathop \in \N} = \tuple {a_1, a_2, a_3, \ldots}$ be a $2$-sequence.
{{begin-eqn}}
{{eqn | l = \norm {\map L {\sequence {a_n}_{n \mathop \in \N} } }_2
| r = \norm { L \tuple {a_1, a_2, a_3, \ldots} }_2
}}
{{eqn | r = \norm {\tuple {a_2, a_3, a_4, \ldots} }_2
| c = {{defof|Left Sh... | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the [[P-Sequence Space with P-Norm forms Normed Vector Space|$2$-sequence space with $2$-norm]].
Let $L : \ell^2 \to \ell^2$ be the [[Definition:Left Shift Operator|left shift operator]].
Then $L$ is [[Definition:Continuous Mapping (Normed Vector Space)/Space/Definiti... | Let $\sequence {a_n}_{n \mathop \in \N} = \tuple {a_1, a_2, a_3, \ldots}$ be a [[Definition:P-Sequence Space|$2$-sequence]].
{{begin-eqn}}
{{eqn | l = \norm {\map L {\sequence {a_n}_{n \mathop \in \N} } }_2
| r = \norm { L \tuple {a_1, a_2, a_3, \ldots} }_2
}}
{{eqn | r = \norm {\tuple {a_2, a_3, a_4, \ldots} ... | Left Shift Operator on 2-Sequence Space is Continuous | https://proofwiki.org/wiki/Left_Shift_Operator_on_2-Sequence_Space_is_Continuous | https://proofwiki.org/wiki/Left_Shift_Operator_on_2-Sequence_Space_is_Continuous | [
"Operator Theory",
"Continuous Mappings"
] | [
"P-Sequence Space with P-Norm forms Normed Vector Space",
"Definition:Left Shift Operator",
"Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1"
] | [
"Definition:P-Sequence Space",
"Continuity of Linear Transformation/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1"
] |
proofwiki-18190 | Right Shift Operator on 2-Sequence Space is Continuous | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the $2$-sequence space with $2$-norm.
Let $R : \ell^2 \to \ell^2$ be the right shift operator.
Then $R$ is continuous on $\struct {\ell^2, \norm {\, \cdot \,}_2}$. | Let $\sequence {a_n}_{n \mathop \in \N} = \tuple {a_1, a_2, a_3, \ldots}$ be a $2$-sequence.
{{begin-eqn}}
{{eqn | l = \norm {\map R {\sequence {a_n}_{n \mathop \in \N} } }_2
| r = \norm { R \tuple {a_1, a_2, a_3, \ldots} }_2
}}
{{eqn | r = \norm {\tuple {0, a_1, a_2, \ldots} }_2
| c = {{defof|Right Shi... | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the [[P-Sequence Space with P-Norm forms Normed Vector Space|$2$-sequence space with $2$-norm]].
Let $R : \ell^2 \to \ell^2$ be the [[Definition:Right Shift Operator|right shift operator]].
Then $R$ is [[Definition:Continuous Mapping (Normed Vector Space)/Space/Defini... | Let $\sequence {a_n}_{n \mathop \in \N} = \tuple {a_1, a_2, a_3, \ldots}$ be a [[Definition:P-Sequence Space|$2$-sequence]].
{{begin-eqn}}
{{eqn | l = \norm {\map R {\sequence {a_n}_{n \mathop \in \N} } }_2
| r = \norm { R \tuple {a_1, a_2, a_3, \ldots} }_2
}}
{{eqn | r = \norm {\tuple {0, a_1, a_2, \ldots} }_... | Right Shift Operator on 2-Sequence Space is Continuous | https://proofwiki.org/wiki/Right_Shift_Operator_on_2-Sequence_Space_is_Continuous | https://proofwiki.org/wiki/Right_Shift_Operator_on_2-Sequence_Space_is_Continuous | [
"Operator Theory",
"Continuous Mappings"
] | [
"P-Sequence Space with P-Norm forms Normed Vector Space",
"Definition:Right Shift Operator",
"Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1"
] | [
"Definition:P-Sequence Space",
"Continuity of Linear Transformation/Normed Vector Space",
"Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1"
] |
proofwiki-18191 | Point at Infinity of Intersection of Parallel Lines | Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$ such that $\LL_1$ and $\LL_2$ are parallel.
By Condition for Straight Lines in Plane to be Parallel, $\LL_1$ and $\LL_2$ can be expressed as the general equations:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = l x + m y + n_1
| r = 0
}... | Let $\LL_1$ be expressed in the form:
:$l x + m y + n = 0$
Hence let $\LL_2$ be expressed in the form:
:$l x + m y + k n = 0$
where $k \ne 1$.
Let their point of intersection be expressed in homogeneous Cartesian coordinates as $\tuple {X, Y, Z}$
Then:
{{begin-eqn}}
{{eqn | l = \tuple {X, Y, Z}
| r = \tuple {m n ... | Let $\LL_1$ and $\LL_2$ be [[Definition:Straight Line|straight lines]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] $\CC$ such that $\LL_1$ and $\LL_2$ are [[Definition:Parallel Lines|parallel]].
By [[Condition for Straight Lines in Plane to be Parallel]], $\LL_1$ and $\LL_2$ can be expressed as the [[... | Let $\LL_1$ be expressed in the form:
:$l x + m y + n = 0$
Hence let $\LL_2$ be expressed in the form:
:$l x + m y + k n = 0$
where $k \ne 1$.
Let their [[Definition:Point|point]] of [[Definition:Intersection (Geometry)|intersection]] be expressed in [[Definition:Homogeneous Cartesian Coordinates|homogeneous Cartesi... | Point at Infinity of Intersection of Parallel Lines | https://proofwiki.org/wiki/Point_at_Infinity_of_Intersection_of_Parallel_Lines | https://proofwiki.org/wiki/Point_at_Infinity_of_Intersection_of_Parallel_Lines | [
"Point at Infinity",
"Parallel Lines"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Parallel (Geometry)/Lines",
"Condition for Straight Lines in Plane to be Parallel",
"Equation of Straight Line in Plane/General Equation",
"Definition:Point at Infinity",
"Definition:Homogeneous Cartesian Coordinates"
] | [
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Homogeneous Cartesian Coordinates"
] |
proofwiki-18192 | Intersection of Straight Line in Homogeneous Cartesian Coordinates with Axes | Let $\LL$ be a straight line embedded in a cartesian plane $\CC$.
Let $\LL$ be given in homogeneous Cartesian coordinates by the equation:
:$l X + m Y + n Z = 0$
such that $l$ and $m$ are not both zero.
Then $\LL$ intersects:
:the $x$-axis $Y = 0$ at the point $\tuple {-n, 0, l}$
:the $y$-axis $X = 0$ at the point $\tu... | The intersection of $\LL$ with the $x$-axis is the point $\tuple {X, Y, Z}$ satisfied by:
{{begin-eqn}}
{{eqn | l = l X + m Y + n Z
| r = 0
| c =
}}
{{eqn | l = Y
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = l X + n Z
| r = 0
| c =
}}
{{end-eqn}}
which is satisfied by setting... | Let $\LL$ be a [[Definition:Straight Line|straight line]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] $\CC$.
Let $\LL$ be given in [[Definition:Homogeneous Cartesian Coordinates|homogeneous Cartesian coordinates]] by the [[Equation of Straight Line in Plane/Homogeneous Cartesian Coordinates|equation]]... | The [[Definition:Intersection (Geometry)|intersection]] of $\LL$ with the [[Definition:X-Axis|$x$-axis]] is the point $\tuple {X, Y, Z}$ satisfied by:
{{begin-eqn}}
{{eqn | l = l X + m Y + n Z
| r = 0
| c =
}}
{{eqn | l = Y
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = l X + n Z
| r... | Intersection of Straight Line in Homogeneous Cartesian Coordinates with Axes | https://proofwiki.org/wiki/Intersection_of_Straight_Line_in_Homogeneous_Cartesian_Coordinates_with_Axes | https://proofwiki.org/wiki/Intersection_of_Straight_Line_in_Homogeneous_Cartesian_Coordinates_with_Axes | [
"Homogeneous Cartesian Coordinates"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Homogeneous Cartesian Coordinates",
"Equation of Straight Line in Plane/Homogeneous Cartesian Coordinates",
"Definition:Zero (Number)",
"Definition:Intersection (Geometry)",
"Definition:Axis/X-Axis",
"Definition:Point",
"Defi... | [
"Definition:Intersection (Geometry)",
"Definition:Axis/X-Axis",
"Definition:Intersection (Geometry)",
"Definition:Axis/Y-Axis"
] |
proofwiki-18193 | Equation of Straight Line through Intersection of Two Straight Lines | Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed using the general equations:
{{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 $\LL_3$ be a third straight line... | Let $P = \tuple {x, y}$ be the point of intersection of $\LL_1$ and $\LL_2$.
We have that:
{{begin-eqn}}
{{eqn | l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | l = l_2 x + m_2 y + n_2
| r = 0
}}
{{eqn | ll= \leadsto
| l = k \paren {l_2 x + m_2 y + n_2}
| r = 0
}}
{{eqn | ll= \leadsto
| l = \pa... | Let $\LL_1$ and $\LL_2$ be [[Definition:Straight Line|straight lines]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] $\CC$, expressed using the [[Equation of Straight Line in Plane/General Equation|general equations]]:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = l_1 x + m_1 y + n_1
| r = 0
}}
{... | Let $P = \tuple {x, y}$ be the [[Definition:Point|point]] of [[Definition:Intersection (Geometry)|intersection]] of $\LL_1$ and $\LL_2$.
We have that:
{{begin-eqn}}
{{eqn | l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | l = l_2 x + m_2 y + n_2
| r = 0
}}
{{eqn | ll= \leadsto
| l = k \paren {l_2 x + m_2 ... | Equation of Straight Line through Intersection of Two Straight Lines | https://proofwiki.org/wiki/Equation_of_Straight_Line_through_Intersection_of_Two_Straight_Lines | https://proofwiki.org/wiki/Equation_of_Straight_Line_through_Intersection_of_Two_Straight_Lines | [
"Equations of Straight Lines in Plane",
"Equation of Straight Line through Intersection of Two Straight Lines"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Line/Straight Line",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Equation of Straight Line in Plane/General Equation"
] | [
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Point",
"Definition:Real Number",
"Equation of Straight Line in Plane/General Equation"
] |
proofwiki-18194 | Pencil of Straight Lines through Intersection of Two Straight Lines | Let $u = l_1 x + m_1 y + n_1$.
Let $v = l_2 x + m_2 y + n_2$.
Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed using the general equations:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = u
| r = 0
}}
{{eqn | lll = \LL_2:
| l = v
| r = 0
}}
{{end-eqn}}
The pencil... | Let $\LL$ denote an arbitrary straight line through the point of intersection of $\LL_1$ and $\LL_2$.
From Equation of Straight Line through Intersection of Two Straight Lines, $\LL$ can be given by an equation of the form:
:$u + k v = 0$
It remains to be seen that the complete pencil of lines through the point of inte... | Let $u = l_1 x + m_1 y + n_1$.
Let $v = l_2 x + m_2 y + n_2$.
Let $\LL_1$ and $\LL_2$ be [[Definition:Straight Line|straight lines]] embedded in a [[Definition:Cartesian Plane|cartesian plane]] $\CC$, expressed using the [[Equation of Straight Line in Plane/General Equation|general equations]]:
{{begin-eqn}}
{{eqn ... | Let $\LL$ denote an arbitrary [[Definition:Straight Line|straight line]] through the [[Definition:Point|point]] of [[Definition:Intersection (Geometry)|intersection]] of $\LL_1$ and $\LL_2$.
From [[Equation of Straight Line through Intersection of Two Straight Lines]], $\LL$ can be given by an equation of the form:
:... | Pencil of Straight Lines through Intersection of Two Straight Lines | https://proofwiki.org/wiki/Pencil_of_Straight_Lines_through_Intersection_of_Two_Straight_Lines | https://proofwiki.org/wiki/Pencil_of_Straight_Lines_through_Intersection_of_Two_Straight_Lines | [
"Pencils",
"Straight Lines"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Pencil/Straight Lines",
"Definition:Point",
"Definition:Intersection (Geometry)"
] | [
"Definition:Line/Straight Line",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Equation of Straight Line through Intersection of Two Straight Lines",
"Definition:Pencil/Straight Lines",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Real Number",
"Definition:Slope... |
proofwiki-18195 | Condition for Concurrency of Three Straight Lines | Let $3$ straight lines $\LL_1$, $\LL_2$ and $\LL_3$ be embedded in a cartesian plane $\CC$, expressed using the general equations:
{{begin-eqn}}
{{eqn | l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | l = l_2 x + m_2 y + n_2
| r = 0
}}
{{eqn | l = l_3 x + m_3 y + n_3
| r = 0
}}
{{end-eqn}}
Then $\LL_1$, $\... | === Necessary Condition ===
Let $\LL_1$, $\LL_2$ and $\LL_3$ be concurrent.
From Equation of Straight Line through Intersection of Two Straight Lines, $\LL_1$, $\LL_2$ and $\LL_3$ are concurrent {{iff}} $\LL_3$ has an equation of the form:
:$\paren {l_1 x + m_1 y + n_1} - k \paren {l_2 x + m_2 y + n_2} = 0$
That is:
:$... | Let $3$ [[Definition:Straight Line|straight lines]] $\LL_1$, $\LL_2$ and $\LL_3$ be embedded in a [[Definition:Cartesian Plane|cartesian plane]] $\CC$, expressed using the [[Equation of Straight Line in Plane/General Equation|general equations]]:
{{begin-eqn}}
{{eqn | l = l_1 x + m_1 y + n_1
| r = 0
}}
{{eqn | l... | === Necessary Condition ===
Let $\LL_1$, $\LL_2$ and $\LL_3$ be [[Definition:Concurrent Lines|concurrent]].
From [[Equation of Straight Line through Intersection of Two Straight Lines]], $\LL_1$, $\LL_2$ and $\LL_3$ are [[Definition:Concurrent Lines|concurrent]] {{iff}} $\LL_3$ has an [[Equation of Straight Line in P... | Condition for Concurrency of Three Straight Lines | https://proofwiki.org/wiki/Condition_for_Concurrency_of_Three_Straight_Lines | https://proofwiki.org/wiki/Condition_for_Concurrency_of_Three_Straight_Lines | [
"Straight Lines",
"Concurrency"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Concurrent Lines",
"Definition:Determinant"
] | [
"Definition:Concurrent Lines",
"Equation of Straight Line through Intersection of Two Straight Lines",
"Definition:Concurrent Lines",
"Equation of Straight Line in Plane/General Equation",
"Determinant of Transpose",
"Multiple of Column Added to Column of Determinant",
"Definition:Matrix/Column",
"Def... |
proofwiki-18196 | Bisectors of Angles between Two Straight Lines/Normal Form | Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed in normal form as:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = x \cos \alpha + y \sin \alpha
| r = p
}}
{{eqn | lll = \LL_2:
| l = x \cos \beta + y \sin \beta
| r = q
}}
{{end-eqn}}
The angle bisectors of the a... | Let $A'SA$ and $B'SB$ be the straight lines $\LL_1$ and $\LL_2$ respectively, intersecting at the point $S$.
Let $P = \tuple {x, y}$ be an arbitrary point on either of the angle bisectors of $\angle ASB$ or $\angle BSA'$.
:500px
Drop perpendiculars $PM$ from $P$ to $SA$ and $PN$ from $P$ to $SB$.
Because:
:$\angle PSM ... | 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/Normal Form|normal form]] as:
{{begin-eqn}}
{{eqn | lll = \LL_1:
| l = x \cos \alpha + y \sin \alpha
| r = p
}}
{{eqn ... | Let $A'SA$ and $B'SB$ be the [[Definition:Straight Line|straight lines]] $\LL_1$ and $\LL_2$ respectively, [[Definition:Intersection (Geometry)|intersecting]] at the [[Definition:Point|point]] $S$.
Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] on either of the [[Definition:Angle Bisector|angle bis... | Bisectors of Angles between Two Straight Lines/Normal Form | https://proofwiki.org/wiki/Bisectors_of_Angles_between_Two_Straight_Lines/Normal_Form | https://proofwiki.org/wiki/Bisectors_of_Angles_between_Two_Straight_Lines/Normal_Form | [
"Angle Bisectors"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/Normal Form",
"Definition:Angle Bisector",
"Definition:Angle",
"Definition:Point",
"Definition:Intersection (Geometry)"
] | [
"Definition:Line/Straight Line",
"Definition:Intersection (Geometry)",
"Definition:Point",
"Definition:Point",
"Definition:Angle Bisector",
"File:Bisectors-of-angles.png",
"Definition:Right Angle/Perpendicular",
"Definition:Right Angle/Perpendicular",
"Equation of Straight Line in Plane/Normal Form"... |
proofwiki-18197 | Fourier Transform of 1 | Let:
:$\map f x = 1$
Then:
:$\map {\hat f} s = \map \delta s$
where $\map {\hat f} s$ is the Fourier transform of $\map f x$. | 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} 1 \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\infty \paren {\map \cos {2 \pi s x} - i \map \sin {... | Let:
:$\map f x = 1$
Then:
:$\map {\hat f} s = \map \delta s$
where $\map {\hat f} s$ is the [[Definition:Fourier Transform of Real Function|Fourier transform]] of $\map f x$. | 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} 1 \rd x
| c =
}}
{{eqn | r = \int_{-\infty}^\... | Fourier Transform of 1 | https://proofwiki.org/wiki/Fourier_Transform_of_1 | https://proofwiki.org/wiki/Fourier_Transform_of_1 | [
"Examples of Fourier Transforms"
] | [
"Definition:Fourier Transform/Real Function"
] | [
"Definition:Fourier Transform/Real Function",
"Euler's Formula/Corollary",
"Linear Combination of Integrals/Definite",
"Primitive of Sine Function",
"Primitive of Cosine Function",
"Cosine of Conjugate Angle",
"Sine of Conjugate Angle"
] |
proofwiki-18198 | Integral of Dirac Delta Function over Reals | Let $\map \delta x$ denote the Dirac delta function.
Then:
:$\ds \int_{-\infty}^{+\infty} \map \delta x \rd x = 1$ | We have that:
:$\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin {cases} 0 & : x < -\epsilon \\ \dfrac 1 {2 \epsilon} & : -\epsilon \le x \le \epsilon \\ 0 & : x > \epsilon \end {cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_{-\infty}^{+\infty} \map {F_... | Let $\map \delta x$ denote the [[Definition:Dirac Delta Function|Dirac delta function]].
Then:
:$\ds \int_{-\infty}^{+\infty} \map \delta x \rd x = 1$ | We have that:
:$\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin {cases} 0 & : x < -\epsilon \\ \dfrac 1 {2 \epsilon} & : -\epsilon \le x \le \epsilon \\ 0 & : x > \epsilon \end {cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_{-\infty}^{+\infty} \m... | Integral of Dirac Delta Function over Reals | https://proofwiki.org/wiki/Integral_of_Dirac_Delta_Function_over_Reals | https://proofwiki.org/wiki/Integral_of_Dirac_Delta_Function_over_Reals | [
"Dirac Delta Function"
] | [
"Definition:Dirac Delta Function"
] | [
"Integral of Constant/Definite"
] |
proofwiki-18199 | Integral of Dirac Delta Function by Continuous Function over Reals | Let $\map \delta x$ denote the Dirac delta function.
Let $g$ be a continuous real function.
Then:
:$\ds \int_{- \infty}^{+ \infty} \map \delta x \map g x \rd x = \map g 0$ | We have that:
:$\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin {cases} 0 & : x < -\epsilon \\ \dfrac 1 {2 \epsilon} & : -\epsilon \le x \le \epsilon \\ 0 & : x > \epsilon \end {cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_{- \infty}^{+ \infty} \map {... | 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:
:$\ds \int_{- \infty}^{+ \infty} \map \delta x \map g x \rd x = \map g 0$ | We have that:
:$\map \delta x = \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
where:
:$\map {F_\epsilon} x = \begin {cases} 0 & : x < -\epsilon \\ \dfrac 1 {2 \epsilon} & : -\epsilon \le x \le \epsilon \\ 0 & : x > \epsilon \end {cases}$
We have that:
{{begin-eqn}}
{{eqn | l = \int_{- \infty}^{+ \infty} ... | Integral of Dirac Delta Function by Continuous Function over Reals | https://proofwiki.org/wiki/Integral_of_Dirac_Delta_Function_by_Continuous_Function_over_Reals | https://proofwiki.org/wiki/Integral_of_Dirac_Delta_Function_by_Continuous_Function_over_Reals | [
"Dirac Delta Function"
] | [
"Definition:Dirac Delta Function",
"Definition:Continuous Real Function"
] | [
"Integral of Constant/Definite",
"Darboux's Theorem",
"Definition:Maximal/Element",
"Definition:Minimal/Element",
"Squeeze Theorem",
"Category:Dirac Delta Function"
] |
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