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-11200 | Linear Second Order ODE/y'' - 2 y' + y = 2 x | The second order ODE:
:$(1): \quad y' ' - 2 y' + y = 2 x$
has the general solution:
:$y = C_1 e^x + C_2 x e^x + 2 x + 4$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = 1$
:$\map R x = 2 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y' ' - 2 y' + y = 0$
From Linear Seco... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 2 y' + y = 2 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 x e^x + 2 x + 4$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = 1$
:$\map R x = 2 x$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homogeneous Linear... | Linear Second Order ODE/y'' - 2 y' + y = 2 x/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_2_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_2_x/Proof_1 | [
"Examples of Constant Coefficient LSOODEs",
"Linear Second Order ODE/y'' - 2 y' + y = 2 x"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 2 y' + y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"De... |
proofwiki-11201 | Linear Second Order ODE/y'' - 2 y' + y = 2 x | The second order ODE:
:$(1): \quad y' ' - 2 y' + y = 2 x$
has the general solution:
:$y = C_1 e^x + C_2 x e^x + 2 x + 4$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = 1$
:$\map R x = 2 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y' ' - 2 y' + y = 0$
From Linear Seco... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 2 y' + y = 2 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 x e^x + 2 x + 4$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = 1$
:$\map R x = 2 x$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homogeneous Linear... | Linear Second Order ODE/y'' - 2 y' + y = 2 x/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_2_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_2_x/Proof_2 | [
"Examples of Constant Coefficient LSOODEs",
"Linear Second Order ODE/y'' - 2 y' + y = 2 x"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 2 y' + y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"De... |
proofwiki-11202 | Linear Second Order ODE/y'' - 2 y' + y = 2 x | The second order ODE:
:$(1): \quad y' ' - 2 y' + y = 2 x$
has the general solution:
:$y = C_1 e^x + C_2 x e^x + 2 x + 4$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = 1$
:$\map R x = 2 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y' ' - 2 y... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 2 y' + y = 2 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + C_2 x e^x + 2 x + 4$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = 1$
:$\map R x = 2 x$
First we establish the solu... | Linear Second Order ODE/y'' - 2 y' + y = 2 x/Proof 3 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_2_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_+_y_=_2_x/Proof_3 | [
"Examples of Constant Coefficient LSOODEs",
"Linear Second Order ODE/y'' - 2 y' + y = 2 x"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 2 y' + y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Method of Undete... |
proofwiki-11203 | Linear Second Order ODE/y'' - y' - 6 y = exp -x | The second order ODE:
:$(1): \quad y'' - y' - 6 y = e^{-x}$
has the general solution:
:$y = C_1 e^{3 x} + C_2 e^{-2 x} - \dfrac {e^{-x} } 4$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -1$
:$q = -6$
:$\map R x = e^{-x}$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' - y' - 6 y = 0... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - y' - 6 y = e^{-x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{3 x} + C_2 e^{-2 x} - \dfrac {e^{-x} } 4$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -1$
:$q = -6$
:$\map R x = e^{-x}$
First we establish the s... | Linear Second Order ODE/y'' - y' - 6 y = exp -x/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y'_-_6_y_=_exp_-x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y'_-_6_y_=_exp_-x/Proof_1 | [
"Examples of Constant Coefficient LSOODEs",
"Linear Second Order ODE/y'' - y' - 6 y = exp -x"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - y' - 6 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Method of Undete... |
proofwiki-11204 | Linear Second Order ODE/y'' - y' - 6 y = exp -x | The second order ODE:
:$(1): \quad y'' - y' - 6 y = e^{-x}$
has the general solution:
:$y = C_1 e^{3 x} + C_2 e^{-2 x} - \dfrac {e^{-x} } 4$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -1$
:$q = -6$
:$\map R x = e^{-x}$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' - y' - 6 y = 0$
From Linear Second Order ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - y' - 6 y = e^{-x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{3 x} + C_2 e^{-2 x} - \dfrac {e^{-x} } 4$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = -1$
:$q = -6$
:$\map R x = e^{-x}$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homogeneous Lin... | Linear Second Order ODE/y'' - y' - 6 y = exp -x/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y'_-_6_y_=_exp_-x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y'_-_6_y_=_exp_-x/Proof_2 | [
"Examples of Constant Coefficient LSOODEs",
"Linear Second Order ODE/y'' - y' - 6 y = exp -x"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - y' - 6 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"De... |
proofwiki-11205 | Linear Second Order ODE/y'' - y' - 6 y = exp -x | The second order ODE:
:$(1): \quad y'' - y' - 6 y = e^{-x}$
has the general solution:
:$y = C_1 e^{3 x} + C_2 e^{-2 x} - \dfrac {e^{-x} } 4$ | Taking Laplace transforms:
:$\laptrans {y'' - y' - 6 y} = \laptrans {e^{-x} }$
We have:
{{begin-eqn}}
{{eqn | l = \laptrans {y'' - y' - 6 y}
| r = \laptrans {y''} - \laptrans {y'} - 6 \laptrans y
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = s^2 \laptrans y - s \map y 0 - \map {y'} 0 - \paren {s \la... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' - y' - 6 y = e^{-x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{3 x} + C_2 e^{-2 x} - \dfrac {e^{-x} } 4$ | Taking [[Definition:Laplace Transform|Laplace transforms]]:
:$\laptrans {y'' - y' - 6 y} = \laptrans {e^{-x} }$
We have:
{{begin-eqn}}
{{eqn | l = \laptrans {y'' - y' - 6 y}
| r = \laptrans {y''} - \laptrans {y'} - 6 \laptrans y
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = s^2 \laptrans y - ... | Linear Second Order ODE/y'' - y' - 6 y = exp -x/Proof 3 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y'_-_6_y_=_exp_-x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_y'_-_6_y_=_exp_-x/Proof_3 | [
"Examples of Constant Coefficient LSOODEs",
"Linear Second Order ODE/y'' - y' - 6 y = exp -x"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Laplace Transform",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential",
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential"
] |
proofwiki-11206 | Filtered iff Finite Subsets have Lower Bounds | Let $\struct {S, \precsim}$ be a preordered set.
Let $H$ be a non-empty subset of $S$.
Then $H$ is filtered {{iff}}:
:for every finite subset $A$ of $H$:
::$\exists h \in H: \forall a \in A: h \precsim a$ | This follows by {{mutatis}} of the proof of Directed iff Finite Subsets have Upper Bounds.
{{qed}} | Let $\struct {S, \precsim}$ be a [[Definition:Preordered Set|preordered set]].
Let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then $H$ is [[Definition:Filtered Subset|filtered]] {{iff}}:
:for every [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] $A$ of $H$:
::$... | This follows by {{mutatis}} of the proof of [[Directed iff Finite Subsets have Upper Bounds]].
{{qed}} | Filtered iff Finite Subsets have Lower Bounds | https://proofwiki.org/wiki/Filtered_iff_Finite_Subsets_have_Lower_Bounds | https://proofwiki.org/wiki/Filtered_iff_Finite_Subsets_have_Lower_Bounds | [
"Preorder Theory"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Filtered Subset",
"Definition:Finite Set",
"Definition:Subset"
] | [
"Directed iff Finite Subsets have Upper Bounds"
] |
proofwiki-11207 | Tangent of Half Angle plus Quarter Pi | :$\map \tan {\dfrac x 2 + \dfrac \pi 4} = \tan x + \sec x$ | Firstly, we have:
{{begin-eqn}}
{{eqn | n = 1
| l = \tan x
| r = \frac {2 \tan \frac x 2} {1 - \tan ^2 \frac x 2}
| c = Double Angle Formula for Tangent
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \map \tan {\frac x 2 + \frac \pi 4}
| r = \frac {\tan \frac x 2 + \tan \frac \pi 4} {1 - \tan \f... | :$\map \tan {\dfrac x 2 + \dfrac \pi 4} = \tan x + \sec x$ | Firstly, we have:
{{begin-eqn}}
{{eqn | n = 1
| l = \tan x
| r = \frac {2 \tan \frac x 2} {1 - \tan ^2 \frac x 2}
| c = [[Double Angle Formula for Tangent]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \map \tan {\frac x 2 + \frac \pi 4}
| r = \frac {\tan \frac x 2 + \tan \frac \pi 4} {1 - \t... | Tangent of Half Angle plus Quarter Pi | https://proofwiki.org/wiki/Tangent_of_Half_Angle_plus_Quarter_Pi | https://proofwiki.org/wiki/Tangent_of_Half_Angle_plus_Quarter_Pi | [
"Trigonometric Identities",
"Tangent Function",
"Secant Function"
] | [] | [
"Double Angle Formulas/Tangent",
"Tangent of Sum",
"Tangent of 45 Degrees",
"Difference of Two Squares",
"Square of Sum",
"Double Angle Formulas/Tangent",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Double Angle Formulas/Cosine",
"Sum of Squares of Sine and Cosine",
"Sec... |
proofwiki-11208 | Linear Second Order ODE/y'' + 4 y = tan 2 x | The second order ODE:
:$(1): \quad y'' + 4 y = \tan 2 x$
has the general solution:
:$y = C_1 \cos 2 x + C_2 \sin 2 x - \dfrac 1 4 \cos 2 x \map \ln {\sec 2 x + \tan 2 x}$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = \tan 2 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' + 4 y = 0$
From Linear Second Order ODE: ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 4 y = \tan 2 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \cos 2 x + C_2 \sin 2 x - \dfrac 1 4 \cos 2 x \map \ln {\sec 2 x + \tan 2 x}$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 4$
:$\map R x = \tan 2 x$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homogeneous Lin... | Linear Second Order ODE/y'' + 4 y = tan 2 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_tan_2_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_4_y_=_tan_2_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Variation of Parameters"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 4 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"Derivat... |
proofwiki-11209 | Linear Second Order ODE/y'' + 2 y' + y = exp -x log x | The second order ODE:
:$(1): \quad y'' + 2 y' + y = e^{-x} \ln x$
has the general solution:
:$y = C_1 e^{-x} + C_2 x e^{-x} - \dfrac {x^2 e^{-x} \ln x} 2 - \dfrac 3 4 x^2 e^{-x}$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 2$
:$q = 1$
:$\map R x = e^{-x} \ln x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' + 2 y' + y = 0$
From Linear Second Or... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 2 y' + y = e^{-x} \ln x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{-x} + C_2 x e^{-x} - \dfrac {x^2 e^{-x} \ln x} 2 - \dfrac 3 4 x^2 e^{-x}$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 2$
:$q = 1$
:$\map R x = e^{-x} \ln x$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homogeneous... | Linear Second Order ODE/y'' + 2 y' + y = exp -x log x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_y_=_exp_-x_log_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_y_=_exp_-x_log_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Variation of Parameters"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 2 y' + y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"De... |
proofwiki-11210 | Linear Second Order ODE/y'' - 2 y' - 3 y = 0 | The second order ODE:
:$(1): \quad y' ' - 2 y' - 3 y = 0$
has the general solution:
:$y = C_1 e^{3 x} + C_2 e^{-x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 - 2 m - 3 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = 3$
:$m_2 = -1$
These are real and unequal.
So from Solution of Constant Coeffic... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 2 y' - 3 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{3 x} + C_2 e^{-x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 - 2 m - 3 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]],... | Linear Second Order ODE/y'' - 2 y' - 3 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_3_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_3_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equ... |
proofwiki-11211 | Linear Second Order ODE/y'' - 2 y' - 3 y = 64 x exp -x | The second order ODE:
:$(1): \quad y' ' - 2 y' - 3 y = 64 x e^{-x}$
has the general solution:
:$y = C_1 e^{3 x} + C_2 e^{-x} - x e^{-x} \paren {8 x + 4}$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = -3$
:$\map R x = 64 x e^{-x}$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y' ' - 2 y' - 3 y = 0$
From Linear Seco... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 2 y' - 3 y = 64 x e^{-x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{3 x} + C_2 e^{-x} - x e^{-x} \paren {8 x + 4}$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = -3$
:$\map R x = 64 x e^{-x}$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homogeneo... | Linear Second Order ODE/y'' - 2 y' - 3 y = 64 x exp -x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_3_y_=_64_x_exp_-x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_3_y_=_64_x_exp_-x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Variation of Parameters"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 2 y' - 3 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Primitive (Calculus)/Constant of Integration",
"Defini... |
proofwiki-11212 | Linear Second Order ODE/y'' + 2 y' + 5 y = 0 | The second order ODE:
:$(1): \quad y' ' + 2 y' + 5 y = 0$
has the general solution:
:$y = e^{-x} \paren {C_1 \cos 2 x + C_2 \sin 2 x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad m^2 + 2 m + 5 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = -1 + 2 i$
:$m_2 = -1 - 2 i$
So from Solution of Constant Coefficient Homogeneous... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 2 y' + 5 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^{-x} \paren {C_1 \cos 2 x + C_2 \sin 2 x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad m^2 + 2 m + 5 = 0$
From [[Solution to Quadratic Equation with Real Coefficients]], ... | Linear Second Order ODE/y'' + 2 y' + 5 y = 0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_5_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_5_y_=_0 | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential Equation/Solution/General Solut... |
proofwiki-11213 | Linear Second Order ODE/y'' + 2 y' + 5 y = exp -x secant 2 x | The second order ODE:
:$(1): \quad y'' + 2 y' + 5 y = e^{-x} \sec 2 x$
has the general solution:
:$y = e^{-x} \paren {C_1 \cos 2 x + C_2 \sin 2 x} + \dfrac {x e^{-x} \sin 2 x} 2 + \dfrac {e^{-x} \cos 2 x \ln \cos 2 x} 4$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 2$
:$q = 5$
:$\map R x = e^{-x} \sec 2 x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' + 2 y' + 5 y = 0$
From Linear Seco... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 2 y' + 5 y = e^{-x} \sec 2 x$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^{-x} \paren {C_1 \cos 2 x + C_2 \sin 2 x} + \dfrac {x e^{-x} \sin 2 x} 2 + \dfrac {e^{-x} \cos 2 x \ln \cos 2 x} 4$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 2$
:$q = 5$
:$\map R x = e^{-x} \sec 2 x$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homogene... | Linear Second Order ODE/y'' + 2 y' + 5 y = exp -x secant 2 x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_5_y_=_exp_-x_secant_2_x | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_y'_+_5_y_=_exp_-x_secant_2_x | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Variation of Parameters"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 2 y' + 5 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"... |
proofwiki-11214 | Singleton is Directed and Filtered Subset | Let $\struct {S, \precsim}$ be a preordered set.
Let $x$ be an element of $S$.
Then $\set x$ is directed and filtered subset of $S$. | Let $y, z \in \set x$.
By definition of singleton:
:$ y = x \land z = x$
By definition of reflexivity:
:$y \precsim x \land z \precsim x$
Thus:
:$\exists h \in \set x: y \precsim h \land z \precsim h$
Thus by definition:
:$\set x$ is a directed subset of $S$.
$\set x$ is a filtered subset of $S$ by {{mutatis}}.
{{qed}} | Let $\struct {S, \precsim}$ be a [[Definition:Preordered Set|preordered set]].
Let $x$ be an [[Definition:Element|element]] of $S$.
Then $\set x$ is [[Definition:Directed Subset|directed]] and [[Definition:Filtered Subset|filtered subset]] of $S$. | Let $y, z \in \set x$.
By definition of [[Definition:Singleton|singleton]]:
:$ y = x \land z = x$
By definition of [[Definition:Reflexivity|reflexivity]]:
:$y \precsim x \land z \precsim x$
Thus:
:$\exists h \in \set x: y \precsim h \land z \precsim h$
Thus by definition:
:$\set x$ is a [[Definition:Directed Subset... | Singleton is Directed and Filtered Subset | https://proofwiki.org/wiki/Singleton_is_Directed_and_Filtered_Subset | https://proofwiki.org/wiki/Singleton_is_Directed_and_Filtered_Subset | [
"Preorder Theory"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Element",
"Definition:Directed Subset",
"Definition:Filtered Subset"
] | [
"Definition:Singleton",
"Definition:Reflexivity",
"Definition:Directed Subset",
"Definition:Filtered Subset"
] |
proofwiki-11215 | Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = 0 | The second order ODE:
:$(1): \quad \paren {x^2 - 1} y' ' - 2 x y' + 2 y = 0$
has the general solution:
:$y = C_1 x + C_2 \paren {x^2 + 1}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = 1
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = {y_1}' '
| r = 0
| c = Derivative of Constant
}}
{{end-eqn}}
and so by inspection:
:$y_1 = x$
is a particula... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad \paren {x^2 - 1} y' ' - 2 x y' + 2 y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 x + C_2 \paren {x^2 + 1}$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = 1
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = {y_1}' '
| r = 0
| c = [[Derivative of Constant]]
}}
{{end-eqn}}
and so by inspection:
:$y_1 = x$
is a... | Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = 0 | https://proofwiki.org/wiki/Second_Order_ODE/(x^2_-_1)_y''_-_2_x_y'_+_2_y_=_0 | https://proofwiki.org/wiki/Second_Order_ODE/(x^2_-_1)_y''_-_2_x_y'_+_2_y_=_0 | [
"Examples of Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Power Rule for Derivatives",
"Derivative of Constant",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Primitive of Function under its Der... |
proofwiki-11216 | Linear Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = (x^2 - 1)^2 | The second order ODE:
:$(1): \quad \paren {x^2 - 1} y' ' - 2 x y' + 2 y = \paren {x^2 - 1}^2$
has the general solution:
:$y = C_1 x + C_2 \paren {x^2 + 1} + \dfrac {x^4} 6 - \dfrac {x^2} 2$ | $(1)$ can be manipulated into the form:
:$y' ' - \dfrac {2 x} {x^2 - 1} y' + \dfrac 2 {x^2 - 1} y = x^2 - 1$
It can be seen that this is a nonhomogeneous linear second order ODE in the form:
:$y' ' + \map P x y' + \map Q x y = \map R x$
where:
:$\map P x = -\dfrac {2 x} {x^2 - 1}$
:$\map Q x = \dfrac 2 {x^2 - 1}$
:$\ma... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad \paren {x^2 - 1} y' ' - 2 x y' + 2 y = \paren {x^2 - 1}^2$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 x + C_2 \paren {x^2 + 1} + \dfrac {x^4} 6 - \dfrac {x^2} 2$ | $(1)$ can be manipulated into the form:
:$y' ' - \dfrac {2 x} {x^2 - 1} y' + \dfrac 2 {x^2 - 1} y = x^2 - 1$
It can be seen that this is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y' ' + \map P x y' + \map Q x y = \map R x$
where:
:$\map P x = -\dfrac ... | Linear Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = (x^2 - 1)^2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/(x^2_-_1)_y''_-_2_x_y'_+_2_y_=_(x^2_-_1)^2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/(x^2_-_1)_y''_-_2_x_y'_+_2_y_=_(x^2_-_1)^2 | [
"Examples of Linear Second Order ODEs",
"Examples of Method of Variation of Parameters"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE",
"Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/Particular Solution",
"Power Rule for Derivat... |
proofwiki-11217 | Directed iff Lower Closure Directed | Let $\struct {S, \precsim}$ be a preordered set.
Let $H$ be a non-empty subset of $S$.
Then $H$ is directed {{iff}}:
:$H^\precsim$ is directed
where $H^\precsim$ denotes the lower closure of $H$ with respect to $\precsim$. | === Sufficient Condition ===
Let us assume that $H$ is directed.
Let $x, y \in H^\precsim$.
By definition of lower closure:
:$\exists x' \in H: x \precsim x'$
and
:$\exists y' \in H: y \precsim y'$
By definition of directed subset:
:$\exists z \in H: x' \precsim z \land y' \precsim z$
By definition of reflexivity;
:$z ... | Let $\struct {S, \precsim}$ be a [[Definition:Preordered Set|preordered set]].
Let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then $H$ is [[Definition:Directed Subset|directed]] {{iff}}:
:$H^\precsim$ is [[Definition:Directed Subset|directed]]
where $H^\precsim$ denotes the ... | === Sufficient Condition ===
Let us assume that $H$ is [[Definition:Directed Subset|directed]].
Let $x, y \in H^\precsim$.
By definition of [[Definition:Lower Closure of Subset|lower closure]]:
:$\exists x' \in H: x \precsim x'$
and
:$\exists y' \in H: y \precsim y'$
By definition of [[Definition:Directed Subset|di... | Directed iff Lower Closure Directed | https://proofwiki.org/wiki/Directed_iff_Lower_Closure_Directed | https://proofwiki.org/wiki/Directed_iff_Lower_Closure_Directed | [
"Preorder Theory"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Lower Closure/Set"
] | [
"Definition:Directed Subset",
"Definition:Lower Closure/Set",
"Definition:Directed Subset",
"Definition:Reflexivity",
"Definition:Lower Closure/Set",
"Definition:Transitive",
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Reflexivity",
"Definition:Lower Closure/Set",
"De... |
proofwiki-11218 | Filtered iff Upper Closure Filtered | Let $\struct {S, \precsim}$ be a preordered set.
Let $H$ be a non-empty subset of $S$.
Then $H$ is filtered {{iff}}
$H^\succsim$ is filtered
where $H^\succsim$ denotes the upper closure of $H$. | This follows by {{mutatis}} of the proof of Directed iff Lower Closure Directed.
{{qed}} | Let $\struct {S, \precsim}$ be a [[Definition:Preordered Set|preordered set]].
Let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then $H$ is [[Definition:Filtered Subset|filtered]] {{iff}}
$H^\succsim$ is [[Definition:Filtered Subset|filtered]]
where $H^\succsim$ denotes the [[... | This follows by {{mutatis}} of the proof of [[Directed iff Lower Closure Directed]].
{{qed}} | Filtered iff Upper Closure Filtered | https://proofwiki.org/wiki/Filtered_iff_Upper_Closure_Filtered | https://proofwiki.org/wiki/Filtered_iff_Upper_Closure_Filtered | [
"Preorder Theory"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Filtered Subset",
"Definition:Filtered Subset",
"Definition:Upper Closure/Set"
] | [
"Directed iff Lower Closure Directed"
] |
proofwiki-11219 | Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0 | The second order ODE:
:$(1): \quad \paren {x^2 + x} y' ' + \paren {2 - x^2} y' - \paren {2 + x} y = 0$
has the general solution:
:$y = C_1 e^x + \dfrac {C_2} x$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = \frac 1 x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = -\frac 1 {x^2}
| c = Power Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = {y_1}' '
| r = \frac 2 {x^3}
| c = Power Rule for Derivatives
}}
{{end-eqn}}
and so:
{{... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad \paren {x^2 + x} y' ' + \paren {2 - x^2} y' - \paren {2 + x} y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + \dfrac {C_2} x$ | Note that:
{{begin-eqn}}
{{eqn | l = y_1
| r = \frac 1 x
| c =
}}
{{eqn | ll= \leadsto
| l = {y_1}'
| r = -\frac 1 {x^2}
| c = [[Power Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = {y_1}' '
| r = \frac 2 {x^3}
| c = [[Power Rule for Derivatives]]
}}
{{end-eqn}}
... | Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/(x^2_+_x)_y''_+_(2_-_x^2)_y'_-_(2_+_x)_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/(x^2_+_x)_y''_+_(2_-_x^2)_y'_-_(2_+_x)_y_=_0/Proof_1 | [
"Examples of Homogeneous LSOODEs",
"Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Power Rule for Derivatives",
"Power Rule for Derivatives",
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Primitive of Reciprocal of x by... |
proofwiki-11220 | Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0 | The second order ODE:
:$(1): \quad \paren {x^2 + x} y' ' + \paren {2 - x^2} y' - \paren {2 + x} y = 0$
has the general solution:
:$y = C_1 e^x + \dfrac {C_2} x$ | Note that:
:$\paren {x^2 + x} + \paren {2 - x^2} - \paren {2 + x} = 0$
so $\map {y_1} x$ such that $y_1 = {y_1}' = {y_1}' '$ satisfies $(1)$.
Hence:
:$y_1 = e^x$
is a particular solution of $(1)$.
$(1)$ can be expressed as:
:$(2): \quad y' ' + \dfrac {2 - x^2} {x^2 + x} y' - \dfrac {2 + x} {x^2 + x} y = 0$
which is in ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad \paren {x^2 + x} y' ' + \paren {2 - x^2} y' - \paren {2 + x} y = 0$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + \dfrac {C_2} x$ | Note that:
:$\paren {x^2 + x} + \paren {2 - x^2} - \paren {2 + x} = 0$
so $\map {y_1} x$ such that $y_1 = {y_1}' = {y_1}' '$ satisfies $(1)$.
Hence:
:$y_1 = e^x$
is a [[Definition:Particular Solution to Differential Equation|particular solution]] of $(1)$.
$(1)$ can be expressed as:
:$(2): \quad y' ' + \dfrac {2 -... | Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/(x^2_+_x)_y''_+_(2_-_x^2)_y'_-_(2_+_x)_y_=_0 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/(x^2_+_x)_y''_+_(2_-_x^2)_y'_-_(2_+_x)_y_=_0/Proof_2 | [
"Examples of Homogeneous LSOODEs",
"Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another",
"Definition:Differential Equation/Solution/Particular Solution",
"Primitive of Reciprocal of x by a x + b",
"Primitive of x over a x + b",
"Primitive of Expo... |
proofwiki-11221 | Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = x (x + 1)^2 | The second order ODE:
:$(1): \quad \paren {x^2 + x} y' ' + \paren {2 - x^2} y' - \paren {2 + x} y = x \paren {x + 1}^2$
has the general solution:
:$y = C_1 e^x + \dfrac {C_2} x - x - 1 - \dfrac {x^2} 3$ | $(1)$ can be manipulated into the form:
:$y' ' + \dfrac {2 - x^2} {x^2 + x} y' - \dfrac {2 + x} {x^2 + x} y = x + 1$
It can be seen that this is a nonhomogeneous linear second order ODE in the form:
:$y' ' + \map P x y' + \map Q x y = \map R x$
where:
:$\map P x = \dfrac {2 - x^2} {x^2 + x}$
:$\map Q x = -\dfrac {2 + x... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad \paren {x^2 + x} y' ' + \paren {2 - x^2} y' - \paren {2 + x} y = x \paren {x + 1}^2$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^x + \dfrac {C_2} x - x - 1 - \dfrac {x^2} 3$ | $(1)$ can be manipulated into the form:
:$y' ' + \dfrac {2 - x^2} {x^2 + x} y' - \dfrac {2 + x} {x^2 + x} y = x + 1$
It can be seen that this is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y' ' + \map P x y' + \map Q x y = \map R x$
where:
:$\map P x = ... | Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = x (x + 1)^2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/(x^2_+_x)_y''_+_(2_-_x^2)_y'_-_(2_+_x)_y_=_x_(x_+_1)^2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/(x^2_+_x)_y''_+_(2_-_x^2)_y'_-_(2_+_x)_y_=_x_(x_+_1)^2 | [
"Examples of Linear Second Order ODEs",
"Examples of Method of Variation of Parameters"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation/Solution/P... |
proofwiki-11222 | Directed in Join Semilattice | Let $\struct {S, \preceq}$ be a join semilattice.
Let $H$ be a non-empty lower section of $S$.
Then $H$ is directed {{iff}}
:$\forall x, y \in H: x \vee y \in H$ | === Sufficient Condition ===
Let us assume that
:$H$ is directed.
Let $x, y \in H$.
By definition of directed:
:$\exists z \in H: x \preceq z \land y \preceq z$
By definition
:$z$ is upper bound of $\set {x, y}$
By definitions of supremum and join:
:$x \vee y = \sup \set {x, y} \preceq z$
Thus by definition of lower se... | Let $\struct {S, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Lower Section|lower section]] of $S$.
Then $H$ is [[Definition:Directed Subset|directed]] {{iff}}
:$\forall x, y \in H: x \vee y \in H$ | === Sufficient Condition ===
Let us assume that
:$H$ is [[Definition:Directed Subset|directed]].
Let $x, y \in H$.
By definition of [[Definition:Directed Subset|directed]]:
:$\exists z \in H: x \preceq z \land y \preceq z$
By definition
:$z$ is [[Definition:Upper Bound of Set|upper bound]] of $\set {x, y}$
By defi... | Directed in Join Semilattice | https://proofwiki.org/wiki/Directed_in_Join_Semilattice | https://proofwiki.org/wiki/Directed_in_Join_Semilattice | [
"Join and Meet Semilattices",
"Lower Sections"
] | [
"Definition:Join Semilattice",
"Definition:Non-Empty Set",
"Definition:Lower Section",
"Definition:Directed Subset"
] | [
"Definition:Directed Subset",
"Definition:Directed Subset",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set",
"Definition:Join (Order Theory)",
"Definition:Lower Section",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Definition:Directed Subset"
] |
proofwiki-11223 | Canonical Form of Underdamped Oscillatory System | Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
:$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ underdamped.
Then the value of $x$ can be expressed in the form:
:$x = \dfrac {x_0 a... | From Solution of Constant Coefficient Homogeneous LSOODE: Complex Roots of Auxiliary Equation, the general solution of $(1)$ is:
:$x = e^{-b t} \paren {C_1 \cos \alpha t + C_2 \sin \alpha t}$
where:
:$\alpha = \sqrt {a^2 - b^2}$
This is a homogeneous linear second order ODE with constant coefficients.
Let $m_1$ and $m_... | Consider a [[Definition:Physical System|physical system]] $S$ whose behaviour can be described with the [[Definition:Second Order ODE|second order ODE]] in the form:
:$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ [[Definition:Underda... | From [[Solution of Constant Coefficient Homogeneous LSOODE/Complex Roots of Auxiliary Equation|Solution of Constant Coefficient Homogeneous LSOODE: Complex Roots of Auxiliary Equation]], the [[Definition:General Solution to Differential Equation|general solution]] of $(1)$ is:
:$x = e^{-b t} \paren {C_1 \cos \alpha t ... | Canonical Form of Underdamped Oscillatory System | https://proofwiki.org/wiki/Canonical_Form_of_Underdamped_Oscillatory_System | https://proofwiki.org/wiki/Canonical_Form_of_Underdamped_Oscillatory_System | [
"Mechanics"
] | [
"Definition:Physical System",
"Definition:Second Order Ordinary Differential Equation",
"Definition:Underdamped",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Solution of Constant Coefficient Homogeneous LSOODE/Complex Roots of Auxiliary Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Root of Polynomial",
"Definition:Auxiliary Equation",
"Solution ... |
proofwiki-11224 | Period of Oscillation of Underdamped System is Regular | Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
:$\dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ underdamped.
Then the period of its movement is well-defined, in the sense that its zeroes are ... | Let the position of $S$ be described in the canonical form:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where $\alpha = \sqrt {a^2 - b^2}$.
The zeroes of $(1)$ occur exactly where:
:$\map \cos {\alpha t - \theta} = 0$
Thus the period $T$ of $\map \cos {\alpha t - \theta}$ is given ... | Consider a [[Definition:Physical System|physical system]] $S$ whose behaviour can be described with the [[Definition:Second Order ODE|second order ODE]] in the form:
:$\dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ [[Definition:Underdamped|underd... | Let the position of $S$ be described in the [[Canonical Form of Underdamped Oscillatory System|canonical form]]:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where $\alpha = \sqrt {a^2 - b^2}$.
The [[Definition:Zero of Function|zeroes]] of $(1)$ occur exactly where:
:$\map \cos {\a... | Period of Oscillation of Underdamped System is Regular | https://proofwiki.org/wiki/Period_of_Oscillation_of_Underdamped_System_is_Regular | https://proofwiki.org/wiki/Period_of_Oscillation_of_Underdamped_System_is_Regular | [
"Mechanics"
] | [
"Definition:Physical System",
"Definition:Second Order Ordinary Differential Equation",
"Definition:Underdamped",
"Definition:Period of Underdamped Oscillation",
"Definition:Root of Mapping"
] | [
"Canonical Form of Underdamped Oscillatory System",
"Definition:Root of Mapping",
"Definition:Periodic Real Function/Period"
] |
proofwiki-11225 | Natural Frequency of Underdamped System | Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
:$\dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ underdamped.
Then the natural frequency of $S$ is given by:
:$\nu = \dfrac {\sqrt {a^2 - b^2} }... | Let the position of $S$ be described in the canonical form:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where $\alpha = \sqrt {a^2 - b^2}$.
From Period of Oscillation of Underdamped System is Regular:
:$T = \dfrac {2 \pi} {\sqrt {a^2 - b^2} }$
where $T$ is the period of the movemen... | Consider a [[Definition:Physical System|physical system]] $S$ whose behaviour can be described with the [[Definition:Second Order ODE|second order ODE]] in the form:
:$\dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ [[Definition:Underdamped|underd... | Let the position of $S$ be described in the [[Canonical Form of Underdamped Oscillatory System|canonical form]]:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where $\alpha = \sqrt {a^2 - b^2}$.
From [[Period of Oscillation of Underdamped System is Regular]]:
:$T = \dfrac {2 \pi} {\... | Natural Frequency of Underdamped System | https://proofwiki.org/wiki/Natural_Frequency_of_Underdamped_System | https://proofwiki.org/wiki/Natural_Frequency_of_Underdamped_System | [
"Mechanics"
] | [
"Definition:Physical System",
"Definition:Second Order Ordinary Differential Equation",
"Definition:Underdamped",
"Definition:Natural Frequency"
] | [
"Canonical Form of Underdamped Oscillatory System",
"Period of Oscillation of Underdamped System is Regular",
"Definition:Period of Underdamped Oscillation",
"Definition:Natural Frequency"
] |
proofwiki-11226 | Linear Second Order ODE/y'' + 2 b y' + a^2 y = 0/b less than a | The second order ODE:
:$(1): \quad y' ' + 2 b y' + a^2 y = 0$ where $b^2 < a^2$
has the general solution:
:$y = e^{-b x} \paren {C_1 \, \map \cos {\sqrt {a^2 - b^2} } x + C_2 \, \map \sin {\sqrt {a^2 - b^2} } x}$ | It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + 2 b m + a^2 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
:$m_1 = - b + \sqrt {b^2 - a^2}$
:$m_2 = - b - \sqrt {b^2 - a^2}$
As $b^2 < a^2$, t... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + 2 b y' + a^2 y = 0$ where $b^2 < a^2$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^{-b x} \paren {C_1 \, \map \cos {\sqrt {a^2 - b^2} } x + C_2 \, \map \sin {\sqrt {a^2 - b^2} } x}$ | It can be seen that $(1)$ is a [[Definition:Constant Coefficient Homogeneous Linear Second Order ODE|constant coefficient homogeneous linear second order ODE]].
Its [[Definition:Auxiliary Equation|auxiliary equation]] is:
:$(2): \quad: m^2 + 2 b m + a^2 = 0$
From [[Solution to Quadratic Equation with Real Coefficient... | Linear Second Order ODE/y'' + 2 b y' + a^2 y = 0/b less than a | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_b_y'_+_a^2_y_=_0/b_less_than_a | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_b_y'_+_a^2_y_=_0/b_less_than_a | [
"Examples of Constant Coefficient Homogeneous LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Definition:Auxiliary Equation",
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Solution of Constant Coefficient Homogeneous LSOODE",
"Definition:Differential ... |
proofwiki-11227 | Linear Second Order ODE/y'' + 2 b y' + a^2 y = K cosine omega x/b less than a | The second order ODE:
:$(1): \quad y'' + 2 b y' + a^2 y = K \cos \omega x$ where $b^2 < a^2$
has the general solution:
:$y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$
where:
:$\alpha = \sqrt {a^2 - b^2}$
:$\phi =... | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 2 b$
:$q = a^2$
:$\map R x = K \cos \omega x$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y'' + 2... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' + 2 b y' + a^2 y = K \cos \omega x$ where $b^2 < a^2$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 -... | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y'' + p y' + q y = \map R x$
where:
:$p = 2 b$
:$q = a^2$
:$\map R x = K \cos \omega x$
First we esta... | Linear Second Order ODE/y'' + 2 b y' + a^2 y = K cosine omega x/b less than a | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_b_y'_+_a^2_y_=_K_cosine_omega_x/b_less_than_a | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_2_b_y'_+_a^2_y_=_K_cosine_omega_x/b_less_than_a | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + 2 b y' + a^2 y = 0/b less than a",
"Definition:Differential Equation/Solution/General Solution",... |
proofwiki-11228 | Condition for Resonance in Forced Vibration of Underdamped System | Consider a physical system $S$ whose behaviour is defined by the second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$
where:
:$K \in \R: k > 0$
:$a, b \in \R_{>0}: b < a$
Then $S$ is in resonance when:
:$\omega = \sqrt {a^2 - 2 b^2}$
and thus the resonance freque... | From Linear Second Order ODE: $y' ' + 2 b y' + a^2 y = K \cos \omega x$ where $b < a$ the general solution of $(1)$ is:
:$(2): \quad y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$
where:
:$\alpha = \sqrt {a^2 - b^... | Consider a [[Definition:Physical System|physical system]] $S$ whose behaviour is defined by the [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$
where:
:$K \in \R: k > 0$
:$a, b \in \R_{>0}: b < a$
Then $S$ is in [[Definition... | From [[Linear Second Order ODE/y'' + 2 b y' + a^2 y = K cosine omega x/b less than a|Linear Second Order ODE: $y' ' + 2 b y' + a^2 y = K \cos \omega x$ where $b < a$]] the [[Definition:General Solution to Differential Equation|general solution]] of $(1)$ is:
:$(2): \quad y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \si... | Condition for Resonance in Forced Vibration of Underdamped System | https://proofwiki.org/wiki/Condition_for_Resonance_in_Forced_Vibration_of_Underdamped_System | https://proofwiki.org/wiki/Condition_for_Resonance_in_Forced_Vibration_of_Underdamped_System | [
"Resonance",
"Mathematical Physics"
] | [
"Definition:Physical System",
"Definition:Second Order Ordinary Differential Equation",
"Definition:Resonance",
"Definition:Resonance Frequency",
"Definition:Resonance Frequency"
] | [
"Linear Second Order ODE/y'' + 2 b y' + a^2 y = K cosine omega x/b less than a",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Resonance",
"Definition:Simple Harmonic Motion/Amplitude",
"Definition:Steady-State/Second Order ODE",
"Definition:Maximum Value of Real Function/Absol... |
proofwiki-11229 | Filtered in Meet Semilattice | Let $\struct {S, \preceq}$ be a meet semilattice.
Let $H$ be a non-empty upper section of $S$.
Then $H$ is filtered {{iff}}
:$\forall x, y \in H: x \wedge y \in H$ | This follows by {{mutatis}} of the proof of Directed in Join Semilattice.
{{qed}} | Let $\struct {S, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Upper Section|upper section]] of $S$.
Then $H$ is [[Definition:Filtered Subset|filtered]] {{iff}}
:$\forall x, y \in H: x \wedge y \in H$ | This follows by {{mutatis}} of the proof of [[Directed in Join Semilattice]].
{{qed}} | Filtered in Meet Semilattice | https://proofwiki.org/wiki/Filtered_in_Meet_Semilattice | https://proofwiki.org/wiki/Filtered_in_Meet_Semilattice | [
"Join and Meet Semilattices",
"Upper Sections"
] | [
"Definition:Meet Semilattice",
"Definition:Non-Empty Set",
"Definition:Upper Section",
"Definition:Filtered Subset"
] | [
"Directed in Join Semilattice"
] |
proofwiki-11230 | Directed in Join Semilattice with Finite Suprema | Let $\struct {S, \preceq}$ be a join semilattice.
Let $H$ be a non-empty lower section of $S$.
Then $H$ is directed {{iff}}
:for every non-empty finite subset $A$ of $H$, $\sup A \in H$ | === Sufficient Condition ===
Let us assume that
:$H$ is directed.
Let $A$ be a non-empty finite subset of $H$.
By Directed iff Finite Subsets have Upper Bounds:
:$\exists h \in H: \forall a \in A: a \preceq h$
By definition
:$z$ is upper bound of $A$
By Existence of Non-Empty Finite Suprema in Join Semilattice:
:$\sup ... | Let $\struct {S, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Lower Section|lower section]] of $S$.
Then $H$ is [[Definition:Directed Subset|directed]] {{iff}}
:for every [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set... | === Sufficient Condition ===
Let us assume that
:$H$ is [[Definition:Directed Subset|directed]].
Let $A$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of $H$.
By [[Directed iff Finite Subsets have Upper Bounds]]:
:$\exists h \in H: \forall a \in A: a \prece... | Directed in Join Semilattice with Finite Suprema | https://proofwiki.org/wiki/Directed_in_Join_Semilattice_with_Finite_Suprema | https://proofwiki.org/wiki/Directed_in_Join_Semilattice_with_Finite_Suprema | [
"Join and Meet Semilattices",
"Lower Sections"
] | [
"Definition:Join Semilattice",
"Definition:Non-Empty Set",
"Definition:Lower Section",
"Definition:Directed Subset",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset"
] | [
"Definition:Directed Subset",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset",
"Directed iff Finite Subsets have Upper Bounds",
"Definition:Upper Bound of Set",
"Existence of Non-Empty Finite Suprema in Join Semilattice",
"Definition:Supremum of Set",
"Definition:Lower Secti... |
proofwiki-11231 | Resonance Frequency is less than Natural Frequency | Consider a physical system $S$ whose behaviour is defined by the second order ODE:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$
where:
:$K \in \R: k > 0$
:$a, b \in \R_{>0}: b < a$
Then the resonance frequency of $S$ is smaller than the natural frequency of the associated ... | From Natural Frequency of Underdamped System, the natural frequency of $(2)$ is:
:$\nu = \dfrac {\sqrt {a^2 - b^2} } {2 \pi}$
From Condition for Resonance in Forced Vibration of Underdamped System, the resonance frequency of $S$ is:
:$\omega = \sqrt {a^2 - 2 b^2}$
:$\nu_R = \dfrac {\sqrt {a^2 - 2 b^2} } {2 \pi}$
We hav... | Consider a [[Definition:Physical System|physical system]] $S$ whose behaviour is defined by the [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$
where:
:$K \in \R: k > 0$
:$a, b \in \R_{>0}: b < a$
Then the [[Definition:Resona... | From [[Natural Frequency of Underdamped System]], the [[Definition:Natural Frequency|natural frequency]] of $(2)$ is:
:$\nu = \dfrac {\sqrt {a^2 - b^2} } {2 \pi}$
From [[Condition for Resonance in Forced Vibration of Underdamped System]], the [[Definition:Resonance Frequency|resonance frequency]] of $S$ is:
:$\omega =... | Resonance Frequency is less than Natural Frequency | https://proofwiki.org/wiki/Resonance_Frequency_is_less_than_Natural_Frequency | https://proofwiki.org/wiki/Resonance_Frequency_is_less_than_Natural_Frequency | [
"Mathematical Physics"
] | [
"Definition:Physical System",
"Definition:Second Order Ordinary Differential Equation",
"Definition:Resonance Frequency",
"Definition:Natural Frequency",
"Definition:Second Order Ordinary Differential Equation"
] | [
"Natural Frequency of Underdamped System",
"Definition:Natural Frequency",
"Condition for Resonance in Forced Vibration of Underdamped System",
"Definition:Resonance Frequency",
"Condition for Resonance in Forced Vibration of Underdamped System",
"Definition:Resonance"
] |
proofwiki-11232 | Filtered in Meet Semilattice with Finite Infima | Let $\struct {S, \preceq}$ be a meet semilattice.
Let $H$ be a non-empty upper section of $S$.
Then $H$ is filtered {{iff}}
:for every non-empty finite subset $A$ of $H$, $\inf A \in H$ | This follows by {{mutatis}} of the proof of Directed in Join Semilattice with Finite Suprema.
{{qed}} | Let $\struct {S, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Upper Section|upper section]] of $S$.
Then $H$ is [[Definition:Filtered Subset|filtered]] {{iff}}
:for every [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set... | This follows by {{mutatis}} of the proof of [[Directed in Join Semilattice with Finite Suprema]].
{{qed}} | Filtered in Meet Semilattice with Finite Infima | https://proofwiki.org/wiki/Filtered_in_Meet_Semilattice_with_Finite_Infima | https://proofwiki.org/wiki/Filtered_in_Meet_Semilattice_with_Finite_Infima | [
"Join and Meet Semilattices",
"Upper Sections"
] | [
"Definition:Meet Semilattice",
"Definition:Non-Empty Set",
"Definition:Upper Section",
"Definition:Filtered Subset",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset"
] | [
"Directed in Join Semilattice with Finite Suprema"
] |
proofwiki-11233 | Interval between Local Maxima for Underdamped Free Vibration | Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
:$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ underdamped.
:600px
Let $T$ be the period of oscillation of $S$.
Then the successive... | Let the position of $S$ be described in the canonical form:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where:
:$\alpha = \sqrt {a^2 - b^2}$.
:$\theta = \map \arctan {\dfrac b \alpha}$
From Period of Oscillation of Underdamped System is Regular, the period of oscillation $T$ is giv... | Consider a [[Definition:Physical System|physical system]] $S$ whose behaviour can be described with the [[Definition:Second Order ODE|second order ODE]] in the form:
:$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ [[Definition:Underda... | Let the position of $S$ be described in the [[Canonical Form of Underdamped Oscillatory System|canonical form]]:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where:
:$\alpha = \sqrt {a^2 - b^2}$.
:$\theta = \map \arctan {\dfrac b \alpha}$
From [[Period of Oscillation of Underdamped... | Interval between Local Maxima for Underdamped Free Vibration | https://proofwiki.org/wiki/Interval_between_Local_Maxima_for_Underdamped_Free_Vibration | https://proofwiki.org/wiki/Interval_between_Local_Maxima_for_Underdamped_Free_Vibration | [
"Mathematical Physics"
] | [
"Definition:Physical System",
"Definition:Second Order Ordinary Differential Equation",
"Definition:Underdamped",
"File:UnderdampedPeriodAmplitude.png",
"Definition:Period of Underdamped Oscillation",
"Definition:Maximum Value of Real Function/Local"
] | [
"Canonical Form of Underdamped Oscillatory System",
"Period of Oscillation of Underdamped System is Regular",
"Definition:Period of Underdamped Oscillation",
"Definition:Differentiation",
"Interior Extremum Theorem",
"Definition:Maximum Value of Real Function/Local",
"Definition:Minimum Value of Real Fu... |
proofwiki-11234 | Ratio of Successive Local Maxima for Underdamped Free Vibration | Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form:
:$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ underdamped.
:600px
Let $T$ be the period of oscillation of $S$.
Let $x_1$ and $x_2$... | Let the position of $S$ be described in the canonical form:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where:
:$\alpha = \sqrt {a^2 - b^2}$
:$\theta = \map \arctan {\dfrac b \alpha}$
From Period of Oscillation of Underdamped System is Regular, the period of oscillation $T$ is give... | Consider a [[Definition:Physical System|physical system]] $S$ whose behaviour can be described with the [[Definition:Second Order ODE|second order ODE]] in the form:
:$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$
for $a, b \in \R_{>0}$.
Let $b < a$, so as to make $S$ [[Definition:Underda... | Let the position of $S$ be described in the [[Canonical Form of Underdamped Oscillatory System|canonical form]]:
:$(1): \quad x = \dfrac {x_0 \, a} \alpha e^{-b t} \map \cos {\alpha t - \theta}$
where:
:$\alpha = \sqrt {a^2 - b^2}$
:$\theta = \map \arctan {\dfrac b \alpha}$
From [[Period of Oscillation of Underdamped ... | Ratio of Successive Local Maxima for Underdamped Free Vibration | https://proofwiki.org/wiki/Ratio_of_Successive_Local_Maxima_for_Underdamped_Free_Vibration | https://proofwiki.org/wiki/Ratio_of_Successive_Local_Maxima_for_Underdamped_Free_Vibration | [
"Mathematical Physics"
] | [
"Definition:Physical System",
"Definition:Second Order Ordinary Differential Equation",
"Definition:Underdamped",
"File:UnderdampedPeriodAmplitude.png",
"Definition:Period of Underdamped Oscillation",
"Definition:Maximum Value of Real Function/Local"
] | [
"Canonical Form of Underdamped Oscillatory System",
"Period of Oscillation of Underdamped System is Regular",
"Definition:Period of Underdamped Oscillation",
"Interval between Local Maxima for Underdamped Free Vibration",
"Definition:Maximum Value of Real Function/Local",
"Definition:Differentiation"
] |
proofwiki-11235 | Motion of Particle in Polar Coordinates | Consider a particle $p$ of mass $m$ moving in the plane under the influence of a force $\mathbf F$.
Let the position of $p$ at time $t$ be given in polar coordinates as $\polar {r, \theta}$.
Let $\mathbf F$ be expressed as:
:$\mathbf F = F_r \mathbf u_r + F_\theta \mathbf u_\theta$
where:
:$\mathbf u_r$ is the unit vec... | Let the radius vector $\mathbf r$ from the origin to $p$ be expressed as:
:$(1): \quad \mathbf r = r \mathbf u_r$
From Velocity Vector in Polar Coordinates, the velocity $\mathbf v$ of $p$ can be expressed in vector form as:
:$\mathbf v = r \dfrac {\d \theta} {\d t} \mathbf u_\theta + \dfrac {\d r} {\d t} \mathbf u_r$
... | Consider a [[Definition:Particle|particle]] $p$ of [[Definition:Mass|mass]] $m$ moving in the [[Definition:Plane|plane]] under the influence of a [[Definition:Force|force]] $\mathbf F$.
Let the [[Definition:Position|position]] of $p$ at [[Definition:Time|time]] $t$ be given in [[Definition:Polar Coordinates|polar coor... | Let the [[Definition:Radius Vector|radius vector]] $\mathbf r$ from the [[Definition:Origin|origin]] to $p$ be expressed as:
:$(1): \quad \mathbf r = r \mathbf u_r$
From [[Velocity Vector in Polar Coordinates]], the [[Definition:Velocity|velocity]] $\mathbf v$ of $p$ can be expressed in [[Definition:Vector|vector]] fo... | Motion of Particle in Polar Coordinates | https://proofwiki.org/wiki/Motion_of_Particle_in_Polar_Coordinates | https://proofwiki.org/wiki/Motion_of_Particle_in_Polar_Coordinates | [
"Mathematical Physics"
] | [
"Definition:Particle",
"Definition:Mass",
"Definition:Plane Surface",
"Definition:Force",
"Definition:Position",
"Definition:Time",
"Definition:Polar Coordinates",
"Definition:Unit Vector",
"Definition:Polar Coordinates/Radial Coordinate",
"Definition:Unit Vector",
"Definition:Polar Coordinates/... | [
"Definition:Position Vector",
"Definition:Coordinate System/Origin",
"Velocity Vector in Polar Coordinates",
"Definition:Velocity",
"Definition:Vector",
"Acceleration Vector in Polar Coordinates",
"Definition:Acceleration",
"Newton's Laws of Motion/Second Law"
] |
proofwiki-11236 | Derivative of Angular Component under Central Force | Let a point mass $p$ of mass $m$ be under the influence of a central force $\mathbf F$.
Let the position of $p$ at time $t$ be given in polar coordinates as $\polar {r, \theta}$.
Let $\mathbf r$ be the radius vector from the origin to $p$.
Then the rate of change of the angular coordinate of $p$ is inversely proportion... | Let $\mathbf F$ be expressed as:
:$\mathbf F = F_r \mathbf u_r + F_\theta \mathbf u_\theta$
where:
:$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$
:$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$
:$F_r$ and $F_\theta$ are the magnitudes of the c... | Let a [[Definition:Particle|point mass]] $p$ of [[Definition:Mass|mass]] $m$ be under the influence of a [[Definition:Central Force|central force]] $\mathbf F$.
Let the [[Definition:Position|position]] of $p$ at [[Definition:Time|time]] $t$ be given in [[Definition:Polar Coordinates|polar coordinates]] as $\polar {r, ... | Let $\mathbf F$ be expressed as:
:$\mathbf F = F_r \mathbf u_r + F_\theta \mathbf u_\theta$
where:
:$\mathbf u_r$ is the [[Definition:Unit Vector|unit vector]] in the direction of the [[Definition:Radial Coordinate|radial coordinate]] of $p$
:$\mathbf u_\theta$ is the [[Definition:Unit Vector|unit vector]] in the direc... | Derivative of Angular Component under Central Force | https://proofwiki.org/wiki/Derivative_of_Angular_Component_under_Central_Force | https://proofwiki.org/wiki/Derivative_of_Angular_Component_under_Central_Force | [
"Mechanics"
] | [
"Definition:Particle",
"Definition:Mass",
"Definition:Central Force",
"Definition:Position",
"Definition:Time",
"Definition:Polar Coordinates",
"Definition:Position Vector",
"Definition:Coordinate System/Origin",
"Definition:Rate of Change/Time",
"Definition:Polar Coordinates/Angular Coordinate",
... | [
"Definition:Unit Vector",
"Definition:Polar Coordinates/Radial Coordinate",
"Definition:Unit Vector",
"Definition:Polar Coordinates/Angular Coordinate",
"Definition:Magnitude",
"Motion of Particle in Polar Coordinates",
"Definition:Second Order Ordinary Differential Equation",
"Definition:Force",
"D... |
proofwiki-11237 | Linear Second Order ODE/y'' + y = K | The second order ODE:
:$(1): \quad y' ' + y = K$
has the general solution:
:$y = C_1 \sin x + C_2 \cos x + K$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 1$
:$\map R x = K$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$(2): \quad y' ' + y = 0$... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + y = K$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 \sin x + C_2 \cos x + K$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] with [[Definition:Constant|constant]] [[Definition:Coefficient|coefficients]] in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = 0$
:$q = 1$
:$\map R x = K$
First we establish the solutio... | Linear Second Order ODE/y'' + y = K | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_K | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_y_=_K | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Undetermined Coefficients"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Constant",
"Definition:Coefficient",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' + y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Method of Undetermined ... |
proofwiki-11238 | Velocity Vector in Polar Coordinates | Consider a particle $p$ moving in the plane.
Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$.
Then the velocity $\mathbf v$ of $p$ can be expressed as:
:$\mathbf v = r \dfrac {\d \theta} {\d t} \mathbf u_\theta + \dfrac {\d r} {\d t} \mathbf u_r$
where:
:$\mat... | Let the radius vector $\mathbf r$ from the origin to $p$ be expressed as:
:$(1): \quad \mathbf r = r \mathbf u_r$
:600px
From Derivatives of Unit Vectors in Polar Coordinates:
{{begin-eqn}}
{{eqn | n = 2
| l = \dfrac {\d \mathbf u_r} {\d \theta}
| r = \mathbf u_\theta
| c =
}}
{{eqn | n = 3
| l... | Consider a [[Definition:Particle|particle]] $p$ moving in the [[Definition:Plane|plane]].
Let the [[Definition:Position|position]] of $p$ at [[Definition:Time|time]] $t$ be given in [[Definition:Polar Coordinates|polar coordinates]] as $\left\langle{r, \theta}\right\rangle$.
Then the [[Definition:Velocity|velocity]]... | Let the [[Definition:Radius Vector|radius vector]] $\mathbf r$ from the [[Definition:Origin|origin]] to $p$ be expressed as:
:$(1): \quad \mathbf r = r \mathbf u_r$
:[[File:MotionInPolarPlane.png|600px]]
From [[Derivatives of Unit Vectors in Polar Coordinates]]:
{{begin-eqn}}
{{eqn | n = 2
| l = \dfrac {\d \... | Velocity Vector in Polar Coordinates | https://proofwiki.org/wiki/Velocity_Vector_in_Polar_Coordinates | https://proofwiki.org/wiki/Velocity_Vector_in_Polar_Coordinates | [
"Polar Coordinates"
] | [
"Definition:Particle",
"Definition:Plane Surface",
"Definition:Position",
"Definition:Time",
"Definition:Polar Coordinates",
"Definition:Velocity",
"Definition:Unit Vector",
"Definition:Polar Coordinates/Radial Coordinate",
"Definition:Unit Vector",
"Definition:Polar Coordinates/Angular Coordinate... | [
"Definition:Position Vector",
"Definition:Coordinate System/Origin",
"File:MotionInPolarPlane.png",
"Derivatives of Unit Vectors in Polar Coordinates",
"Definition:Velocity",
"Definition:Rate of Change/Time",
"Definition:Position",
"Product Rule for Derivatives",
"Derivative of Composite Function"
] |
proofwiki-11239 | Derivatives of Unit Vectors in Polar Coordinates | Consider a particle $p$ moving in the plane.
Let the position of $p$ be given in polar coordinates as $\polar {r, \theta}$.
Let:
:$\mathbf u_r$ be the unit vector in the direction of the radial coordinate of $p$
:$\mathbf u_\theta$ be the unit vector in the direction of the angular coordinate of $p$
Then the derivative... | By definition of sine and cosine:
{{begin-eqn}}
{{eqn | n = 1
| l = \mathbf u_r
| r = \mathbf i \cos \theta + \mathbf j \sin \theta
}}
{{eqn | n = 2
| l = \mathbf u_\theta
| r = -\mathbf i \sin \theta + \mathbf j \cos \theta
}}
{{end-eqn}}
where $\mathbf i$ and $\mathbf j$ are the unit vectors i... | Consider a [[Definition:Particle|particle]] $p$ moving in the [[Definition:Plane|plane]].
Let the [[Definition:Position|position]] of $p$ be given in [[Definition:Polar Coordinates|polar coordinates]] as $\polar {r, \theta}$.
Let:
:$\mathbf u_r$ be the [[Definition:Unit Vector|unit vector]] in the direction of the [[... | By definition of [[Definition:Sine|sine]] and [[Definition:Cosine|cosine]]:
{{begin-eqn}}
{{eqn | n = 1
| l = \mathbf u_r
| r = \mathbf i \cos \theta + \mathbf j \sin \theta
}}
{{eqn | n = 2
| l = \mathbf u_\theta
| r = -\mathbf i \sin \theta + \mathbf j \cos \theta
}}
{{end-eqn}}
where $\math... | Derivatives of Unit Vectors in Polar Coordinates | https://proofwiki.org/wiki/Derivatives_of_Unit_Vectors_in_Polar_Coordinates | https://proofwiki.org/wiki/Derivatives_of_Unit_Vectors_in_Polar_Coordinates | [
"Polar Coordinates"
] | [
"Definition:Particle",
"Definition:Plane Surface",
"Definition:Position",
"Definition:Polar Coordinates",
"Definition:Unit Vector",
"Definition:Polar Coordinates/Radial Coordinate",
"Definition:Unit Vector",
"Definition:Polar Coordinates/Angular Coordinate",
"Definition:Derivative"
] | [
"Definition:Sine",
"Definition:Cosine",
"Definition:Unit Vector",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"File:DerivativesOfUnitPolarVectors.png",
"Definition:Differentiation"
] |
proofwiki-11240 | Acceleration Vector in Polar Coordinates | Consider a particle $p$ moving in the plane.
Let the position of $p$ at time $t$ be given in polar coordinates as $\polar {r, \theta}$.
Then the acceleration $\mathbf a$ of $p$ can be expressed as:
:$\mathbf a = \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} } \mathbf u_\thet... | Let the radius vector $\mathbf r$ from the origin to $p$ be expressed as:
:$(1): \quad \mathbf r = r \mathbf u_r$
:600px
From Derivatives of Unit Vectors in Polar Coordinates:
{{begin-eqn}}
{{eqn | n = 2
| l = \dfrac {\d \mathbf u_r} {\d \theta}
| r = \mathbf u_\theta
| c =
}}
{{eqn | n = 3
| l... | Consider a [[Definition:Particle|particle]] $p$ moving in the [[Definition:Plane|plane]].
Let the [[Definition:Position|position]] of $p$ at [[Definition:Time|time]] $t$ be given in [[Definition:Polar Coordinates|polar coordinates]] as $\polar {r, \theta}$.
Then the [[Definition:Acceleration|acceleration]] $\mathbf ... | Let the [[Definition:Radius Vector|radius vector]] $\mathbf r$ from the [[Definition:Origin|origin]] to $p$ be expressed as:
:$(1): \quad \mathbf r = r \mathbf u_r$
:[[File:MotionInPolarPlane.png|600px]]
From [[Derivatives of Unit Vectors in Polar Coordinates]]:
{{begin-eqn}}
{{eqn | n = 2
| l = \dfrac {\d \... | Acceleration Vector in Polar Coordinates | https://proofwiki.org/wiki/Acceleration_Vector_in_Polar_Coordinates | https://proofwiki.org/wiki/Acceleration_Vector_in_Polar_Coordinates | [
"Polar Coordinates",
"Acceleration"
] | [
"Definition:Particle",
"Definition:Plane Surface",
"Definition:Position",
"Definition:Time",
"Definition:Polar Coordinates",
"Definition:Acceleration",
"Definition:Unit Vector",
"Definition:Polar Coordinates/Radial Coordinate",
"Definition:Unit Vector",
"Definition:Polar Coordinates/Angular Coordi... | [
"Definition:Position Vector",
"Definition:Coordinate System/Origin",
"File:MotionInPolarPlane.png",
"Derivatives of Unit Vectors in Polar Coordinates",
"Velocity Vector in Polar Coordinates",
"Definition:Velocity",
"Definition:Acceleration",
"Definition:Rate of Change/Time",
"Definition:Velocity",
... |
proofwiki-11241 | Eccentricity of Orbit indicates its Total Energy | Consider a planet $p$ of mass $m$ orbiting a star $S$ of mass $M$ under the influence of the gravitational field which the two bodies give rise to.
Then the total energy of the system determines the eccentricity of the orbit of $p$ around $S$. | Let:
:$\mathbf u_r$ be the unit vector in the direction of the radial coordinate of $p$
:$\mathbf u_\theta$ be the unit vector in the direction of the angular coordinate of $p$.
By Kinetic Energy of Motion, the kinetic energy of $p$ is:
:$K = \dfrac {m v^2} 2$
where $v$ is the magnitude of the velocity of $p$.
Thus:
{{... | Consider a [[Definition:Planet|planet]] $p$ of [[Definition:Mass|mass]] $m$ orbiting a [[Definition:Star (Physics)|star]] $S$ of [[Definition:Mass|mass]] $M$ under the influence of the [[Definition:Gravitational Field|gravitational field]] which the two [[Definition:Body|bodies]] give rise to.
Then the total [[Definit... | Let:
:$\mathbf u_r$ be the [[Definition:Unit Vector|unit vector]] in the direction of the [[Definition:Radial Coordinate|radial coordinate]] of $p$
:$\mathbf u_\theta$ be the [[Definition:Unit Vector|unit vector]] in the direction of the [[Definition:Angular Coordinate|angular coordinate]] of $p$.
By [[Kinetic Energy... | Eccentricity of Orbit indicates its Total Energy | https://proofwiki.org/wiki/Eccentricity_of_Orbit_indicates_its_Total_Energy | https://proofwiki.org/wiki/Eccentricity_of_Orbit_indicates_its_Total_Energy | [
"Celestial Mechanics",
"Eccentricity of Conic Section"
] | [
"Definition:Planet",
"Definition:Mass",
"Definition:Star (Physics)",
"Definition:Mass",
"Definition:Gravitational Field",
"Definition:Body",
"Definition:Energy",
"Definition:Conic Section/Eccentricity",
"Definition:Orbit (Physics)"
] | [
"Definition:Unit Vector",
"Definition:Polar Coordinates/Radial Coordinate",
"Definition:Unit Vector",
"Definition:Polar Coordinates/Angular Coordinate",
"Kinetic Energy of Motion",
"Definition:Kinetic Energy",
"Definition:Magnitude",
"Definition:Velocity",
"Dot Product of Vector with Itself",
"Vel... |
proofwiki-11242 | Existence of Non-Empty Finite Suprema in Join Semilattice | Let $\struct {S, \preceq}$ be a join semilattice.
Let $A$ be a non-empty finite subset of $S$.
Then $A$ admits a supremum in $\struct {S, \preceq}$. | We will prove by induction of the cardinality of finite subset of $H$. | Let $\struct {S, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $A$ be a [[Definition:Empty Set|non-empty]] [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of $S$.
Then $A$ admits a [[Definition:Supremum of Set|supremum]] in $\struct {S, \preceq}$. | We will prove by [[Principle of Mathematical Induction|induction]] of the [[Definition:Cardinality|cardinality]] of [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of $H$. | Existence of Non-Empty Finite Suprema in Join Semilattice | https://proofwiki.org/wiki/Existence_of_Non-Empty_Finite_Suprema_in_Join_Semilattice | https://proofwiki.org/wiki/Existence_of_Non-Empty_Finite_Suprema_in_Join_Semilattice | [
"Join and Meet Semilattices"
] | [
"Definition:Join Semilattice",
"Definition:Empty Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Supremum of Set"
] | [
"Principle of Mathematical Induction",
"Definition:Cardinality",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Cardinality",
"Definition:Subset",
"Definition:Subset",
"Definition:Subset"
] |
proofwiki-11243 | Variance of Exponential Distribution | Let $X$ be a continuous random variable with the exponential distribution with parameter $\beta$.
Then the variance of $X$ is:
:$\var X = \beta^2$ | From Variance as Expectation of Square minus Square of Expectation:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
From Expectation of Exponential Distribution:
:$\expect X = \beta$
The expectation of $X^2$ is:
{{begin-eqn}}
{{eqn | l = \expect {X^2}
| r = \int_{x \mathop \in \Omega_X} x^2 \, \map {f_X} x \rd x... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$.
Then the [[Definition:Variance of Discrete Random Variable|variance]] of $X$ is:
:$\var X = \beta^2$ | From [[Variance as Expectation of Square minus Square of Expectation]]:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
From [[Expectation of Exponential Distribution]]:
:$\expect X = \beta$
The expectation of $X^2$ is:
{{begin-eqn}}
{{eqn | l = \expect {X^2}
| r = \int_{x \mathop \in \Omega_X} x^2 \, \map {f... | Variance of Exponential Distribution/Proof 1 | https://proofwiki.org/wiki/Variance_of_Exponential_Distribution | https://proofwiki.org/wiki/Variance_of_Exponential_Distribution/Proof_1 | [
"Variance of Exponential Distribution",
"Exponential Distribution",
"Variance"
] | [
"Definition:Random Variable/Continuous",
"Definition:Exponential Distribution",
"Definition:Variance/Discrete"
] | [
"Variance as Expectation of Square minus Square of Expectation",
"Expectation of Exponential Distribution",
"Probability Density Function of Exponential Distribution",
"Integration by Parts",
"Expectation of Exponential Distribution",
"Definition:Variance/Discrete"
] |
proofwiki-11244 | Variance of Exponential Distribution | Let $X$ be a continuous random variable with the exponential distribution with parameter $\beta$.
Then the variance of $X$ is:
:$\var X = \beta^2$ | By Moment Generating Function of Exponential Distribution, the moment generating function $M_X$ of $X$ is given by:
:$\map {M_X} t = \dfrac 1 {1 - \beta t}$
From Variance as Expectation of Square minus Square of Expectation:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
From Moment in terms of Moment Generating F... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with the [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$.
Then the [[Definition:Variance of Discrete Random Variable|variance]] of $X$ is:
:$\var X = \beta^2$ | By [[Moment Generating Function of Exponential Distribution]], the [[Definition:Moment Generating Function|moment generating function]] $M_X$ of $X$ is given by:
:$\map {M_X} t = \dfrac 1 {1 - \beta t}$
From [[Variance as Expectation of Square minus Square of Expectation]]:
:$\var X = \expect {X^2} - \paren {\expe... | Variance of Exponential Distribution/Proof 2 | https://proofwiki.org/wiki/Variance_of_Exponential_Distribution | https://proofwiki.org/wiki/Variance_of_Exponential_Distribution/Proof_2 | [
"Variance of Exponential Distribution",
"Exponential Distribution",
"Variance"
] | [
"Definition:Random Variable/Continuous",
"Definition:Exponential Distribution",
"Definition:Variance/Discrete"
] | [
"Moment Generating Function of Exponential Distribution",
"Definition:Moment Generating Function",
"Variance as Expectation of Square minus Square of Expectation",
"Moment in terms of Moment Generating Function",
"Expectation of Exponential Distribution/Proof 2",
"Derivative of Composite Function",
"Pow... |
proofwiki-11245 | Cardinality of Singleton | Let $A$ be a set.
Then $\card A = 1$ {{iff}} $\exists a: A = \set a$
where $\card A$ denotes the cardinality of $A$. | === Sufficient Condition ===
Assume that
:$\card A = 1$
By definition of cardinality of finite set:
:$A \sim \N_{< 1} = \set 0$
where $\sim$ denotes set equivalence.
By Set Equivalence behaves like Equivalence Relation:
:$\set 0 \sim A$
By definition of set equivalence there exists a bijection:
:$f: \set 0 \to A$
By de... | Let $A$ be a [[Definition:Set|set]].
Then $\card A = 1$ {{iff}} $\exists a: A = \set a$
where $\card A$ denotes the [[Definition:Cardinality|cardinality]] of $A$. | === Sufficient Condition ===
Assume that
:$\card A = 1$
By definition of [[Definition:Cardinality of Finite Set|cardinality of finite set]]:
:$A \sim \N_{< 1} = \set 0$
where $\sim$ denotes [[Definition:Set Equivalence|set equivalence]].
By [[Set Equivalence behaves like Equivalence Relation]]:
:$\set 0 \sim A$
By ... | Cardinality of Singleton | https://proofwiki.org/wiki/Cardinality_of_Singleton | https://proofwiki.org/wiki/Cardinality_of_Singleton | [
"Cardinals",
"Cardinality"
] | [
"Definition:Set",
"Definition:Cardinality"
] | [
"Definition:Cardinality/Finite",
"Definition:Set Equivalence",
"Set Equivalence behaves like Equivalence Relation",
"Definition:Set Equivalence",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Surjection",
"Image of Singleton under Mapping",
"Definition:Surjection",
"Definition:Biject... |
proofwiki-11246 | Existence of Non-Empty Finite Infima in Meet Semilattice | Let $\struct {S, \preceq}$ be a meet semilattice.
Let $A$ be a non-empty finite subset of $S$.
Then $A$ admits a infimum in $\struct {S, \preceq}$. | This follows by {{mutatis}} of the proof of Existence of Non-Empty Finite Suprema in Join Semilattice.
{{qed}} | Let $\struct {S, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $A$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] of $S$.
Then $A$ admits a [[Definition:Infimum of Set|infimum]] in $\struct {S, \preceq}$. | This follows by {{mutatis}} of the proof of [[Existence of Non-Empty Finite Suprema in Join Semilattice]].
{{qed}} | Existence of Non-Empty Finite Infima in Meet Semilattice | https://proofwiki.org/wiki/Existence_of_Non-Empty_Finite_Infima_in_Meet_Semilattice | https://proofwiki.org/wiki/Existence_of_Non-Empty_Finite_Infima_in_Meet_Semilattice | [
"Join and Meet Semilattices"
] | [
"Definition:Meet Semilattice",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Infimum of Set"
] | [
"Existence of Non-Empty Finite Suprema in Join Semilattice"
] |
proofwiki-11247 | Circle is Bisected by Diameter | A circle is bisected by a diameter. | Let $AB$ be a diameter of a circle $ADBE$ whose center is at $C$.
By definition of diameter, $AB$ passes through $C$.
{{AimForCont}} that $AB$ does not bisect $ADBE$, but that $ADBC$ is larger than $AEBC$.
:400px
Thus it will be possible to find a diameter $DE$ passing through $C$ such that $DC \ne CE$.
Both $DC$ and $... | A [[Definition:Circle|circle]] is [[Definition:Bisection|bisected]] by a [[Definition:Diameter of Circle|diameter]]. | Let $AB$ be a [[Definition:Diameter of Circle|diameter]] of a [[Definition:Circle|circle]] $ADBE$ whose [[Definition:Center of Circle|center]] is at $C$.
By definition of [[Definition:Diameter of Circle|diameter]], $AB$ passes through $C$.
{{AimForCont}} that $AB$ does not [[Definition:Bisection|bisect]] $ADBE$, but ... | Circle is Bisected by Diameter/Proof 1 | https://proofwiki.org/wiki/Circle_is_Bisected_by_Diameter | https://proofwiki.org/wiki/Circle_is_Bisected_by_Diameter/Proof_1 | [
"Circles",
"Circle is Bisected by Diameter"
] | [
"Definition:Circle",
"Definition:Bisection",
"Definition:Circle/Diameter"
] | [
"Definition:Circle/Diameter",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Diameter",
"Definition:Bisection",
"File:CircleBisectedByDiameter.png",
"Definition:Circle/Diameter",
"Definition:Circle/Radius",
"Definition:Euclid's Definitions - Book I/15 - Circle",
"Definition:Ci... |
proofwiki-11248 | Circle is Bisected by Diameter | A circle is bisected by a diameter. | :400px
Let $AB$ be a diameter of a circle whose center is at $O$.
By definition of diameter, $AB$ passes through $O$.
$\angle AOB\cong\angle BOA$ because they are both straight angles.
Thus, the arcs are congruent by Equal Angles in Equal Circles:
:$\stackrel{\frown}{AB}\cong\stackrel{\frown}{BA}$
Hence, a circle is sp... | A [[Definition:Circle|circle]] is [[Definition:Bisection|bisected]] by a [[Definition:Diameter of Circle|diameter]]. | :[[File:Diameter Bisects Circle.png|400px]]
Let $AB$ be a [[Definition:Diameter of Circle|diameter]] of a [[Definition:Circle|circle]] whose [[Definition:Center of Circle|center]] is at $O$.
By definition of [[Definition:Diameter of Circle|diameter]], $AB$ passes through $O$.
$\angle AOB\cong\angle BOA$ because they... | Circle is Bisected by Diameter/Proof 2 | https://proofwiki.org/wiki/Circle_is_Bisected_by_Diameter | https://proofwiki.org/wiki/Circle_is_Bisected_by_Diameter/Proof_2 | [
"Circles",
"Circle is Bisected by Diameter"
] | [
"Definition:Circle",
"Definition:Bisection",
"Definition:Circle/Diameter"
] | [
"File:Diameter Bisects Circle.png",
"Definition:Circle/Diameter",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Diameter",
"Equal Angles in Equal Circles",
"Definition:Circle",
"Definition:Circle/Arc",
"Definition:Circle/Diameter"
] |
proofwiki-11249 | External Angle of Triangle equals Sum of other Internal Angles | The external angle of a triangle equals the sum of the other two internal angles.
{{:Euclid:Proposition/I/32}} | :300px
Let $\triangle ABC$ be a triangle.
Let $BC$ be extended to a point $D$.
Construct $CE$ through the point $C$ parallel to the straight line $AB$.
We have that $AB \parallel CE$ and $AC$ is a transversal that cuts them.
From Parallelism implies Equal Alternate Angles:
:$\angle BAC = \angle ACE$
Similarly, we have ... | The [[Definition:External Angle|external angle]] of a [[Definition:Triangle (Geometry)|triangle]] equals the sum of the other two [[Definition:Internal Angle|internal angles]].
{{:Euclid:Proposition/I/32}} | :[[File:Euclid-I-32.png|300px]]
Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $BC$ be extended to a point $D$.
[[Construction of Parallel Line|Construct $CE$]] through the point $C$ [[Definition:Parallel Lines|parallel]] to the [[Definition:Straight Line|straight line]] $AB$.
We have tha... | External Angle of Triangle equals Sum of other Internal Angles/Proof 1 | https://proofwiki.org/wiki/External_Angle_of_Triangle_equals_Sum_of_other_Internal_Angles | https://proofwiki.org/wiki/External_Angle_of_Triangle_equals_Sum_of_other_Internal_Angles/Proof_1 | [
"External Angle of Triangle equals Sum of other Internal Angles",
"Triangles",
"Euclid Book I",
"External Angles"
] | [
"Definition:Polygon/External Angle",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Internal Angle"
] | [
"File:Euclid-I-32.png",
"Definition:Triangle (Geometry)",
"Construction of Parallel Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line",
"Definition:Transversal (Geometry)",
"Parallelism implies Equal Alternate Angles",
"Definition:Transversal (Geometry)",
"Parallelism imp... |
proofwiki-11250 | External Angle of Triangle equals Sum of other Internal Angles | The external angle of a triangle equals the sum of the other two internal angles.
{{:Euclid:Proposition/I/32}} | Let $\triangle ABC$ be a triangle.
From Sum of Angles of Triangle equals Two Right Angles: Proof 2:
:$\paren 1: \angle ABC + \angle BCA + \angle CAB = 180^\circ$
Extend $AB$ to $D$.
By Two Angles on Straight Line make Two Right Angles:
:$\paren 1: \angle ABC + \angle CBD = 180^\circ$
Combining $\paren 1$ and $\paren 2$... | The [[Definition:External Angle|external angle]] of a [[Definition:Triangle (Geometry)|triangle]] equals the sum of the other two [[Definition:Internal Angle|internal angles]].
{{:Euclid:Proposition/I/32}} | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
From [[Sum of Angles of Triangle equals Two Right Angles/Proof 2|Sum of Angles of Triangle equals Two Right Angles: Proof 2]]:
:$\paren 1: \angle ABC + \angle BCA + \angle CAB = 180^\circ$
Extend $AB$ to $D$.
By [[Two Angles on Straight Line make ... | External Angle of Triangle equals Sum of other Internal Angles/Proof 2 | https://proofwiki.org/wiki/External_Angle_of_Triangle_equals_Sum_of_other_Internal_Angles | https://proofwiki.org/wiki/External_Angle_of_Triangle_equals_Sum_of_other_Internal_Angles/Proof_2 | [
"External Angle of Triangle equals Sum of other Internal Angles",
"Triangles",
"Euclid Book I",
"External Angles"
] | [
"Definition:Polygon/External Angle",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Internal Angle"
] | [
"Definition:Triangle (Geometry)",
"Sum of Angles of Triangle equals Two Right Angles/Proof 2",
"Two Angles on Straight Line make Two Right Angles",
"Equality is Transitive",
"Axiom:Euclid's Common Notions",
"Equality is Symmetric"
] |
proofwiki-11251 | First Pappus-Guldinus Theorem | Let $C$ be a plane figure that lies entirely on one side of a straight line $\LL$.
Let $S$ be the solid of revolution generated by $C$ around $\LL$.
Then the volume of $S$ is equal to the area of $C$ multiplied by the distance travelled by the centroid of $C$ around $\LL$ when generating $S$. | Let $V$ denote the volume of $S$
Let $\AA$ denote the area of $C$.
Let $C$ be embedded in a Cartesian plane such that $\LL$ coincides with the $x$-axis.
Let $\tuple {\overline x, \overline y}$ be the coordinates of the centroid of $C$.
Consider a rectangle $R$ bounded by the lines:
:$y = 0$
:$x = \xi$
:$x = \xi + \delt... | Let $C$ be a [[Definition:Plane Figure|plane figure]] that lies entirely on one side of a [[Definition:Straight Line|straight line]] $\LL$.
Let $S$ be the [[Definition:Solid of Revolution|solid of revolution]] generated by $C$ around $\LL$.
Then the [[Definition:Volume|volume]] of $S$ is equal to the [[Definition:Ar... | Let $V$ denote the [[Definition:Volume|volume]] of $S$
Let $\AA$ denote the [[Definition:Area|area]] of $C$.
Let $C$ be embedded in a [[Definition:Cartesian Plane|Cartesian plane]] such that $\LL$ coincides with the [[Definition:X-Axis|$x$-axis]].
Let $\tuple {\overline x, \overline y}$ be the [[Definition:Cartesian... | First Pappus-Guldinus Theorem | https://proofwiki.org/wiki/First_Pappus-Guldinus_Theorem | https://proofwiki.org/wiki/First_Pappus-Guldinus_Theorem | [
"Pappus's Theorems",
"Solids of Revolution"
] | [
"Definition:Geometric Figure/Plane Figure",
"Definition:Line/Straight Line",
"Definition:Solid of Revolution",
"Definition:Volume",
"Definition:Area",
"Definition:Arc Distance",
"Definition:Centroid"
] | [
"Definition:Volume",
"Definition:Area",
"Definition:Cartesian Plane",
"Definition:Axis/X-Axis",
"Definition:Cartesian Coordinate System",
"Definition:Centroid",
"Definition:Quadrilateral/Rectangle",
"Definition:Moment",
"Definition:Axis/Y-Axis",
"Area under Curve",
"Definition:Moment",
"Defini... |
proofwiki-11252 | Pythagoras's Theorem for Parallelograms | Let $\triangle ABC$ be a triangle.
Let $ACDE$ and $BCFG$ be parallelograms constructed on the sides $AC$ and $BC$ of $\triangle ABC$.
Let $DE$ and $FG$ be produced to intersect at $H$.
Let $AJ$ and $BI$ be constructed on $A$ and $B$ parallel to and equal to $HC$.
Then the area of the parallelogram $ABIJ$ equals the sum... | :400px
From Parallelograms with Same Base and Same Height have Equal Area:
:$ACDE = ACHR = ATUJ$
and:
:$BCFG = BCHS = BIUT$
Hence the result.
{{qed}} | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $ACDE$ and $BCFG$ be [[Definition:Parallelogram|parallelograms]] constructed on the [[Definition:Side of Polygon|sides]] $AC$ and $BC$ of $\triangle ABC$.
Let $DE$ and $FG$ be [[Definition:Production|produced]] to [[Definition:Intersection (Geo... | :[[File:PappusPythagorasExtension.png|400px]]
From [[Parallelograms with Same Base and Same Height have Equal Area]]:
:$ACDE = ACHR = ATUJ$
and:
:$BCFG = BCHS = BIUT$
Hence the result.
{{qed}} | Pythagoras's Theorem for Parallelograms | https://proofwiki.org/wiki/Pythagoras's_Theorem_for_Parallelograms | https://proofwiki.org/wiki/Pythagoras's_Theorem_for_Parallelograms | [
"Triangles",
"Parallelograms"
] | [
"Definition:Triangle (Geometry)",
"Definition:Quadrilateral/Parallelogram",
"Definition:Polygon/Side",
"Definition:Production",
"Definition:Intersection (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Definition:Area",
"Definition:Quadrilateral/Parallelogram",
"Definition:Area",
"Definition:... | [
"File:PappusPythagorasExtension.png",
"Parallelograms with Same Base and Same Height have Equal Area"
] |
proofwiki-11253 | Mapping Preserves Finite and Filtered Infima | Let $\struct {S_1, \preceq_1}$, $\struct {S_2, \preceq_2}$ be meet semilattices.
Let $f: S_1 \to S_2$ be a mapping.
Let $f$ preserve finite infima and preserve filtered infima.
Then $f$ also preserves all infima | Assume that
:$(1): \quad f$ preserves finite infima
and
:$(2): \quad f$ preserves filtered infima.
Let $X$ be a subset of $S_1$.
Let $X$ admits an infimum in $\struct {S_1, \preceq_1}$
Define $Z := \set {\inf A: A \in \map {\operatorname {Fin} } X \land A \ne \O}$
where
:$\inf A$ denotes the infimum of $A$ in $\struct ... | Let $\struct {S_1, \preceq_1}$, $\struct {S_2, \preceq_2}$ be [[Definition:Meet Semilattice|meet semilattices]].
Let $f: S_1 \to S_2$ be a [[Definition:Mapping|mapping]].
Let $f$ [[Definition:Mapping Preserves Finite Infimum|preserve finite infima]] and [[Definition:Mapping Preserves Filtered Infimum|preserve filter... | Assume that
:$(1): \quad f$ [[Definition:Mapping Preserves Finite Infimum|preserves finite infima]]
and
:$(2): \quad f$ [[Definition:Mapping Preserves Filtered Infimum|preserves filtered infima]].
Let $X$ be a [[Definition:Subset|subset]] of $S_1$.
Let $X$ admits an [[Definition:Infimum of Set|infimum]] in $\struct {... | Mapping Preserves Finite and Filtered Infima | https://proofwiki.org/wiki/Mapping_Preserves_Finite_and_Filtered_Infima | https://proofwiki.org/wiki/Mapping_Preserves_Finite_and_Filtered_Infima | [
"Order Theory",
"Join and Meet Semilattices"
] | [
"Definition:Meet Semilattice",
"Definition:Mapping",
"Definition:Mapping Preserves Infimum/Finite",
"Definition:Mapping Preserves Infimum/Filtered",
"Definition:Mapping Preserves Infimum/All"
] | [
"Definition:Mapping Preserves Infimum/Finite",
"Definition:Mapping Preserves Infimum/Filtered",
"Definition:Subset",
"Definition:Infimum of Set",
"Definition:Infimum of Set",
"Definition:Finite Set",
"Definition:Subset",
"Existence of Non-Empty Finite Infima in Meet Semilattice",
"Definition:Non-Emp... |
proofwiki-11254 | Volume of Sphere from Surface Area | The volume $V$ of a sphere of radius $r$ is given by:
:$V = \dfrac {r A} 3$
where $A$ is the surface area of the sphere. | Let the surface of the sphere of radius $r$ be divided into many small areas.
If they are made small enough, they can be approximated to plane figures.
Let the areas of these plane figures be denoted:
:$a_1, a_2, a_3, \ldots$
Let the sphere of radius $r$ be divided into as many pyramids whose apices are at the center a... | The [[Definition:Volume|volume]] $V$ of a [[Definition:Sphere (Geometry)|sphere]] of [[Definition:Radius of Sphere|radius]] $r$ is given by:
:$V = \dfrac {r A} 3$
where $A$ is the [[Definition:Surface Area|surface area]] of the [[Definition:Sphere (Geometry)|sphere]]. | Let the [[Definition:Surface of Solid Figure|surface]] of the [[Definition:Sphere (Geometry)|sphere]] of [[Definition:Radius of Sphere|radius]] $r$ be divided into many small areas.
If they are made small enough, they can be approximated to [[Definition:Plane Figure|plane figures]].
Let the [[Definition:Area|areas]] ... | Volume of Sphere from Surface Area | https://proofwiki.org/wiki/Volume_of_Sphere_from_Surface_Area | https://proofwiki.org/wiki/Volume_of_Sphere_from_Surface_Area | [
"Volume of Sphere",
"Spheres",
"Volume Formulas"
] | [
"Definition:Volume",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radius",
"Definition:Surface Area",
"Definition:Sphere/Geometry"
] | [
"Definition:Surface of Solid Figure",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radius",
"Definition:Geometric Figure/Plane Figure",
"Definition:Area",
"Definition:Geometric Figure/Plane Figure",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radius",
"Definition:Pyramid",... |
proofwiki-11255 | Factors of Mersenne Number M67 | The Mersenne number $M_{67}$ has the factors:
:$193 \, 707 \, 721$
:$761 \, 838 \, 257 \, 287$ | First we calculate $M_{67}$, which is $2^{67} - 1$:
:$2^2 = 2 \times 2 = 4$
{{begin-eqn}}
{{eqn | l = 2^4
| r = 2^2 \times 2^2
| c =
}}
{{eqn | r = 4 \times 4
| c =
}}
{{eqn | r = 16
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 2^8
| r = 2^4 \times 2^4
| c =
}}
{{eqn | r = 16 \... | The [[Definition:Mersenne Number|Mersenne number]] $M_{67}$ has the [[Definition:Divisor of Integer|factors]]:
:$193 \, 707 \, 721$
:$761 \, 838 \, 257 \, 287$ | First we calculate $M_{67}$, which is $2^{67} - 1$:
:$2^2 = 2 \times 2 = 4$
{{begin-eqn}}
{{eqn | l = 2^4
| r = 2^2 \times 2^2
| c =
}}
{{eqn | r = 4 \times 4
| c =
}}
{{eqn | r = 16
| c =
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 2^8
| r = 2^4 \times 2^4
| c =
}}
{{eqn | r =... | Factors of Mersenne Number M67 | https://proofwiki.org/wiki/Factors_of_Mersenne_Number_M67 | https://proofwiki.org/wiki/Factors_of_Mersenne_Number_M67 | [
"Mersenne Numbers",
"Classic Problems"
] | [
"Definition:Mersenne Number",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-11256 | Lower Bound is Lower Bound for Subset | Let $\struct {S, \preceq}$ be a preordered set.
Let $A, B$ be subsets of $S$ such that
:$B \subseteq A$
Let $L$ be an element of $S$.
Let $L$ be a lower bound for $A$.
Then $L$ is a lower bound for $B$. | Let $L$ be a lower bound for $A$.
By definition of lower bound:
:$\forall x \in A: L \preceq x$
By definition of subset:
:$\forall x \in B: x \in A$
Hence:
:$\forall x \in B: L \preceq x$
Thus by definition:
:$L$ is a lower bound for $B$.
{{qed}} | Let $\struct {S, \preceq}$ be a [[Definition:Preordered Set|preordered set]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$ such that
:$B \subseteq A$
Let $L$ be an [[Definition:Element|element]] of $S$.
Let $L$ be a [[Definition:Lower Bound of Set|lower bound]] for $A$.
Then $L$ is a [[Definition:Lower Bound... | Let $L$ be a [[Definition:Lower Bound of Set|lower bound]] for $A$.
By definition of [[Definition:Lower Bound of Set|lower bound]]:
:$\forall x \in A: L \preceq x$
By definition of [[Definition:Subset|subset]]:
:$\forall x \in B: x \in A$
Hence:
:$\forall x \in B: L \preceq x$
Thus by definition:
:$L$ is a [[Defini... | Lower Bound is Lower Bound for Subset | https://proofwiki.org/wiki/Lower_Bound_is_Lower_Bound_for_Subset | https://proofwiki.org/wiki/Lower_Bound_is_Lower_Bound_for_Subset | [
"Preorder Theory"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Subset",
"Definition:Element",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set"
] | [
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Subset",
"Definition:Lower Bound of Set"
] |
proofwiki-11257 | Upper Bound is Upper Bound for Subset | Let $\left({S, \preceq}\right)$ be a preordered set.
Let $A, B$ be subsets of $S$ such that
:$B \subseteq A$
Let $U$ be an upper bound for $A$.
Then $U$ is an upper bound for $B$. | Assume that:
: $U$ is upper bound for $A$.
By definition of upper bound:
:$\forall x \in A: x \preceq U$
By definition of subset:
:$\forall x \in B: x \in A$
Hence:
:$\forall x \in B: x \preceq U$
Thus by definition
: $U$ is sn upper bound for $B$.
{{qed}} | Let $\left({S, \preceq}\right)$ be a [[Definition:Preordered Set|preordered set]].
Let $A, B$ be [[Definition:Subset|subsets]] of $S$ such that
:$B \subseteq A$
Let $U$ be an [[Definition:Upper Bound of Set|upper bound]] for $A$.
Then $U$ is an [[Definition:Upper Bound of Set|upper bound]] for $B$. | Assume that:
: $U$ is [[Definition:Upper Bound of Set|upper bound]] for $A$.
By definition of [[Definition:Upper Bound of Set|upper bound]]:
:$\forall x \in A: x \preceq U$
By definition of [[Definition:Subset|subset]]:
:$\forall x \in B: x \in A$
Hence:
:$\forall x \in B: x \preceq U$
Thus by definition
: $U$ is s... | Upper Bound is Upper Bound for Subset | https://proofwiki.org/wiki/Upper_Bound_is_Upper_Bound_for_Subset | https://proofwiki.org/wiki/Upper_Bound_is_Upper_Bound_for_Subset | [
"Preorder Theory"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Subset",
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set"
] | [
"Definition:Upper Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Subset",
"Definition:Upper Bound of Set"
] |
proofwiki-11258 | Largest Rectangle with Given Perimeter is Square | Let $\SS$ be the set of all rectangles with a given perimeter $L$.
The element of $\SS$ with the largest area is the square with length of side $\dfrac L 4$. | Consider an arbitrary element $R$ of $\SS$.
Let $B$ be half the perimeter of $R$.
Let $x$ be the length of one side of $R$.
Then the length of an adjacent side is $B - x$.
The area $\AA$ of $R$ is then given by:
:$\AA = x \paren {B - x}$
Let $\AA$ be expressed in functional notation as:
:$\map f x = x \paren {B - x}$
W... | Let $\SS$ be the [[Definition:Set|set]] of all [[Definition:Rectangle|rectangles]] with a given [[Definition:Perimeter|perimeter]] $L$.
The element of $\SS$ with the largest [[Definition:Area|area]] is the [[Definition:Square (Geometry)|square]] with [[Definition:Length (Linear Measure)|length]] of [[Definition:Side o... | Consider an arbitrary element $R$ of $\SS$.
Let $B$ be half the [[Definition:Perimeter|perimeter]] of $R$.
Let $x$ be the [[Definition:Length (Linear Measure)|length]] of one [[Definition:Side of Polygon|side]] of $R$.
Then the [[Definition:Length (Linear Measure)|length]] of an [[Definition:Adjacent Sides|adjacent ... | Largest Rectangle with Given Perimeter is Square | https://proofwiki.org/wiki/Largest_Rectangle_with_Given_Perimeter_is_Square | https://proofwiki.org/wiki/Largest_Rectangle_with_Given_Perimeter_is_Square | [
"Squares",
"Rectangles"
] | [
"Definition:Set",
"Definition:Quadrilateral/Rectangle",
"Definition:Perimeter",
"Definition:Area",
"Definition:Quadrilateral/Square",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side"
] | [
"Definition:Perimeter",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Polygon/Adjacent/Sides",
"Definition:Area",
"Interior Extremum Theorem",
"Definition:Area",
"Definition:Maximum Value of Real Function/Absolute",
"Definition:Minim... |
proofwiki-11259 | Mapping Preserves Finite and Directed Suprema | Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be join semilattices.
Let $f: S_1 \to S_2$ be a mapping.
Let $f$ preserve finite suprema and preserve directed suprema.
Then $f$ also preserves all suprema | This follows by {{mutatis}} of the proof of Mapping Preserves Finite and Filtered Infima.
{{qed}} | Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be [[Definition:Join Semilattice|join semilattices]].
Let $f: S_1 \to S_2$ be a [[Definition:Mapping|mapping]].
Let $f$ [[Definition:Mapping Preserves Supremum/Finite|preserve finite suprema]] and [[Definition:Mapping Preserves Supremum/Directed|preserve ... | This follows by {{mutatis}} of the proof of [[Mapping Preserves Finite and Filtered Infima]].
{{qed}} | Mapping Preserves Finite and Directed Suprema | https://proofwiki.org/wiki/Mapping_Preserves_Finite_and_Directed_Suprema | https://proofwiki.org/wiki/Mapping_Preserves_Finite_and_Directed_Suprema | [
"Order Theory",
"Join and Meet Semilattices"
] | [
"Definition:Join Semilattice",
"Definition:Mapping",
"Definition:Mapping Preserves Supremum/Finite",
"Definition:Mapping Preserves Supremum/Directed",
"Definition:Mapping Preserves Supremum/All"
] | [
"Mapping Preserves Finite and Filtered Infima"
] |
proofwiki-11260 | Integer as Sum of Polygonal Numbers | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then $n$ is:
:$(1): \quad$ Either triangular or the sum of $2$ or $3$ triangular numbers
:$(2): \quad$ Either square or the sum of $2$, $3$ or $4$ square numbers
:$(3): \quad$ Either pentagonal or the sum of $2$, $3$, $4$ or $5$ pentagonal numbers
:and so on.
That i... | First some lemmata: | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then $n$ is:
:$(1): \quad$ Either [[Definition:Triangular Number|triangular]] or the [[Definition:Integer Addition|sum]] of $2$ or $3$ [[Definition:Triangular Number|triangular numbers]]
:$(2): \quad$ Either [[Definition:Squ... | First some [[Definition:Lemma|lemmata]]: | Integer as Sum of Polygonal Numbers | https://proofwiki.org/wiki/Integer_as_Sum_of_Polygonal_Numbers | https://proofwiki.org/wiki/Integer_as_Sum_of_Polygonal_Numbers | [
"Polygonal Numbers",
"Integer as Sum of Polygonal Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Triangular Number",
"Definition:Addition/Integers",
"Definition:Triangular Number",
"Definition:Square Number",
"Definition:Addition/Integers",
"Definition:Square Number",
"Definition:Pentagonal Number",
"Definition:Addition/Integers",
"Definitio... | [
"Definition:Lemma"
] |
proofwiki-11261 | Infima Preserving Mapping on Filters is Increasing | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets.
Let $f: S \to T$ be a mapping.
For every filter $F$ in $\struct {S, \preceq}$, let $f$ preserve the infimum on $F$.
Then $f$ is increasing. | Let $x, y \in S$ such that:
:$x \preceq y$
By Infimum of Singleton:
:$\set x$ and $\set y$ admit infima in $\struct {S, \preceq}$
By Infimum of Upper Closure of Set:
:$\set x^\succeq$ and $\set y^\succeq$ admit infima in $\struct {S, \preceq}$
where $\set x^\succeq$ denotes the upper closure of $\set x$.
By Upper Closu... | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
For every [[Definition:Filter in Ordered Set|filter]] $F$ in $\struct {S, \preceq}$, let $f$ [[Definition:Mapping Preserves Infimum/Subset|preserve the infimum]] on $F... | Let $x, y \in S$ such that:
:$x \preceq y$
By [[Infimum of Singleton]]:
:$\set x$ and $\set y$ admit [[Definition:Infimum of Set|infima]] in $\struct {S, \preceq}$
By [[Infimum of Upper Closure of Set]]:
:$\set x^\succeq$ and $\set y^\succeq$ admit [[Definition:Infimum of Set|infima]] in $\struct {S, \preceq}$
where ... | Infima Preserving Mapping on Filters is Increasing | https://proofwiki.org/wiki/Infima_Preserving_Mapping_on_Filters_is_Increasing | https://proofwiki.org/wiki/Infima_Preserving_Mapping_on_Filters_is_Increasing | [
"Order Theory",
"Increasing Mappings"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Filter in Ordered Set",
"Definition:Mapping Preserves Infimum/Subset",
"Definition:Increasing/Mapping"
] | [
"Infimum of Singleton",
"Definition:Infimum of Set",
"Infimum of Upper Closure of Set",
"Definition:Infimum of Set",
"Definition:Upper Closure/Set",
"Upper Closure of Singleton",
"Definition:Infimum of Set",
"Upper Closure of Element is Filter",
"Definition:Filter in Ordered Set",
"Definition:Mapp... |
proofwiki-11262 | Infimum of Upper Closure of Set | Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$.
Let $U = T^\succeq$ be the upper closure of $T$ in $S$.
Let $s \in S$.
Then $s$ is the infimum of $T$ {{iff}} it is the infimum of $U$. | This follows by {{mutatis}} of the proof of Supremum of Lower Closure of Set.
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $T \subseteq S$.
Let $U = T^\succeq$ be the [[Definition:Upper Closure of Subset|upper closure]] of $T$ in $S$.
Let $s \in S$.
Then $s$ is the [[Definition:Infimum of Set|infimum]] of $T$ {{iff}} it is the [[Definition:Infimum of Set|infi... | This follows by {{mutatis}} of the proof of [[Supremum of Lower Closure of Set]].
{{qed}} | Infimum of Upper Closure of Set | https://proofwiki.org/wiki/Infimum_of_Upper_Closure_of_Set | https://proofwiki.org/wiki/Infimum_of_Upper_Closure_of_Set | [
"Upper Closures"
] | [
"Definition:Ordered Set",
"Definition:Upper Closure/Set",
"Definition:Infimum of Set",
"Definition:Infimum of Set"
] | [
"Supremum of Lower Closure of Set"
] |
proofwiki-11263 | Cavalieri's Principle/Extension | Let two solid figures $S_1$ and $S_2$ have equal height.
Let the areas of the sections made by planes parallel to their bases and at equal distances from the bases always have the same ratio.
Then the volumes of $S_1$ and $S_2$ are in that same ratio. | {{ProofWanted}}
{{Namedfor|Bonaventura Francesco Cavalieri|cat = Cavalieri}} | Let two [[Definition:Solid Figure|solid figures]] $S_1$ and $S_2$ have equal [[Definition:Height (Linear Measure)|height]].
Let the [[Definition:Area|areas]] of the [[Definition:Intersection (Geometry)|sections]] made by [[Definition:Plane|planes]] [[Definition:Parallel Planes|parallel]] to their [[Definition:Base of ... | {{ProofWanted}}
{{Namedfor|Bonaventura Francesco Cavalieri|cat = Cavalieri}} | Cavalieri's Principle/Extension | https://proofwiki.org/wiki/Cavalieri's_Principle/Extension | https://proofwiki.org/wiki/Cavalieri's_Principle/Extension | [
"Cavalieri's Principle",
"Volume",
"Solid Geometry"
] | [
"Definition:Geometric Figure/Three-Dimensional Figure",
"Definition:Linear Measure/Height",
"Definition:Area",
"Definition:Intersection (Geometry)",
"Definition:Plane Surface",
"Definition:Parallel (Geometry)/Planes",
"Definition:Base of Solid Figure",
"Definition:Distance between Parallel Planes",
... | [] |
proofwiki-11264 | Cavalieri's Principle | Let two solid figures $S_1$ and $S_2$ have equal height.
Let sections made by planes parallel to their bases and at equal distances from the bases always have equal area.
Then the volumes of $S_1$ and $S_2$ are equal. | Let $H$ be the common height of the two figures.
The volume of a solid figure is its Lebesgue Measure in $\R^3$.
{{TheoremWanted|The above needs to be demonstrated as it is in fact is the core of the principle}}
Therefore:
{{begin-eqn}}
{{eqn | l = \map V {S_1}
| r = \map {\lambda^3} {S_1}
}}
{{eqn | r = \int_{\R... | Let two [[Definition:Solid Figure|solid figures]] $S_1$ and $S_2$ have equal [[Definition:Height (Linear Measure)|height]].
Let [[Definition:Intersection (Geometry)|sections]] made by [[Definition:Plane|planes]] [[Definition:Parallel Planes|parallel]] to their [[Definition:Base of Solid Figure|bases]] and at equal [[D... | Let $H$ be the common [[Definition:Height (Linear Measure)|height]] of the two [[Definition:Solid Figure|figures]].
The [[Definition:Volume|volume]] of a [[Definition:Solid Figure|solid figure]] is its [[Definition:Lebesgue Measure|Lebesgue Measure]] in $\R^3$.
{{TheoremWanted|The above needs to be demonstrated as it... | Cavalieri's Principle | https://proofwiki.org/wiki/Cavalieri's_Principle | https://proofwiki.org/wiki/Cavalieri's_Principle | [
"Cavalieri's Principle",
"Volume",
"Solid Geometry"
] | [
"Definition:Geometric Figure/Three-Dimensional Figure",
"Definition:Linear Measure/Height",
"Definition:Intersection (Geometry)",
"Definition:Plane Surface",
"Definition:Parallel (Geometry)/Planes",
"Definition:Base of Solid Figure",
"Definition:Distance between Parallel Planes",
"Definition:Base of S... | [
"Definition:Linear Measure/Height",
"Definition:Geometric Figure/Three-Dimensional Figure",
"Definition:Volume",
"Definition:Geometric Figure/Three-Dimensional Figure",
"Definition:Lebesgue Measure",
"Fubini's Theorem",
"Definition:Plane Surface",
"Definition:Area",
"Definition:Geometric Figure/Plan... |
proofwiki-11265 | Volume of Cone | Let $K$ be a cone whose base is of area $A$ and whose height is $h$.
Then the volume of $K$ is given by:
:$V_K = \dfrac {A h} 3$ | Let $V_K$ be the volume of $K$.
Let $V_C$ be the volume of a cylinder of base $A$ and of height $h$.
From Volume of Cylinder in terms of Height and Base Area:
:$V_C = A h$
From Volume of Cone is Third of Cylinder on Same Base and of Same Height:
{{begin-eqn}}
{{eqn | l = V_K
| r = \dfrac {V_C} 3
| c =
}}
{... | Let $K$ be a [[Definition:Cone (Geometry)|cone]] whose [[Definition:Base of Cone|base]] is of [[Definition:Area|area]] $A$ and whose [[Definition:Height of Cone|height]] is $h$.
Then the [[Definition:Volume|volume]] of $K$ is given by:
:$V_K = \dfrac {A h} 3$ | Let $V_K$ be the [[Definition:Volume|volume]] of $K$.
Let $V_C$ be the [[Definition:Volume|volume]] of a [[Definition:Cylinder|cylinder]] of [[Definition:Base of Cylinder|base]] $A$ and of [[Definition:Height of Cylinder|height]] $h$.
From [[Volume of Cylinder in terms of Height and Base Area]]:
:$V_C = A h$
From [[... | Volume of Cone | https://proofwiki.org/wiki/Volume_of_Cone | https://proofwiki.org/wiki/Volume_of_Cone | [
"Cones",
"Volume Formulas"
] | [
"Definition:Cone (Geometry)",
"Definition:Cone (Geometry)/Base",
"Definition:Area",
"Definition:Cone (Geometry)/Height",
"Definition:Volume"
] | [
"Definition:Volume",
"Definition:Volume",
"Definition:Cylinder",
"Definition:Cylinder/Base",
"Definition:Cylinder/Height",
"Volume of Cylinder/Height and Base Area",
"Volume of Cone is Third of Cylinder on Same Base and of Same Height"
] |
proofwiki-11266 | Volume of Cone | Let $K$ be a cone whose base is of area $A$ and whose height is $h$.
Then the volume of $K$ is given by:
:$V_K = \dfrac {A h} 3$ | :300px
Let a cone and a cylinder have the same base $ABCD$.
Let them be of equal height.
Suppose the cylinder is not three times the volume of the cone.
Then the cylinder will either be greater than or less than three times the volume of the cone.
First suppose that the cylinder is greater than three times the volume o... | Let $K$ be a [[Definition:Cone (Geometry)|cone]] whose [[Definition:Base of Cone|base]] is of [[Definition:Area|area]] $A$ and whose [[Definition:Height of Cone|height]] is $h$.
Then the [[Definition:Volume|volume]] of $K$ is given by:
:$V_K = \dfrac {A h} 3$ | :[[File:Euclid-XII-10.png|300px]]
Let a [[Definition:Right Circular Cone|cone]] and a [[Definition:Right Circular Cylinder|cylinder]] have the same [[Definition:Base of Solid Figure|base]] $ABCD$.
Let them be of equal [[Definition:Height of Solid Figure|height]].
Suppose the [[Definition:Right Circular Cylinder|cyl... | Volume of Cone is Third of Cylinder on Same Base and of Same Height/Proof 1 | https://proofwiki.org/wiki/Volume_of_Cone | https://proofwiki.org/wiki/Volume_of_Cone_is_Third_of_Cylinder_on_Same_Base_and_of_Same_Height/Proof_1 | [
"Cones",
"Volume Formulas"
] | [
"Definition:Cone (Geometry)",
"Definition:Cone (Geometry)/Base",
"Definition:Area",
"Definition:Cone (Geometry)/Height",
"Definition:Volume"
] | [
"File:Euclid-XII-10.png",
"Definition:Right Circular Cone",
"Definition:Right Circular Cylinder",
"Definition:Base of Solid Figure",
"Definition:Height of Solid Figure",
"Definition:Right Circular Cylinder",
"Definition:Volume",
"Definition:Right Circular Cone",
"Definition:Right Circular Cylinder",... |
proofwiki-11267 | Volume of Cone | Let $K$ be a cone whose base is of area $A$ and whose height is $h$.
Then the volume of $K$ is given by:
:$V_K = \dfrac {A h} 3$ | Let the cone be of height $h$.
Let the area of the base of the cone be $A$.
From Volume of Cylinder in terms of Height and Base Area, the volume of a cylinder of base $A$ and height $h$ is $A h$.
Let the cone be divided by planes parallel to its base each positioned some small distance $d$ apart.
Let $d$ be sufficientl... | Let $K$ be a [[Definition:Cone (Geometry)|cone]] whose [[Definition:Base of Cone|base]] is of [[Definition:Area|area]] $A$ and whose [[Definition:Height of Cone|height]] is $h$.
Then the [[Definition:Volume|volume]] of $K$ is given by:
:$V_K = \dfrac {A h} 3$ | Let the [[Definition:Cone (Geometry)|cone]] be of [[Definition:Height of Cone|height]] $h$.
Let the [[Definition:Area|area]] of the [[Definition:Base of Cone|base]] of the [[Definition:Cone (Geometry)|cone]] be $A$.
From [[Volume of Cylinder in terms of Height and Base Area]], the [[Definition:Volume|volume]] of a [[... | Volume of Cone is Third of Cylinder on Same Base and of Same Height/Proof 2 | https://proofwiki.org/wiki/Volume_of_Cone | https://proofwiki.org/wiki/Volume_of_Cone_is_Third_of_Cylinder_on_Same_Base_and_of_Same_Height/Proof_2 | [
"Cones",
"Volume Formulas"
] | [
"Definition:Cone (Geometry)",
"Definition:Cone (Geometry)/Base",
"Definition:Area",
"Definition:Cone (Geometry)/Height",
"Definition:Volume"
] | [
"Definition:Cone (Geometry)",
"Definition:Cone (Geometry)/Height",
"Definition:Area",
"Definition:Cone (Geometry)/Base",
"Definition:Cone (Geometry)",
"Volume of Cylinder/Height and Base Area",
"Definition:Volume",
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cylinder/Base",
"D... |
proofwiki-11268 | Volume of Right Circular Cylinder | The volume $V_C$ of a right circular cylinder whose bases are circles of radius $r$ and whose height is $h$ is given by the formula:
:$V_C = \pi r^2 h$ | Consider a right circular cylinder $C$ whose base is a circle of radius $r$ and whose height is $h$.
Let $V_C$ denote the volume of $C$.
From Volume of Cylinder in terms of Height and Base Area:
:$V_C = A h$
where $A$ is the area of the base of $C$.
From Area of Circle, the area of each base is:
:$A = \pi r^2$
Hence:
:... | The [[Definition:Volume|volume]] $V_C$ of a [[Definition:Right Circular Cylinder|right circular cylinder]] whose [[Definition:Base of Right Circular Cylinder|bases]] are [[Definition:Circle|circles]] of [[Definition:Radius of Circle|radius]] $r$ and whose [[Definition:Height of Cylinder|height]] is $h$ is given by the ... | Consider a [[Definition:Right Circular Cylinder|right circular cylinder]] $C$ whose [[Definition:Base of Right Circular Cylinder|base]] is a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ and whose [[Definition:Height of Cylinder|height]] is $h$.
Let $V_C$ denote the [[Definition:Volume|vol... | Volume of Right Circular Cylinder/Proof 1 | https://proofwiki.org/wiki/Volume_of_Right_Circular_Cylinder | https://proofwiki.org/wiki/Volume_of_Right_Circular_Cylinder/Proof_1 | [
"Volume of Right Circular Cylinder",
"Right Circular Cylinders",
"Volume Formulas"
] | [
"Definition:Volume",
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cylinder/Base",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cylinder/Height"
] | [
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cylinder/Base",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cylinder/Height",
"Definition:Volume",
"Volume of Cylinder/Height and Base Area",
"Definition:Area",
"Definition:Right Circular Cylinder/Base",
"Area of Cir... |
proofwiki-11269 | Volume of Right Circular Cylinder | The volume $V_C$ of a right circular cylinder whose bases are circles of radius $r$ and whose height is $h$ is given by the formula:
:$V_C = \pi r^2 h$ | :600px
Consider a right circular cylinder $C$ whose base is a circle of radius $r$ and whose height is $h$.
Consider a cuboid $K$ whose height is $h$ and whose base has the same area as the base of $C$.
Let the area of those bases be $A$.
Let $C$ be positioned with its base in the same plane as the base of $K$.
By Cava... | The [[Definition:Volume|volume]] $V_C$ of a [[Definition:Right Circular Cylinder|right circular cylinder]] whose [[Definition:Base of Right Circular Cylinder|bases]] are [[Definition:Circle|circles]] of [[Definition:Radius of Circle|radius]] $r$ and whose [[Definition:Height of Cylinder|height]] is $h$ is given by the ... | :[[File:VolumeOfCylinder.png|600px]]
Consider a [[Definition:Right Circular Cylinder|right circular cylinder]] $C$ whose [[Definition:Base of Right Circular Cylinder|base]] is a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$ and whose [[Definition:Height of Cylinder|height]] is $h$.
Consid... | Volume of Right Circular Cylinder/Proof 2 | https://proofwiki.org/wiki/Volume_of_Right_Circular_Cylinder | https://proofwiki.org/wiki/Volume_of_Right_Circular_Cylinder/Proof_2 | [
"Volume of Right Circular Cylinder",
"Right Circular Cylinders",
"Volume Formulas"
] | [
"Definition:Volume",
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cylinder/Base",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cylinder/Height"
] | [
"File:VolumeOfCylinder.png",
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cylinder/Base",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cylinder/Height",
"Definition:Cuboid",
"Definition:Cylinder/Height",
"Definition:Base of Solid Figure",
"Definition:Area",
"D... |
proofwiki-11270 | Slope of Tangent to Cycloid | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
The slope of the tangent to $C$ at the point $\tuple {x, y}$ is given by:
:$\dfrac {\d y} {\d x} = \cot \dfrac \theta 2$ | By Derivative of Curve at Point, the tangent to $C$ at the point $\tuple {x, y}$ is the derivative of its equation at that point.
Thus:
{{begin-eqn}}
{{eqn | l = \frac {\d x} {\d \theta}
| r = a \paren {1 - \cos \theta}
| c =
}}
{{eqn | l = \frac {\d y} {\d \theta}
| r = a \sin \theta
| c =
}}... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
The [[Definition:Slope of Straight Line|slope]] of the [[Definition:Tangent Line|tangent]] to $C$ at the point $\tuple {x, y}$ is given by:
:$\dfrac {\d y} {\d x} = \cot \dfr... | By [[Derivative of Curve at Point]], the [[Definition:Tangent Line|tangent]] to $C$ at the point $\tuple {x, y}$ is the [[Definition:Derivative of Real Function at Point|derivative]] of its equation at that point.
Thus:
{{begin-eqn}}
{{eqn | l = \frac {\d x} {\d \theta}
| r = a \paren {1 - \cos \theta}
| ... | Slope of Tangent to Cycloid/Proof 1 | https://proofwiki.org/wiki/Slope_of_Tangent_to_Cycloid | https://proofwiki.org/wiki/Slope_of_Tangent_to_Cycloid/Proof_1 | [
"Slope of Tangent to Cycloid",
"Cycloids"
] | [
"Definition:Cycloid",
"Definition:Slope/Straight Line",
"Definition:Tangent Line"
] | [
"Derivative of Curve at Point",
"Definition:Tangent Line",
"Definition:Derivative/Real Function/Derivative at Point"
] |
proofwiki-11271 | Slope of Tangent to Cycloid | Let $C$ be a cycloid generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
The slope of the tangent to $C$ at the point $\tuple {x, y}$ is given by:
:$\dfrac {\d y} {\d x} = \cot \dfrac \theta 2$ | Consider a polygon $ABCD$ being rolled along a straight line in the same way as the generating circle of $C$.
Let $A', B', C', D'$ be the points around which the $ABCD$ rotates while rolling.
:600px
The point $A$ traces out in succession several arcs of circles with centers $B', C', D'$.
The tangent to each of these ar... | Let $C$ be a [[Definition:Cycloid|cycloid]] generated by the equations:
:$x = a \paren {\theta - \sin \theta}$
:$y = a \paren {1 - \cos \theta}$
The [[Definition:Slope of Straight Line|slope]] of the [[Definition:Tangent Line|tangent]] to $C$ at the point $\tuple {x, y}$ is given by:
:$\dfrac {\d y} {\d x} = \cot \dfr... | Consider a [[Definition:Polygon|polygon]] $ABCD$ being rolled along a [[Definition:Straight Line|straight line]] in the same way as the [[Definition:Generating Circle of Cycloid|generating circle]] of $C$.
Let $A', B', C', D'$ be the [[Definition:Point|points]] around which the $ABCD$ rotates while rolling.
:[[File:... | Slope of Tangent to Cycloid/Proof 2 | https://proofwiki.org/wiki/Slope_of_Tangent_to_Cycloid | https://proofwiki.org/wiki/Slope_of_Tangent_to_Cycloid/Proof_2 | [
"Slope of Tangent to Cycloid",
"Cycloids"
] | [
"Definition:Cycloid",
"Definition:Slope/Straight Line",
"Definition:Tangent Line"
] | [
"Definition:Polygon",
"Definition:Line/Straight Line",
"Definition:Cycloid/Generating Circle",
"Definition:Point",
"File:TangentToCycloid-construction.png",
"Definition:Circle/Arc",
"Definition:Circle/Center",
"Definition:Tangent Line",
"Definition:Circle/Arc",
"Definition:Right Angle/Perpendicula... |
proofwiki-11272 | Upper Closure of Singleton | Let $\struct {S, \preceq}$ be an ordered set.
Let $s$ be an element of $S$.
Then:
:$\set s^\succeq = s^\succeq$
where:
:$\set s^\succeq$ denotes the upper closure of $\set s$
:$s^\succeq$ denotes the upper closure of $s$ | {{begin-eqn}}
{{eqn | l = \set s^\succeq
| r = \bigcup \set {t^\succeq: t \in \set s}
| c = {{Defof|Upper Closure of Subset}}
}}
{{eqn | r = \bigcup \set {s^\succeq}
| c = {{Defof|Singleton}}
}}
{{eqn | r = s^\succeq
| c = Union of Singleton
}}
{{end-eqn}}
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $s$ be an [[Definition:Element|element]] of $S$.
Then:
:$\set s^\succeq = s^\succeq$
where:
:$\set s^\succeq$ denotes the [[Definition:Upper Closure of Subset|upper closure]] of $\set s$
:$s^\succeq$ denotes the [[Definition:Upper Closure of... | {{begin-eqn}}
{{eqn | l = \set s^\succeq
| r = \bigcup \set {t^\succeq: t \in \set s}
| c = {{Defof|Upper Closure of Subset}}
}}
{{eqn | r = \bigcup \set {s^\succeq}
| c = {{Defof|Singleton}}
}}
{{eqn | r = s^\succeq
| c = [[Union of Singleton]]
}}
{{end-eqn}}
{{qed}} | Upper Closure of Singleton | https://proofwiki.org/wiki/Upper_Closure_of_Singleton | https://proofwiki.org/wiki/Upper_Closure_of_Singleton | [
"Upper Closures"
] | [
"Definition:Ordered Set",
"Definition:Element",
"Definition:Upper Closure/Set",
"Definition:Upper Closure/Element"
] | [
"Union of Singleton"
] |
proofwiki-11273 | Lower Closure of Singleton | Let $\struct {S, \preceq}$ be an ordered set.
Let $s$ be an element of $S$.
Then:
:$\set s^\preceq = s^\preceq$
where:
:$\set s^\preceq$ denotes the lower closure of $\set s$
:$s^\preceq$ denotes the lower closure of $s$. | {{begin-eqn}}
{{eqn | l = \set s^\preceq
| r = \bigcup \set {t^\preceq: t \in \set s}
| c = {{Defof|Lower Closure of Subset}}
}}
{{eqn | r = \bigcup \set {s^\preceq}
| c = {{Defof|Singleton}}
}}
{{eqn | r = s^\preceq
| c = Union of Singleton
}}
{{end-eqn}}
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $s$ be an [[Definition:Element|element]] of $S$.
Then:
:$\set s^\preceq = s^\preceq$
where:
:$\set s^\preceq$ denotes the [[Definition:Lower Closure of Subset|lower closure]] of $\set s$
:$s^\preceq$ denotes the [[Definition:Lower Closure ... | {{begin-eqn}}
{{eqn | l = \set s^\preceq
| r = \bigcup \set {t^\preceq: t \in \set s}
| c = {{Defof|Lower Closure of Subset}}
}}
{{eqn | r = \bigcup \set {s^\preceq}
| c = {{Defof|Singleton}}
}}
{{eqn | r = s^\preceq
| c = [[Union of Singleton]]
}}
{{end-eqn}}
{{qed}} | Lower Closure of Singleton | https://proofwiki.org/wiki/Lower_Closure_of_Singleton | https://proofwiki.org/wiki/Lower_Closure_of_Singleton | [
"Lower Closures"
] | [
"Definition:Ordered Set",
"Definition:Element",
"Definition:Lower Closure/Set",
"Definition:Lower Closure/Element"
] | [
"Union of Singleton"
] |
proofwiki-11274 | Upper Closure of Element is Filter | Let $\struct {S, \preceq}$ be an ordered set.
Let $s$ be an element of $S$.
Then:
:$s^\succeq$ is a filter in $\struct {S, \preceq}$
where $s^\succeq$ denotes the upper closure of $s$. | By Singleton is Directed and Filtered Subset
:$\set s$ is a filtered subset of $S$
By Filtered iff Upper Closure Filtered:
:$\set s^\succeq$ is a filtered subset of $S$
By Upper Closure is Upper Section:
:$\set s^\succeq$ is a upper section in $S$
By Upper Closure of Singleton
:$\set s^\succeq = s^\succeq$
By definitio... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $s$ be an [[Definition:Element|element]] of $S$.
Then:
:$s^\succeq$ is a [[Definition:Filter in Ordered Set|filter]] in $\struct {S, \preceq}$
where $s^\succeq$ denotes the [[Definition:Upper Closure of Element|upper closure]] of $s$. | By [[Singleton is Directed and Filtered Subset]]
:$\set s$ is a [[Definition:Filtered Subset|filtered subset]] of $S$
By [[Filtered iff Upper Closure Filtered]]:
:$\set s^\succeq$ is a [[Definition:Filtered Subset|filtered subset]] of $S$
By [[Upper Closure is Upper Section]]:
:$\set s^\succeq$ is a [[Definition:Uppe... | Upper Closure of Element is Filter | https://proofwiki.org/wiki/Upper_Closure_of_Element_is_Filter | https://proofwiki.org/wiki/Upper_Closure_of_Element_is_Filter | [
"Upper Closures"
] | [
"Definition:Ordered Set",
"Definition:Element",
"Definition:Filter in Ordered Set",
"Definition:Upper Closure/Element"
] | [
"Singleton is Directed and Filtered Subset",
"Definition:Filtered Subset",
"Filtered iff Upper Closure Filtered",
"Definition:Filtered Subset",
"Upper Closure is Upper Section",
"Definition:Upper Section",
"Upper Closure of Singleton",
"Definition:Reflexivity",
"Definition:Upper Closure/Element",
... |
proofwiki-11275 | Length of Logarithmic Spiral | Consider a logarithmic spiral $S$ given by the equation:
:$r = a e^{b \theta}$
Construct a tangent to $S$ at the point $Q = \tuple {a, 0}$.
Let the tangent cross the $y$-axis at $P$.
Then the length of $PQ$ equals the total length of $S$ from $P$ inwards to the origin. |
:400px
{{ProofWanted}} | Consider a [[Definition:Logarithmic Spiral|logarithmic spiral]] $S$ given by the equation:
:$r = a e^{b \theta}$
Construct a [[Definition:Tangent Line|tangent]] to $S$ at the point $Q = \tuple {a, 0}$.
Let the [[Definition:Tangent Line|tangent]] cross the [[Definition:Y-Axis|$y$-axis]] at $P$.
Then the [[Definition... |
:[[File:LogarithmicSpiralLength.png|400px]]
{{ProofWanted}} | Length of Logarithmic Spiral | https://proofwiki.org/wiki/Length_of_Logarithmic_Spiral | https://proofwiki.org/wiki/Length_of_Logarithmic_Spiral | [
"Logarithmic Spiral"
] | [
"Definition:Logarithmic Spiral",
"Definition:Tangent Line",
"Definition:Tangent Line",
"Definition:Axis/Y-Axis",
"Definition:Linear Measure/Length",
"Definition:Arc Length",
"Definition:Coordinate System/Origin"
] | [
"File:LogarithmicSpiralLength.png"
] |
proofwiki-11276 | Logarithmic Spiral is Equiangular | The logarithmic spiral is '''equiangular''', in the following sense:
Let $P = \polar {r, \theta}$ be a point on a logarithmic spiral $S$ expressed in polar coordinates as:
:$r = a e^{b \theta}$
Then the angle $\psi$ that the tangent makes to the radius vector of $S$ is constant and equal to $\arccot b$. | Consider the logarithmic spiral $S$ expressed as:
:$r = a e^{b \theta}$
:400px
Let $\psi$ be the angle between the tangent to $S$ and the radius vector.
The derivative of $r$ {{WRT|Differentiation}} $\theta$ is:
:$\dfrac {\d r} {\d \theta} = a b e^{b \theta} = b r$
and thus:
{{begin-eqn}}
{{eqn | l = \tan \psi
|... | The [[Definition:Logarithmic Spiral|logarithmic spiral]] is '''[[Definition:Equiangular Spiral|equiangular]]''', in the following sense:
Let $P = \polar {r, \theta}$ be a [[Definition:Point|point]] on a [[Definition:Logarithmic Spiral|logarithmic spiral]] $S$ expressed in [[Definition:Polar Coordinates|polar coordina... | Consider the [[Definition:Logarithmic Spiral|logarithmic spiral]] $S$ expressed as:
:$r = a e^{b \theta}$
:[[File:LogarithmicSpiralAngle.png|400px]]
Let $\psi$ be the [[Definition:Angle|angle]] between the [[Definition:Tangent Line|tangent]] to $S$ and the [[Definition:Radius Vector|radius vector]].
The [[Defini... | Logarithmic Spiral is Equiangular | https://proofwiki.org/wiki/Logarithmic_Spiral_is_Equiangular | https://proofwiki.org/wiki/Logarithmic_Spiral_is_Equiangular | [
"Logarithmic Spiral"
] | [
"Definition:Logarithmic Spiral",
"Definition:Logarithmic Spiral",
"Definition:Point",
"Definition:Logarithmic Spiral",
"Definition:Polar Coordinates",
"Definition:Angle",
"Definition:Tangent Line",
"Definition:Position Vector",
"Definition:Constant"
] | [
"Definition:Logarithmic Spiral",
"File:LogarithmicSpiralAngle.png",
"Definition:Angle",
"Definition:Tangent Line",
"Definition:Position Vector",
"Definition:Derivative",
"Definition:Logarithmic Spiral",
"Definition:Constant"
] |
proofwiki-11277 | Infimum of Upper Closure of Element | Let $\struct {S, \preceq}$ be an ordered set.
Let $s$ be an element of $S$.
Then:
:$\map \inf {s^\succeq} = s$
where $s^\succeq$ denotes the upper closure of $s$. | {{begin-eqn}}
{{eqn | l = \map \inf {s^\succeq}
| r = \map \inf {\set s^\succeq}
| c = Upper Closure of Singleton
}}
{{eqn | r = \map \inf {\set s}
| c = Infimum of Upper Closure of Set
}}
{{eqn | r = s
| c = Infimum of Singleton
}}
{{end-eqn}}
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $s$ be an [[Definition:Element|element]] of $S$.
Then:
:$\map \inf {s^\succeq} = s$
where $s^\succeq$ denotes the [[Definition:Upper Closure of Element|upper closure]] of $s$. | {{begin-eqn}}
{{eqn | l = \map \inf {s^\succeq}
| r = \map \inf {\set s^\succeq}
| c = [[Upper Closure of Singleton]]
}}
{{eqn | r = \map \inf {\set s}
| c = [[Infimum of Upper Closure of Set]]
}}
{{eqn | r = s
| c = [[Infimum of Singleton]]
}}
{{end-eqn}}
{{qed}} | Infimum of Upper Closure of Element | https://proofwiki.org/wiki/Infimum_of_Upper_Closure_of_Element | https://proofwiki.org/wiki/Infimum_of_Upper_Closure_of_Element | [
"Upper Closures"
] | [
"Definition:Ordered Set",
"Definition:Element",
"Definition:Upper Closure/Element"
] | [
"Upper Closure of Singleton",
"Infimum of Upper Closure of Set",
"Infimum of Singleton"
] |
proofwiki-11278 | Volume of Gabriel's Horn | Consider Gabriel's horn, the solid of revolution formed by rotating about the $x$-axis the curve:
:$y = \dfrac 1 x$
Consider the volume $V$ of the space enclosed by the planes $x = 1$, $x = a$ and the portion of Gabriel's horn where $1 \le x \le a$.
Then:
:$V = \pi \paren {1 - \dfrac 1 a}$ | From Volume of Solid of Revolution:
{{begin-eqn}}
{{eqn | l = V
| r = \pi \int_1^a \frac 1 {x^2} \rd x
| c =
}}
{{eqn | r = \pi \intlimits {-\dfrac 1 x} 1 a
| c = Primitive of Power
}}
{{eqn | r = \pi \intlimits {\dfrac 1 x} a 1
| c =
}}
{{eqn | r = \pi \paren {1 - \dfrac 1 a}
| c =
}}
{{e... | Consider [[Definition:Gabriel's Horn|Gabriel's horn]], the [[Definition:Solid of Revolution|solid of revolution]] formed by rotating about the [[Definition:X-Axis|$x$-axis]] the [[Definition:Curve|curve]]:
:$y = \dfrac 1 x$
Consider the [[Definition:Volume|volume]] $V$ of the space enclosed by the [[Definition:Plane... | From [[Volume of Solid of Revolution]]:
{{begin-eqn}}
{{eqn | l = V
| r = \pi \int_1^a \frac 1 {x^2} \rd x
| c =
}}
{{eqn | r = \pi \intlimits {-\dfrac 1 x} 1 a
| c = [[Primitive of Power]]
}}
{{eqn | r = \pi \intlimits {\dfrac 1 x} a 1
| c =
}}
{{eqn | r = \pi \paren {1 - \dfrac 1 a}
| c... | Volume of Gabriel's Horn | https://proofwiki.org/wiki/Volume_of_Gabriel's_Horn | https://proofwiki.org/wiki/Volume_of_Gabriel's_Horn | [
"Gabriel's Horn"
] | [
"Definition:Gabriel's Horn",
"Definition:Solid of Revolution",
"Definition:Axis/X-Axis",
"Definition:Line/Curve",
"Definition:Volume",
"Definition:Plane Surface",
"Definition:Gabriel's Horn"
] | [
"Volume of Solid of Revolution",
"Primitive of Power"
] |
proofwiki-11279 | Pascal's Theorem | Let $ABCDEF$ be a hexagon whose $6$ vertices lie on a conic section and whose opposite sides are not parallel.
Then the points of intersection of the opposite sides, when produced as necessary, all lie on a straight line. | :400px
:500px
{{ProofWanted}} | Let $ABCDEF$ be a [[Definition:Hexagon|hexagon]] whose $6$ [[Definition:Vertex of Polygon|vertices]] lie on a [[Definition:Conic Section|conic section]] and whose [[Definition:Opposite Sides|opposite sides]] are not [[Definition:Parallel Lines|parallel]].
Then the [[Definition:Point|points]] of [[Definition:Intersecti... | :[[File:PascalsTheorem.png|400px]]
:[[File:PascalsTheorem2.png|500px]]
{{ProofWanted}} | Pascal's Theorem | https://proofwiki.org/wiki/Pascal's_Theorem | https://proofwiki.org/wiki/Pascal's_Theorem | [
"Pascal's Theorem",
"Conic Sections",
"Projective Geometry"
] | [
"Definition:Hexagon",
"Definition:Polygon/Vertex",
"Definition:Conic Section",
"Definition:Polygon/Opposite",
"Definition:Parallel (Geometry)/Lines",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Polygon/Opposite",
"Definition:Production",
"Definition:Line/Straight Line"
] | [
"File:PascalsTheorem.png",
"File:PascalsTheorem2.png"
] |
proofwiki-11280 | Lower Closure of Element is Ideal | Let $\struct {S, \preceq}$ ne an ordered set.
Let $s$ be an element of $S$.
Then $s^\preceq$ is ideal in $\struct {S, \preceq}$
where $s^\preceq$ denotes the lower closure of $s$. | By Singleton is Directed and Filtered Subset:
:$\set s$ is a directed subset of $S$
By Directed iff Lower Closure Directed:
:$\set s^\preceq$ is a directed subset of $S$
By Lower Closure is Lower Section:
:$\set s^\preceq$ is a lower section in $S$
By Lower Closure of Singleton
:$\set s^\preceq = s^\preceq$
By definiti... | Let $\struct {S, \preceq}$ ne an [[Definition:Ordered Set|ordered set]].
Let $s$ be an [[Definition:Element|element]] of $S$.
Then $s^\preceq$ is [[Definition:Ideal in Ordered Set|ideal]] in $\struct {S, \preceq}$
where $s^\preceq$ denotes the [[Definition:Lower Closure of Element|lower closure]] of $s$. | By [[Singleton is Directed and Filtered Subset]]:
:$\set s$ is a [[Definition:Directed Subset|directed subset]] of $S$
By [[Directed iff Lower Closure Directed]]:
:$\set s^\preceq$ is a [[Definition:directed Subset|directed subset]] of $S$
By [[Lower Closure is Lower Section]]:
:$\set s^\preceq$ is a [[Definition:Low... | Lower Closure of Element is Ideal | https://proofwiki.org/wiki/Lower_Closure_of_Element_is_Ideal | https://proofwiki.org/wiki/Lower_Closure_of_Element_is_Ideal | [
"Lower Closures"
] | [
"Definition:Ordered Set",
"Definition:Element",
"Definition:Ideal in Ordered Set",
"Definition:Lower Closure/Element"
] | [
"Singleton is Directed and Filtered Subset",
"Definition:Directed Subset",
"Directed iff Lower Closure Directed",
"Definition:directed Subset",
"Lower Closure is Lower Section",
"Definition:Lower Section",
"Lower Closure of Singleton",
"Definition:Reflexivity",
"Definition:Lower Closure/Element",
... |
proofwiki-11281 | Supremum of Lower Closure of Element | Let $\struct {S, \preceq}$ be an ordered set.
Let $s$ be an element of $S$.
Then:
:$\map \sup {s^\preceq} = s$
where $s^\preceq$ denotes the lower closure of $s$. | {{begin-eqn}}
{{eqn | l = \map \sup {s^\preceq}
| r = \map \sup {\set s^\preceq}
| c = Lower Closure of Singleton
}}
{{eqn | r = \map \sup {\set s}
| c = Supremum of Lower Closure of Set
}}
{{eqn | r = s
| c = Supremum of Singleton
}}
{{end-eqn}}
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $s$ be an [[Definition:Element|element]] of $S$.
Then:
:$\map \sup {s^\preceq} = s$
where $s^\preceq$ denotes the [[Definition:Lower Closure of Element|lower closure]] of $s$. | {{begin-eqn}}
{{eqn | l = \map \sup {s^\preceq}
| r = \map \sup {\set s^\preceq}
| c = [[Lower Closure of Singleton]]
}}
{{eqn | r = \map \sup {\set s}
| c = [[Supremum of Lower Closure of Set]]
}}
{{eqn | r = s
| c = [[Supremum of Singleton]]
}}
{{end-eqn}}
{{qed}} | Supremum of Lower Closure of Element | https://proofwiki.org/wiki/Supremum_of_Lower_Closure_of_Element | https://proofwiki.org/wiki/Supremum_of_Lower_Closure_of_Element | [
"Lower Closures"
] | [
"Definition:Ordered Set",
"Definition:Element",
"Definition:Lower Closure/Element"
] | [
"Lower Closure of Singleton",
"Supremum of Lower Closure of Set",
"Supremum of Singleton"
] |
proofwiki-11282 | Upper Closure is Decreasing | Let $\struct {S, \preceq}$ be an ordered set.
Let $x, y$ be elements of $S$ such that
:$x \preceq y$
then $y^\succeq \subseteq x^\succeq$
where $y^\succeq$ denotes the upper closure of $y$. | Let $z \in y^\succeq$.
By definition of upper closure of element:
:$y \preceq z$
By definition of ordering, $\preceq$ is transitive.
From $x \preceq y$ and $y \preceq z$:
:$x \preceq z$
Thus again by definition of upper closure of element:
:$z \in x^\succeq$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y$ be [[Definition:Element|elements]] of $S$ such that
:$x \preceq y$
then $y^\succeq \subseteq x^\succeq$
where $y^\succeq$ denotes the [[Definition:Upper Closure of Element|upper closure]] of $y$. | Let $z \in y^\succeq$.
By definition of [[Definition:Upper Closure of Element|upper closure of element]]:
:$y \preceq z$
By definition of [[Definition:Ordering|ordering]], $\preceq$ is [[Definition:Transitive Relation|transitive]].
From $x \preceq y$ and $y \preceq z$:
:$x \preceq z$
Thus again by definition of [[... | Upper Closure is Decreasing | https://proofwiki.org/wiki/Upper_Closure_is_Decreasing | https://proofwiki.org/wiki/Upper_Closure_is_Decreasing | [
"Upper Closures"
] | [
"Definition:Ordered Set",
"Definition:Element",
"Definition:Upper Closure/Element"
] | [
"Definition:Upper Closure/Element",
"Definition:Ordering",
"Definition:Transitive Relation",
"Definition:Upper Closure/Element"
] |
proofwiki-11283 | Lower Closure is Increasing | Let $\struct {S, \preceq}$ be an ordered set.
Let $x, y$ be elements of $S$ such that
:$x \preceq y$
Then:
:$x^\preceq \subseteq y^\preceq$
where $y^\preceq$ denotes the lower closure of $y$. | Let $z \in x^\preceq$.
By definition of lower closure of element:
:$z \preceq x$
By definition of transitivity:
:$z \preceq y$
Thus again by definition of lower closure of element:
:$z \in y^\preceq$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $x, y$ be [[Definition:Element|elements]] of $S$ such that
:$x \preceq y$
Then:
:$x^\preceq \subseteq y^\preceq$
where $y^\preceq$ denotes the [[Definition:Lower Closure of Element|lower closure]] of $y$. | Let $z \in x^\preceq$.
By definition of [[Definition:Lower Closure of Element|lower closure of element]]:
:$z \preceq x$
By definition of [[Definition:Transitivity|transitivity]]:
:$z \preceq y$
Thus again by definition of [[Definition:Lower Closure of Element|lower closure of element]]:
:$z \in y^\preceq$
{{qed}} | Lower Closure is Increasing | https://proofwiki.org/wiki/Lower_Closure_is_Increasing | https://proofwiki.org/wiki/Lower_Closure_is_Increasing | [
"Lower Closures"
] | [
"Definition:Ordered Set",
"Definition:Element",
"Definition:Lower Closure/Element"
] | [
"Definition:Lower Closure/Element",
"Definition:Transitive",
"Definition:Lower Closure/Element"
] |
proofwiki-11284 | Suprema Preserving Mapping on Ideals is Increasing | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets.
Let $f: S \to T$ be a mapping.
For every ideal $I$ in $\struct {S, \preceq}$, let $f$ preserve the supremum on $I$.
Then $f$ is increasing. | Let $x, y \in S$ such that:
:$x \preceq y$
By Supremum of Singleton:
:$\set x$ and $\set y$ admit suprema in $\struct {S, \preceq}$
By Supremum of Lower Closure of Set:
:$\set x^\preceq$ and $\set y^\preceq$ admit suprema in $\struct {S, \preceq}$
where $\set x^\preceq$ denotes the lower closure of $\set x$
By Lower Cl... | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
For every [[Definition:Ideal in Ordered Set|ideal]] $I$ in $\struct {S, \preceq}$, let $f$ [[Definition:Mapping Preserves Supremum/Subset|preserve the supremum]] on... | Let $x, y \in S$ such that:
:$x \preceq y$
By [[Supremum of Singleton]]:
:$\set x$ and $\set y$ admit [[Definition:Supremum of Set|suprema]] in $\struct {S, \preceq}$
By [[Supremum of Lower Closure of Set]]:
:$\set x^\preceq$ and $\set y^\preceq$ admit [[Definition:Supremum of Set|suprema]] in $\struct {S, \preceq}$
... | Suprema Preserving Mapping on Ideals is Increasing | https://proofwiki.org/wiki/Suprema_Preserving_Mapping_on_Ideals_is_Increasing | https://proofwiki.org/wiki/Suprema_Preserving_Mapping_on_Ideals_is_Increasing | [
"Order Theory",
"Increasing Mappings"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Ideal in Ordered Set",
"Definition:Mapping Preserves Supremum/Subset",
"Definition:Increasing/Mapping"
] | [
"Supremum of Singleton",
"Definition:Supremum of Set",
"Supremum of Lower Closure of Set",
"Definition:Supremum of Set",
"Definition:Lower Closure/Set",
"Lower Closure of Singleton",
"Definition:Supremum of Set",
"Lower Closure of Element is Ideal",
"Definition:Ideal in Ordered Set",
"Definition:M... |
proofwiki-11285 | Imaginary Unit to Power of Itself | :$i^i = e^{-\pi / 2}$
where $i$ is the imaginary unit. | {{begin-eqn}}
{{eqn | l = i^i
| r = \paren {e^{i \pi / 2} }^i
| c = Euler's Formula
}}
{{eqn | r = e^{\pi i^2 / 2}
| c =
}}
{{eqn | r = e^{-\pi / 2}
| c =
}}
{{end-eqn}}
{{qed}} | :$i^i = e^{-\pi / 2}$
where $i$ is the [[Definition:Imaginary Unit|imaginary unit]]. | {{begin-eqn}}
{{eqn | l = i^i
| r = \paren {e^{i \pi / 2} }^i
| c = [[Euler's Formula]]
}}
{{eqn | r = e^{\pi i^2 / 2}
| c =
}}
{{eqn | r = e^{-\pi / 2}
| c =
}}
{{end-eqn}}
{{qed}} | Imaginary Unit to Power of Itself | https://proofwiki.org/wiki/Imaginary_Unit_to_Power_of_Itself | https://proofwiki.org/wiki/Imaginary_Unit_to_Power_of_Itself | [
"Imaginary Unit to Power of Itself",
"Imaginary Unit"
] | [
"Definition:Complex Number/Imaginary Unit"
] | [
"Euler's Formula"
] |
proofwiki-11286 | Imaginary Unit to Power of Itself/Complete | :$i^i = \set {\exp \paren {\dfrac {4 k + 3} 2 \pi}: k \in \Z}$
where $i$ is the imaginary unit. | {{begin-eqn}}
{{eqn | l = i^i
| r = \exp \paren {i \ln \paren i}
| c = {{Defof|Complex Power}}
}}
{{eqn | r = \exp \paren {i \paren {\ln 1 + i \paren {\dfrac \pi 2 + 2 k \pi} } }
| c = {{Defof|Complex Natural Logarithm}}: for all $k \in \Z$
}}
{{eqn | r = \exp \paren {i \paren {i \paren {\dfrac \pi 2 ... | :$i^i = \set {\exp \paren {\dfrac {4 k + 3} 2 \pi}: k \in \Z}$
where $i$ is the [[Definition:Imaginary Unit|imaginary unit]]. | {{begin-eqn}}
{{eqn | l = i^i
| r = \exp \paren {i \ln \paren i}
| c = {{Defof|Complex Power}}
}}
{{eqn | r = \exp \paren {i \paren {\ln 1 + i \paren {\dfrac \pi 2 + 2 k \pi} } }
| c = {{Defof|Complex Natural Logarithm}}: for all $k \in \Z$
}}
{{eqn | r = \exp \paren {i \paren {i \paren {\dfrac \pi 2 ... | Imaginary Unit to Power of Itself/Complete | https://proofwiki.org/wiki/Imaginary_Unit_to_Power_of_Itself/Complete | https://proofwiki.org/wiki/Imaginary_Unit_to_Power_of_Itself/Complete | [
"Imaginary Unit to Power of Itself"
] | [
"Definition:Complex Number/Imaginary Unit"
] | [
"Natural Logarithm of 1 is 0",
"Definition:Integer"
] |
proofwiki-11287 | Riemann Zeta Function of 4 | The Riemann zeta function of $4$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^4} {90}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 08232 \, 3 \ldots
| c =
}}
{{end-... | By Fourier Series of Fourth Power of x, for $x \in \closedint {-\pi} \pi$:
:$\ds x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \map \cos {n \pi} \map \cos {n x}$
Setting $x = \pi$:
{{begin-eqn}}
{{eqn | l = \pi^4
| r = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n... | The [[Definition:Riemann Zeta Function|Riemann zeta function]] of $4$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^4} {90}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 08... | By [[Fourier Series of Fourth Power of x]], for $x \in \closedint {-\pi} \pi$:
:$\ds x^4 = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \frac {8 n^2 \pi^2 - 48} {n^4} \map \cos {n \pi} \map \cos {n x}$
Setting $x = \pi$:
{{begin-eqn}}
{{eqn | l = \pi^4
| r = \frac {\pi^4} 5 + \sum_{n \mathop = 1}^\infty \fra... | Riemann Zeta Function of 4/Proof 1 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_4 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_4/Proof_1 | [
"Riemann Zeta Function of 4",
"Riemann Zeta Function at Even Integers",
"Formulas for Pi",
"Fourth Powers"
] | [
"Definition:Riemann Zeta Function"
] | [
"Fourier Series/Fourth Power of x over Minus Pi to Pi",
"Cosine of Integer Multiple of Pi",
"Basel Problem"
] |
proofwiki-11288 | Riemann Zeta Function of 4 | The Riemann zeta function of $4$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^4} {90}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 08232 \, 3 \ldots
| c =
}}
{{end-... | By Fourier Series of x squared, for $x \in \closedint {-\pi} \pi$:
:$\ds x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$
Hence:
{{begin-eqn}}
{{eqn | l = \frac 1 \pi \int_{-\pi}^\pi x^4 \rd x
| r = \frac 1 2 \paren {\frac {2 \pi^2} 3}^2 + \sum_{n \mathop = 1}^\in... | The [[Definition:Riemann Zeta Function|Riemann zeta function]] of $4$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^4} {90}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 08... | By [[Fourier Series of x squared]], for $x \in \closedint {-\pi} \pi$:
:$\ds x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$
Hence:
{{begin-eqn}}
{{eqn | l = \frac 1 \pi \int_{-\pi}^\pi x^4 \rd x
| r = \frac 1 2 \paren {\frac {2 \pi^2} 3}^2 + \sum_{n \mathop = ... | Riemann Zeta Function of 4/Proof 2 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_4 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_4/Proof_2 | [
"Riemann Zeta Function of 4",
"Riemann Zeta Function at Even Integers",
"Formulas for Pi",
"Fourth Powers"
] | [
"Definition:Riemann Zeta Function"
] | [
"Fourier Series/x squared over Minus Pi to Pi",
"Parseval's Theorem",
"Definite Integral of Even Function"
] |
proofwiki-11289 | Riemann Zeta Function of 4 | The Riemann zeta function of $4$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^4} {90}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 08232 \, 3 \ldots
| c =
}}
{{end-... | {{ProofWanted|$\ds \int_0^1\int_0^1\int_0^1\int_0^1 \frac 1 {1 - x_1x_2x_3x_4} \rd x_1 \rd x_2 \rd x_3 \rd x_4$. Use Beukers-Calabi-Kolk sub.}} | The [[Definition:Riemann Zeta Function|Riemann zeta function]] of $4$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^4} {90}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 08... | {{ProofWanted|[[Riemann Zeta Function as a Multiple Integral|$\ds \int_0^1\int_0^1\int_0^1\int_0^1 \frac 1 {1 - x_1x_2x_3x_4} \rd x_1 \rd x_2 \rd x_3 \rd x_4$]]. Use Beukers-Calabi-Kolk sub.}} | Riemann Zeta Function of 4/Proof 3 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_4 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_4/Proof_3 | [
"Riemann Zeta Function of 4",
"Riemann Zeta Function at Even Integers",
"Formulas for Pi",
"Fourth Powers"
] | [
"Definition:Riemann Zeta Function"
] | [
"Riemann Zeta Function as a Multiple Integral"
] |
proofwiki-11290 | Riemann Zeta Function of 4 | The Riemann zeta function of $4$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^4} {90}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 08232 \, 3 \ldots
| c =
}}
{{end-... | {{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \paren {-1}^3 \dfrac {B_4 2^3 \pi^4} {4!}
| c = Riemann Zeta Function at Even Integers
}}
{{eqn | r = \paren {-1}^3 \paren {-\dfrac 1 {30} } \dfrac {2^3 \pi^4} {4!}
| c = {{Defof|Sequence of Bernoulli Numbers}}
}}
{{eqn | r = \paren {\dfrac 1 {30} } \paren ... | The [[Definition:Riemann Zeta Function|Riemann zeta function]] of $4$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^4} {90}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 08... | {{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \paren {-1}^3 \dfrac {B_4 2^3 \pi^4} {4!}
| c = [[Riemann Zeta Function at Even Integers]]
}}
{{eqn | r = \paren {-1}^3 \paren {-\dfrac 1 {30} } \dfrac {2^3 \pi^4} {4!}
| c = {{Defof|Sequence of Bernoulli Numbers}}
}}
{{eqn | r = \paren {\dfrac 1 {30} } \pa... | Riemann Zeta Function of 4/Proof 4 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_4 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_4/Proof_4 | [
"Riemann Zeta Function of 4",
"Riemann Zeta Function at Even Integers",
"Formulas for Pi",
"Fourth Powers"
] | [
"Definition:Riemann Zeta Function"
] | [
"Riemann Zeta Function at Even Integers"
] |
proofwiki-11291 | Riemann Zeta Function of 4 | The Riemann zeta function of $4$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^4} {90}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 08232 \, 3 \ldots
| c =
}}
{{end-... | Create a multiplication table where the column down the {{LHS}} and the row across the top each contains the terms of zeta function of $2$:
:<nowiki>$\begin {array} {c|cccccccccc}
\paren {\map \zeta 2}^2 & \paren {\dfrac 1 {1^2} } & \paren {\dfrac 1 {2^2} } & \paren {\dfrac 1 {3^2} } & \paren {\dfrac 1 {4^2} } & \cdots... | The [[Definition:Riemann Zeta Function|Riemann zeta function]] of $4$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 4
| r = \dfrac 1 {1^4} + \dfrac 1 {2^4} + \dfrac 1 {3^4} + \dfrac 1 {4^4} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^4} {90}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 08... | Create a [[Definition:Cayley Table|multiplication table]] where the [[Definition:Column of Array|column]] down the {{LHS}} and the [[Definition:Row of Array|row]] across the top each contains the [[Definition:Term of Sequence|terms]] of [[Definition:Riemann Zeta Function|zeta function]] of $2$:
:<nowiki>$\begin {array... | Riemann Zeta Function of 4/Proof 5 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_4 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_4/Proof_5 | [
"Riemann Zeta Function of 4",
"Riemann Zeta Function at Even Integers",
"Formulas for Pi",
"Fourth Powers"
] | [
"Definition:Riemann Zeta Function"
] | [
"Definition:Cayley Table",
"Definition:Array/Column",
"Definition:Array/Row",
"Definition:Term of Sequence",
"Definition:Riemann Zeta Function",
"Definition:Matrix/Diagonal/Main",
"Product of Absolutely Convergent Series",
"Definition:Term of Expression",
"Definition:Coefficient",
"Definition:Term... |
proofwiki-11292 | Riemann Zeta Function of 6 | The Riemann zeta function of $6$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 6
| r = \dfrac 1 {1^6} + \dfrac 1 {2^6} + \dfrac 1 {3^6} + \dfrac 1 {4^6} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^6} {945}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 01734 \, 3 \ldots
| c =
}}
{{end... | By Fourier Series: $x^6$ over $-\pi$ to $\pi$, for $x \in \closedint {-\pi} \pi$:
:$\ds x^6 = \frac {\pi^6} 7 + \sum_{n \mathop = 1}^\infty \frac {12 n^4 \pi^4 - 240 n^2 \pi^2 +1440} {n^6} \, \map \cos {n \pi} \, \map \cos {n x}$
Setting $x = \pi$:
{{begin-eqn}}
{{eqn | l = \pi^6
| r = \frac {\pi^6} 7 + \sum_{n... | The [[Definition:Riemann Zeta Function|Riemann zeta function]] of $6$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 6
| r = \dfrac 1 {1^6} + \dfrac 1 {2^6} + \dfrac 1 {3^6} + \dfrac 1 {4^6} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^6} {945}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 0... | By [[Fourier Series/Sixth Power of x over Minus Pi to Pi|Fourier Series: $x^6$ over $-\pi$ to $\pi$]], for $x \in \closedint {-\pi} \pi$:
:$\ds x^6 = \frac {\pi^6} 7 + \sum_{n \mathop = 1}^\infty \frac {12 n^4 \pi^4 - 240 n^2 \pi^2 +1440} {n^6} \, \map \cos {n \pi} \, \map \cos {n x}$
Setting $x = \pi$:
{{begin-eqn}... | Riemann Zeta Function of 6/Proof 1 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_6 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_6/Proof_1 | [
"Riemann Zeta Function of 6",
"Riemann Zeta Function at Even Integers",
"Formulas for Pi",
"Sixth Powers"
] | [
"Definition:Riemann Zeta Function"
] | [
"Fourier Series/Sixth Power of x over Minus Pi to Pi",
"Cosine of Integer Multiple of Pi",
"Basel Problem",
"Riemann Zeta Function of 4"
] |
proofwiki-11293 | Riemann Zeta Function of 6 | The Riemann zeta function of $6$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 6
| r = \dfrac 1 {1^6} + \dfrac 1 {2^6} + \dfrac 1 {3^6} + \dfrac 1 {4^6} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^6} {945}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 01734 \, 3 \ldots
| c =
}}
{{end... | {{begin-eqn}}
{{eqn | l = \sin x
| r = x \prod_{n \mathop = 1}^\infty \paren {1 - \dfrac {x^2} {n^2 \pi^2} }
| c = Euler Formula for Sine Function
}}
{{eqn | r = x \paren {1 - \dfrac {x^2} {1^2 \pi^2} } \paren {1 - \dfrac {x^2} {2^2 \pi^2} } \paren {1 - \dfrac {x^2} {3^2 \pi^2} } \cdots
}}
{{end-eqn}}
{{beg... | The [[Definition:Riemann Zeta Function|Riemann zeta function]] of $6$ is given by:
{{begin-eqn}}
{{eqn | l = \map \zeta 6
| r = \dfrac 1 {1^6} + \dfrac 1 {2^6} + \dfrac 1 {3^6} + \dfrac 1 {4^6} + \cdots
| c =
}}
{{eqn | r = \dfrac {\pi^6} {945}
| c =
}}
{{eqn | o = \approx
| r = 1 \cdotp 0... | {{begin-eqn}}
{{eqn | l = \sin x
| r = x \prod_{n \mathop = 1}^\infty \paren {1 - \dfrac {x^2} {n^2 \pi^2} }
| c = [[Euler Formula for Sine Function]]
}}
{{eqn | r = x \paren {1 - \dfrac {x^2} {1^2 \pi^2} } \paren {1 - \dfrac {x^2} {2^2 \pi^2} } \paren {1 - \dfrac {x^2} {3^2 \pi^2} } \cdots
}}
{{end-eqn}}
... | Riemann Zeta Function of 6/Proof 2 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_6 | https://proofwiki.org/wiki/Riemann_Zeta_Function_of_6/Proof_2 | [
"Riemann Zeta Function of 6",
"Riemann Zeta Function at Even Integers",
"Formulas for Pi",
"Sixth Powers"
] | [
"Definition:Riemann Zeta Function"
] | [
"Euler Formula for Sine Function",
"Power Series Expansion for Sine Function",
"Basel Problem",
"Basel Problem",
"Basel Problem",
"Basel Problem"
] |
proofwiki-11294 | Infima Preserving Mapping on Filters Preserves Filtered Infima | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets.
Let $f: S \to T$ be a mapping.
For every filter $F$ in $\struct {S, \preceq}$, let $f$ preserve the infimum on $F$.
Then $f$ preserves filtered infima. | Let $F$ be a filtered subset of $S$ such that:
:$F$ admits an infimum in $\struct {S, \preceq}$.
By Filtered iff Upper Closure Filtered:
:$F^\succeq$ is filtered
where $F^\succeq$ denotes the upper closure of $F$.
By Upper Closure is Upper Section:
:$F^\succeq$ is upper.
Because filtered is non-empty, by definition:
:$... | Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
For every [[Definition:Filter in Ordered Set|filter]] $F$ in $\struct {S, \preceq}$, let $f$ [[Definition:Infimum on Subset Preserving Mapping|preserve the infimum]] o... | Let $F$ be a [[Definition:Filtered Subset|filtered]] [[Definition:Subset|subset]] of $S$ such that:
:$F$ admits an [[Definition:Infimum of Set|infimum]] in $\struct {S, \preceq}$.
By [[Filtered iff Upper Closure Filtered]]:
:$F^\succeq$ is [[Definition:Filtered Subset|filtered]]
where $F^\succeq$ denotes the [[Definit... | Infima Preserving Mapping on Filters Preserves Filtered Infima | https://proofwiki.org/wiki/Infima_Preserving_Mapping_on_Filters_Preserves_Filtered_Infima | https://proofwiki.org/wiki/Infima_Preserving_Mapping_on_Filters_Preserves_Filtered_Infima | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Filter in Ordered Set",
"Definition:Mapping Preserves Infimum/Subset",
"Definition:Mapping Preserves Infimum/Filtered"
] | [
"Definition:Filtered Subset",
"Definition:Subset",
"Definition:Infimum of Set",
"Filtered iff Upper Closure Filtered",
"Definition:Filtered Subset",
"Definition:Upper Closure/Set",
"Upper Closure is Upper Section",
"Definition:Upper Section",
"Definition:Filtered Subset",
"Definition:Non-Empty Set... |
proofwiki-11295 | Number of Partitions as Coefficient of Power Series | The number of partitions $\map p n$ of a (strictly) positive integer $n$ is equal to the coefficient of $x^n$ when the expression:
:$\map f n = \dfrac 1 {\paren {1 - x} \paren {1 - x^2} \paren {1 - x^3} \cdots}$
is expanded into a power series.
That is:
:$\map f n = 1 + \map p 1 x + \map p 2 x^2 + \map p 3 x^3 + \cdots... | {{begin-eqn}}
{{eqn | l = \map f n
| r = \dfrac 1 {\paren {1 - x} \paren {1 - x^2} \paren {1 - x^3} \paren {1 - x^4} \cdots}
| c =
}}
{{eqn | r = \frac 1 {\paren {1 - x^1} } \times \frac 1 {\paren {1 - x^2} } \times \frac 1 {\paren {1 - x^3} } \times \frac 1 {\paren {1 - x^4} } \times \cdots
| c =
}... | The [[Definition:Partition Function|number of partitions]] $\map p n$ of a [[Definition:Strictly Positive Integer|(strictly) positive integer]] $n$ is equal to the [[Definition:Coefficient|coefficient]] of $x^n$ when the [[Definition:Expression|expression]]:
:$\map f n = \dfrac 1 {\paren {1 - x} \paren {1 - x^2} \paren... | {{begin-eqn}}
{{eqn | l = \map f n
| r = \dfrac 1 {\paren {1 - x} \paren {1 - x^2} \paren {1 - x^3} \paren {1 - x^4} \cdots}
| c =
}}
{{eqn | r = \frac 1 {\paren {1 - x^1} } \times \frac 1 {\paren {1 - x^2} } \times \frac 1 {\paren {1 - x^3} } \times \frac 1 {\paren {1 - x^4} } \times \cdots
| c =
}... | Number of Partitions as Coefficient of Power Series | https://proofwiki.org/wiki/Number_of_Partitions_as_Coefficient_of_Power_Series | https://proofwiki.org/wiki/Number_of_Partitions_as_Coefficient_of_Power_Series | [
"Partition Theory",
"Number Theory"
] | [
"Definition:Integer Partition/Partition Function",
"Definition:Strictly Positive/Integer",
"Definition:Coefficient",
"Definition:Expression",
"Definition:Power Series"
] | [
"Sum of Infinite Geometric Sequence",
"Exponent Combination Laws/Product of Powers"
] |
proofwiki-11296 | Suprema Preserving Mapping on Ideals Preserves Directed Suprema | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets.
Let $f: S \to T$ be a mapping.
Let every filter $F$ in $\left({S, \preceq}\right)$, $f$ preserve the infimum on $F$.
Then $f$ preserves directed suprema. | This follows by {{mutatis}} of the proof of Infima Preserving Mapping on Filters Preserves Filtered Infima.
{{qed}} | Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be [[Definition:Ordered Set|ordered sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let every [[Definition:Filter in Ordered Set|filter]] $F$ in $\left({S, \preceq}\right)$, $f$ [[Definition:Mapping Preserves Infimum/Subset|preserve the infimum]] o... | This follows by {{mutatis}} of the proof of [[Infima Preserving Mapping on Filters Preserves Filtered Infima]].
{{qed}} | Suprema Preserving Mapping on Ideals Preserves Directed Suprema | https://proofwiki.org/wiki/Suprema_Preserving_Mapping_on_Ideals_Preserves_Directed_Suprema | https://proofwiki.org/wiki/Suprema_Preserving_Mapping_on_Ideals_Preserves_Directed_Suprema | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Filter in Ordered Set",
"Definition:Mapping Preserves Infimum/Subset",
"Definition:Mapping Preserves Supremum/Directed"
] | [
"Infima Preserving Mapping on Filters Preserves Filtered Infima"
] |
proofwiki-11297 | Mellin Transform of Heaviside Step Function/Lemma | Let $t \in \R$.
Let $s \in \C$ with $\map \Re s < 0$.
Then:
:$\ds \lim_{t \mathop \to +\infty} t^s = 0$ | Let $s = a + b i$, where $a, b \in \R$, $a<0$.
{{begin-eqn}}
{{eqn | l = \cmod {\lim_{t \mathop \to + \infty} t^s}
| r = \lim_{t \mathop \to +\infty} \cmod {t^s}
| c =
}}
{{eqn | r = \lim_{t \mathop \to +\infty} \cmod {e^{s \ln t} }
| c = {{Defof|Power to Complex Number}}
}}
{{eqn | r = \lim_{t \math... | Let $t \in \R$.
Let $s \in \C$ with $\map \Re s < 0$.
Then:
:$\ds \lim_{t \mathop \to +\infty} t^s = 0$ | Let $s = a + b i$, where $a, b \in \R$, $a<0$.
{{begin-eqn}}
{{eqn | l = \cmod {\lim_{t \mathop \to + \infty} t^s}
| r = \lim_{t \mathop \to +\infty} \cmod {t^s}
| c =
}}
{{eqn | r = \lim_{t \mathop \to +\infty} \cmod {e^{s \ln t} }
| c = {{Defof|Power to Complex Number}}
}}
{{eqn | r = \lim_{t \mat... | Mellin Transform of Heaviside Step Function/Lemma | https://proofwiki.org/wiki/Mellin_Transform_of_Heaviside_Step_Function/Lemma | https://proofwiki.org/wiki/Mellin_Transform_of_Heaviside_Step_Function/Lemma | [
"Heaviside Step Function"
] | [] | [
"Complex Modulus of Product of Complex Numbers",
"Modulus of Exponential of Imaginary Number is One",
"Sequence of Powers of Reciprocals is Null Sequence/Real Index",
"Category:Heaviside Step Function"
] |
proofwiki-11298 | Up-Complete Lower Bounded Join Semilattice is Complete | Let $\struct {S, \preceq}$ be an up-complete lower bounded join semillattice.
Then $\struct {S, \preceq}$ is complete. | Let $X$ be a subset of $S$.
In the case when $X = \O$:
by definition of lower bounded:
:$\exists L \in S: L$ is lower bound for $S$.
By definition of empty set:
:$L$ is upper bound for $X$.
By definition of lower bound:
:$\forall x \in S: x$ is upper bound for $X \implies L \preceq x$
Thus by definition
:$L$ is a supre... | Let $\struct {S, \preceq}$ be an [[Definition:Up-Complete|up-complete]] [[Definition:Bounded Below Set|lower bounded]] [[Definition:Join Semilattice|join semillattice]].
Then $\struct {S, \preceq}$ is [[Definition:Complete Lattice|complete]]. | Let $X$ be a [[Definition:Subset|subset]] of $S$.
In the case when $X = \O$:
by definition of [[Definition:Bounded Below Set|lower bounded]]:
:$\exists L \in S: L$ is [[Definition:Lower Bound of Set|lower bound]] for $S$.
By definition of [[Definition:Empty Set|empty set]]:
:$L$ is [[Definition:Upper Bound of Set|up... | Up-Complete Lower Bounded Join Semilattice is Complete | https://proofwiki.org/wiki/Up-Complete_Lower_Bounded_Join_Semilattice_is_Complete | https://proofwiki.org/wiki/Up-Complete_Lower_Bounded_Join_Semilattice_is_Complete | [
"Complete Lattices"
] | [
"Definition:Up-Complete",
"Definition:Bounded Below Set",
"Definition:Join Semilattice",
"Definition:Complete Lattice"
] | [
"Definition:Subset",
"Definition:Bounded Below Set",
"Definition:Lower Bound of Set",
"Definition:Empty Set",
"Definition:Upper Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Set of Sets",
"... |
proofwiki-11299 | Gregory Series | For $-\dfrac \pi 4 \le \theta \le \dfrac \pi 4$:
:$\theta = \tan \theta - \dfrac 1 3 \tan^3 \theta + \dfrac 1 5 \tan^5 \theta - \ldots$
This is called the '''Gregory series'''. | {{begin-eqn}}
{{eqn | l = 1
| r = \frac {\sec^2 \theta} {\sec^2 \theta}
| c =
}}
{{eqn | l =
| r = \sec^2 \theta \times \frac 1 {1 - \paren {-\tan^2 \theta} }
| c =
}}
{{eqn | l =
| r = \sec^2 \theta \times \sum_{n \mathop = 0}^\infty \paren {-\tan^2 \theta}^n
| c = Sum of Infini... | For $-\dfrac \pi 4 \le \theta \le \dfrac \pi 4$:
:$\theta = \tan \theta - \dfrac 1 3 \tan^3 \theta + \dfrac 1 5 \tan^5 \theta - \ldots$
This is called the '''[[Gregory Series|Gregory series]]'''. | {{begin-eqn}}
{{eqn | l = 1
| r = \frac {\sec^2 \theta} {\sec^2 \theta}
| c =
}}
{{eqn | l =
| r = \sec^2 \theta \times \frac 1 {1 - \paren {-\tan^2 \theta} }
| c =
}}
{{eqn | l =
| r = \sec^2 \theta \times \sum_{n \mathop = 0}^\infty \paren {-\tan^2 \theta}^n
| c = [[Sum of Infi... | Gregory Series | https://proofwiki.org/wiki/Gregory_Series | https://proofwiki.org/wiki/Gregory_Series | [
"Gregory Series",
"Tangent Function"
] | [
"Gregory Series"
] | [
"Sum of Infinite Geometric Sequence",
"Nth Root Test",
"Definition:Radius of Convergence",
"Definition:Primitive (Calculus)/Integration",
"Primitive of Power of Tangent of a x by Square of Secant of a x"
] |
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