hcju/proofwikips huggingface dataset:
Power Series Expansion for Hyperbolic Sine Function
The hyperbolic sine function has the power series expansion: {{begin-eqn}} {{eqn | l = \sinh x       | r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {\paren {2 n + 1}!}       | c =  }} {{eqn | r = x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} + \cdots       | c =  }} {{end-eqn}} valid for all $x \in \R$.
Power Series Expansion for Hyperbolic Cosine Function
The hyperbolic cosine function has the power series expansion: {{begin-eqn}} {{eqn | l = \cosh x       | r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots       | c =  }} {{end-eqn}} valid for all $x \in \R$.
Power Series Expansion for Hyperbolic Tangent Function
The hyperbolic tangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \tanh x       | r = \sum_{n \mathop = 1}^\infty \frac {2^{2 n} \left({2^{2 n} - 1}\right) B_{2 n} \, x^{2 n - 1} } {\left({2 n}\right)!}       | c =  }} {{eqn | r = x - \frac {x^3} 3 + \frac {2 x^5} {15} - \frac {17 x^7} {315} + \cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $\left|{x}\right| < \dfrac \pi 2$.
Power Series Expansion for Hyperbolic Cotangent Function
The hyperbolic cotangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \coth x       | r = \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} \, x^{2 n - 1} } {\left({2 n}\right)!}       | c =  }} {{eqn | r = \frac 1 x + \frac x 3 - \frac {x^3} {45} + \frac {2 x^5} {45} + \cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $0 < \left|{x}\right| < \pi$.
Power Series Expansion for Hyperbolic Secant Function
The hyperbolic secant function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \sech x       | r = \sum_{n \mathop = 0}^\infty \frac {E_{2 n} x^{2 n} } {\paren {2 n}!}        | c =  }} {{eqn | r = 1 - \frac {x^2} 2 + \frac {5 x^4} {24} - \frac {61 x^6} {720} + \cdots       | c =  }} {{end-eqn}} where $E_{2 n}$ denotes the Euler numbers. This converges for $\size x < \dfrac \pi 2$.
Power Series Expansion for Hyperbolic Cosecant Function
 The hyperbolic cosecant function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \operatorname {csch} x       | r = \sum_{n \mathop = 0}^\infty \dfrac {2 \left({1 - 2^{2 n - 1} }\right) B_{2 n} \,  x^{2 n - 1} } {\left({2 n}\right)!}       | c =  }} {{eqn | r = \frac 1 x - \frac x 6 + \frac {7 x^3} {360} - \frac {31 x^5} {15 \, 120} + \cdots       | c =  }} {{end-eqn}} where $B_n$ denotes the Bernoulli numbers. This converges for $0 < \left\lvert{x}\right\rvert < \pi$.
Sum of Hyperbolic Tangent and Cotangent
:$\tanh x + \coth x = 2 \coth 2 x$
Power Series Expansion for Real Inverse Hyperbolic Sine
The (real) inverse hyperbolic sine function has a Taylor series expansion: :$\sinh^{-1} x = \begin{cases} \displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({-1}\right)^n \left({2 n}\right)!} {2^{2 n} \left({n!}\right)^2} \frac {x^{2 n + 1} } {2 n + 1} & : \left\lvert{x}\right\rvert \le 1 \\  \displaystyle \ln 2 x - \left({\sum_{n \mathop = 1}^\infty \frac {\left({-1}\right)^n \left({2 n}\right)!} {2^{2 n + 1} \left({n!}\right)^2 n x^{2 n} } }\right) & : x \ge 1 \\  \displaystyle -\ln \left({-2 x}\right) + \left({\sum_{n \mathop = 1}^\infty \frac {\left({-1}\right)^n \left({2 n}\right)!} {2^{2 n + 1} \left({n!}\right)^2 n x^{2 n} } }\right) & : x \le -1 \\  \end{cases}$ That is: :$\sinh^{-1} x = \begin{cases} \displaystyle x - \dfrac 1 2 \dfrac {x^3} 3 + \dfrac {1 \times 3} {2 \times 4} \dfrac {x^5} 5 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac {x^7} 7  + \cdots& : \left\lvert{x}\right\rvert \le 1 \\  \displaystyle \ln 2 x - \left({\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots}\right) & : x \ge 1 \\  \displaystyle -\ln \left({-2 x}\right) + \left({\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots}\right) & : x \le -1 \\  \end{cases}$
Power Series Expansion for Real Inverse Hyperbolic Cosine
The (real) inverse hyperbolic cosine function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \cosh^{-1} x       | r = \ln 2 x - \left({\sum_{n \mathop = 1}^\infty \frac {\left({2 n}\right)!} {2^{2 n + 1} \left({n!}\right)^2 n x^{2 n} } }\right)       | c =  }} {{eqn | r = \ln 2 x - \left({\dfrac 1 2 \dfrac 1 {2 x^2} + \dfrac {1 \times 3} {2 \times 4} \dfrac 1 {4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \dfrac 1 {6 x^6} + \cdots}\right)       | c =  }} {{end-eqn}} for $x \ge 1$.
Power Series Expansion for Real Inverse Hyperbolic Tangent
The (real) inverse hyperbolic tangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \tanh^{-1} x       | r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}       | c =  }} {{eqn | r = x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots       | c =  }} {{end-eqn}} for $\size x < 1$.
Power Series Expansion for Real Inverse Hyperbolic Cotangent
The (real) inverse hyperbolic cotangent function has a Taylor series expansion: {{begin-eqn}} {{eqn | l = \coth^{-1} x       | r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1} x^{2 n + 1} }       | c =  }} {{eqn | r = \frac 1 x + \frac 1 {3 x^3} + \frac 1 {5 x^5} + \frac 1 {7 x^7} + \cdots       | c =  }} {{end-eqn}} for $\size x > 1$.
Power Series Expansion for Exponential of Cosine of x
:$e^{\cos x} = e \left({1 - \dfrac {x^2} 2 + \dfrac {x^4} 6 - \dfrac {31 x^6} {720} + \cdots}\right)$ for all $x \in \R$.
Power Series Expansion for Exponential of Tangent of x
:$e^{\tan x} = 1 + x + \dfrac {x^2} 2 + \dfrac {x^3} 2 + \dfrac {3 x^4} 8 + \cdots$ for all $x \in \R$ such that $\size x < \frac \pi 2$.
Power Series Expansion for Exponential of Cosine of x/Proof 2
For all $x \in \R$: :$\displaystyle e^{\cos \left({x}\right)} = e \left({e^{cos \left({x}\right)- 1 } }\right) =  e \left({\sum_{n \mathop = 0}^\infty \frac{ \left({-1}\right)^m P_2 \left({2 m}\right)} {2 m!} x^{2 m} }\right)$ where $P_2 \left({2 m}\right)$ is the partition of the set of size 2m into even blocks.
Sine of x plus Cosine of x
=== Sine Form === {{:Sine of x plus Cosine of x/Sine Form}} === Cosine Form === {{:Sine of x plus Cosine of x/Cosine Form}}
Sine of x plus Cosine of x/Sine Form
:$\sin x + \cos x = \sqrt 2 \sin \left({x + \dfrac \pi 4}\right)$
Cosine of x minus Sine of x
=== Sine Form === {{:Cosine of x minus Sine of x/Sine Form}} === Cosine Form === {{:Cosine of x minus Sine of x/Cosine Form}}
Cosine of x minus Sine of x/Sine Form
:$\cos x - \sin x = \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}$
Cosine of x minus Sine of x/Cosine Form
:$\cos x - \sin x = \sqrt 2 \, \map \cos {x + \dfrac \pi 4}$
Power Series Expansion for Exponential of x by Cosine of x
{{begin-eqn}} {{eqn | l = e^x \cos x       | r = \sum_{n \mathop = 1}^\infty \frac {2^{n / 2} \, \cos \left({n \pi / 4}\right) x^n} {n!}       | c =  }} {{eqn | r = 1 + x - \frac {x^3} 3 - \frac {x^4} 6 + \cdots       | c =  }} {{end-eqn}} for all $x \in \R$.
Power Series Expansion for Logarithm of Cosine of x
{{begin-eqn}} {{eqn | l = \ln \size {\cos x}       | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n 2^{2 n - 1} \{2^{2 n} - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}       | c =  }} {{eqn | r = -\frac {x^2} 2 - \frac {x^4} {12} - \frac {x^6} {45} - \frac {17 x^8} {2520} - \cdots       | c =  }} {{end-eqn}} for all $x \in \R$ such that $\size x < \dfrac \pi 2$.
Expectation of Gaussian Distribution
Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution. Then:  :$\expect X = \mu$
Power Series Expansion for Logarithm of Tangent of x
{{begin-eqn}} {{eqn | l = \ln \left\lvert{\tan x}\right\rvert       | r = \ln \left\lvert{x}\right\rvert + \sum_{n \mathop = 1}^\infty \frac {\left({-1}\right)^{n - 1} 2^{2 n} \left({2^{2 n - 1} - 1}\right) B_{2 n} \, x^{2 n} } {n \left({2 n}\right)!}       | c =  }} {{eqn | r = \ln \left\lvert{x}\right\rvert + \frac {x^2} 3 + \frac {7 x^4} {90} + \frac {62 x^6} {2835} + \cdots       | c =  }} {{end-eqn}} for all $x \in \R$ such that $0 < \left|{x}\right| < \dfrac \pi 2$.
Derivative of Logarithm over Power
:$\dfrac \d {\d x} \dfrac {\ln x} {x^n} = \dfrac {1 - n \ln x} {x^{n + 1} }$
Nth Derivative of Natural Logarithm by Reciprocal
:$\dfrac {\d^n} {\d x^n} \dfrac {\ln x} x = \paren {-1}^{n + 1} n! \dfrac {H_n - \ln x} {x^{n + 1} }$ where $H_n$ denotes the $n$th harmonic number: :$H_n = \displaystyle \sum_{r \mathop = 1}^n \dfrac 1 r = 1 + \dfrac 1 2 + \dfrac 1 3 + \cdots + \dfrac 1 r$
Variance of Gaussian Distribution
Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution. Then:  :$\var X = \sigma^2$
Variance as Expectation of Square minus Square of Expectation/Continuous
Let $X$ be a continuous random variable. Then the variance of $X$ can be expressed as: :$\var X = \expect {X^2} - \paren {\expect X}^2$ That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$.
Variance as Expectation of Square minus Square of Expectation/Discrete
Let $X$ be a discrete random variable. Then the variance of $X$ can be expressed as: :$\var X = \expect {X^2} - \paren {\expect X}^2$ That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$.
Reversion of Power Series
Let $\displaystyle y = \sum_{n \mathop = 1}^\infty c_n x^n$ be a power series. Then: :$\displaystyle x = \sum_{n \mathop = 1}^\infty C_n y^n$ is also a power series, where: {{begin-eqn}} {{eqn | l = c_1 C_1       | r = 1 }} {{eqn | l = {c_1}^3 C_2       | r = -c_2 }} {{eqn | l = {c_1}^5 C_3       | r = 2 {c_2}^2 - c_1 c_3 }} {{eqn | l = {c_1}^7 C_4       | r = 5 c_1 c_2 c_3 - 5 {c_2}^2 - {c_1}^2 c_4 }} {{eqn | l = {c_1}^9 C_5       | r = 6 {c_1}^2 c_2 c_4 + 3 {c_1}^2 {c_3}^2 - {c_1}^3 c_5 + 14 {c_2}^4 - 21 c_1 {c_2}^2 c_3 }} {{eqn | l = {c_1}^{11} C_6       | r = 7 {c_1}^3 c_2 c_5 + 84 c_1 {c_2}^3 c_3 + 7 {c_1}^3 c_3 c_4 - 28 {c_1}^2 c_2 {c_3}^2 - {c_1}^4 c_6 - 28 {c_1}^2 {c_2}^2 c_4 - 42 {c_2}^5 }} {{end-eqn}}
Expectation of Continuous Uniform Distribution
Let $X \sim \ContinuousUniform a b$ for some $a, b \in \R$, $a \ne b$, where $\operatorname U$ is the continuous uniform distribution.  Then:  :$\expect X = \dfrac {a + b} 2$
Variance of Continuous Uniform Distribution
Let $X \sim \ContinuousUniform a b$ for some $a, b \in \R$, $a \ne b$, where $\operatorname U$ is the continuous uniform distribution.  Then:  :$\var X = \dfrac {\paren {b - a}^2} {12}$
Median of Gaussian Distribution
Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution. Then the median of $X$ is equal to $\mu$.
Expectation of Gamma Distribution
Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution. The expectation of $X$ is given by: :$\expect X = \dfrac \alpha \beta$
Variance of Gamma Distribution
Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution. The variance of $X$ is given by: :$\var X = \dfrac \alpha {\beta^2}$
Sum of Bernoulli Numbers by Power of Two and Binomial Coefficient
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^n \dbinom {2 n + 1} {2 k} 2^{2 k} B_{2 k}       | r = \binom {2 n + 1} 2 2^2 B_2 + \binom {2 n + 1} 4 2^4 B_4 + \binom {2 n + 1} 6 2^6 B_6 + \cdots       | c =  }} {{eqn | r = 2 n       | c =  }} {{end-eqn}} where $B_n$ denotes the $n$th Bernoulli number.
Moment in terms of Moment Generating Function
Let $X$ be a random variable.  Let $M_X$ be the moment generating function of $X$. Then:  :$\expect {X^n} = \map {M^{\paren n}_X} 0$ where: :$n$ is a non-negative integer :$M^{\paren n}_X$ denotes the $n$th derivative of $M_X$ :$\expect {X^n}$ denotes the expectation of $X^n$.
Sum of Euler Numbers by Binomial Coefficient
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = E_{2 n}       | r = \sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 n - 2 k}       | c =  }} {{eqn | r = \binom {2 n} 2 E_{2 n - 2} + \binom {2 n} 4 E_{2 n - 4} + \binom {2 n} 6 E_{2 n - 6} + \cdots + 1       | c =  }} {{end-eqn}} where $E_n$ denotes the $n$th Euler number.
Moment Generating Function of Gaussian Distribution
Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution. Then the moment generating function $M_X$ of $X$ is given by:  :$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$
Bernoulli Number in terms of Euler Numbers
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = B_{2 n}       | r = \frac {2n} {2^{2 n} \left({2^{2 n} - 1}\right)} \left({\sum_{k \mathop = 1}^n \dbinom {2 n - 1} {2 k - 1} E_{2 n - 2 k} }\right)       | c =  }} {{eqn | r = \frac {2n} {2^{2 n} \left({2^{2 n} - 1}\right)} \left({\binom {2 n - 1} 1 E_{2 n - 2} + \binom {2 n - 1} 3 E_{2 n - 4} + \binom {2 n - 1} 5 E_{2 n - 6} + \cdots + 1}\right)       | c =  }} {{end-eqn}} where: : $B_n$ denotes the $n$th Bernoulli number : $E_n$ denotes the $n$th Euler number.
Equality of Vector Quantities
Two vector quantities are equal  they have the same magnitude and direction. That is: :$\mathbf a = \mathbf b \iff \paren {\size {\mathbf a} = \size {\mathbf b} \land \hat {\mathbf a} = \hat {\mathbf b} }$ where: :$\hat {\mathbf a}$ denotes the unit vector in the direction of $\mathbf a$ :$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$.
Vector Addition is Commutative
Let $\mathbf a, \mathbf b$ be vector quantities. Then: :$\mathbf a + \mathbf b = \mathbf b + \mathbf a$
Vector Addition is Associative
Let $\mathbf a, \mathbf b, \mathbf c$ be vectors. Then: :$\mathbf a + \paren {\mathbf b + \mathbf c} = \paren {\mathbf a + \mathbf b} + \mathbf c$
Scalar Multiplication of Vectors is Associative
Let $\mathbf a$ be a vector quantity. Let $m, n$ be scalar quantities. Then: :$m \paren {n \mathbf a} = \paren {m n} \mathbf a = n \paren {m \mathbf a}$
Scalar Multiplication of Vectors is Distributive over Scalar Addition
Let $\mathbf a$ be a vector quantity. Let $m, n$ be scalar quantities. Then: :$\paren {m + n} \mathbf a = m \mathbf a + n \mathbf a$
Scalar Multiplication of Vectors is Distributive over Vector Addition
Let $\mathbf a, \mathbf b$ be a vector quantities. Let $m$ be a scalar quantity. Then: :$m \paren {\mathbf a + \mathbf b} = m \mathbf a + m \mathbf b$
Scalar Triple Product equals Determinant
Let $\mathbf a, \mathbf b, \mathbf c$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \mathbf a       | r = a_i \mathbf i + a_j \mathbf j + a_k \mathbf k }} {{eqn | l = \mathbf b       | r = b_i \mathbf i + b_j \mathbf j + b_k \mathbf k }} {{eqn | l = \mathbf c       | r = c_i \mathbf i + c_j \mathbf j + c_k \mathbf k }} {{end-eqn}} where $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of $\mathbf V$. Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$. Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ with $\mathbf b$. Then: :$\mathbf a \cdot \left({\mathbf b \times \mathbf c}\right) = \begin{vmatrix} a_i & a_j & a_k \\ b_i & b_j & b_k \\ c_i & c_j & c_k \end{vmatrix}$
Differential Entropy of Gaussian Distribution
Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution. Then the differential entropy $\map h X$ of $X$ is given by:  :$\map h X = \map \ln {\sigma \sqrt {2 \pi} } + \dfrac 1 2$
Lagrange's Formula/Corollary
:$\left({\mathbf a \times \mathbf b}\right) \times \mathbf c = \left({\mathbf{a \cdot c} }\right) \mathbf b - \left({\mathbf{b \cdot c} }\right) \mathbf a$
Dot Product of Vector Cross Products
Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \mathbf a       | r = a_1 \mathbf e_1 + a_2 \mathbf e_2 + a_3 \mathbf e_3 }} {{eqn | l = \mathbf b       | r = b_1 \mathbf e_1 + b_2 \mathbf e_2 + b_3 \mathbf e_3 }} {{eqn | l = \mathbf c       | r = c_1 \mathbf e_1 + c_2 \mathbf e_2 + c_3 \mathbf e_3 }} {{eqn | l = \mathbf d       | r = d_1 \mathbf e_1 + d_2 \mathbf e_2 + d_3 \mathbf e_3 }} {{end-eqn}} where $\left({\mathbf e_1, \mathbf e_2, \mathbf e_3}\right)$ is the standard ordered basis of $\mathbf V$. Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$. Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ with $\mathbf b$. Then: :$\left({\mathbf a \times \mathbf b}\right) \cdot \left({\mathbf c \times \mathbf d}\right) = \left({\mathbf a \cdot \mathbf c}\right) \left({\mathbf b \cdot \mathbf d}\right) - \left({\mathbf a \cdot \mathbf d}\right) \left({\mathbf b \cdot \mathbf c}\right)$
Vector Cross Product of Vector Cross Products
Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \mathbf a       | r = a_1 \mathbf e_1 + a_2 \mathbf e_2 + a_3 \mathbf e_3 }} {{eqn | l = \mathbf b       | r = b_1 \mathbf e_1 + b_2 \mathbf e_2 + b_3 \mathbf e_3 }} {{eqn | l = \mathbf c       | r = c_1 \mathbf e_1 + c_2 \mathbf e_2 + c_3 \mathbf e_3 }} {{eqn | l = \mathbf d       | r = d_1 \mathbf e_1 + d_2 \mathbf e_2 + d_3 \mathbf e_3 }} {{end-eqn}} where $\left({\mathbf e_1, \mathbf e_2, \mathbf e_3}\right)$ is the standard ordered basis of $\mathbf V$. Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$. Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ with $\mathbf b$. Then: {{begin-eqn}} {{eqn | l = \left({\mathbf a \times \mathbf b}\right) \times \left({\mathbf c \times \mathbf d}\right)       | r = \mathbf c \left({\mathbf a \cdot \left({\mathbf b \times \mathbf d}\right)}\right) - \mathbf d \left({\mathbf a \cdot \left({\mathbf b \times \mathbf c}\right)}\right)       | c =  }} {{eqn | r = \mathbf b \left({\mathbf a \cdot \left({\mathbf c \times \mathbf d}\right)}\right) - \mathbf a \left({\mathbf b \cdot \left({\mathbf c \times \mathbf d}\right)}\right)       | c =  }} {{end-eqn}}
Derivative of Scalar Triple Product of Vector-Valued Functions
Let: : $\mathbf f: x \mapsto \left\langle{f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)}\right\rangle$ : $\mathbf g: x \mapsto \left\langle{g_1 \left({x}\right), g_2 \left({x}\right), \ldots, g_n \left({x}\right)}\right\rangle$ : $\mathbf h: x \mapsto \left\langle{h_1 \left({x}\right), h_2 \left({x}\right), \ldots, h_n \left({x}\right)}\right\rangle$ be differentiable vector-valued functions. The derivative of their scalar triple product is given by: :$\dfrac \d {\d x} \left({\mathbf f \cdot \left({\mathbf g \times \mathbf h}\right)}\right) = \dfrac {\d \mathbf f} {\d x} \cdot \left({\mathbf g \times \mathbf h}\right) + \mathbf f \cdot \left({\dfrac {\d \mathbf g} {\d x} \times \mathbf h}\right) + \mathbf f \cdot \left({\mathbf g \times \dfrac {\d \mathbf h} {\d x} }\right)$
Dot Product of Vector-Valued Function with its Derivative
Let: : $\mathbf f \left({x}\right) = \displaystyle \sum_{k \mathop = 1}^n f_k \left({x}\right) \mathbf e_k$ be a differentiable vector-valued function. The dot product of $\mathbf f$ with its derivative is given by: :$\mathbf f \left({x}\right) \cdot \dfrac {\d \mathbf f \left({x}\right)} {\d x} = \left\lvert{\mathbf f \left({x}\right)}\right\rvert \dfrac {\d \left\lvert{\mathbf f \left({x}\right)}\right\rvert} {\d x}$ where $\left\lvert{\mathbf f \left({x}\right)}\right\rvert \ne 0$.
Dot Product of Constant Magnitude Vector-Valued Function with its Derivative is Zero
Let: :$\map {\mathbf f} x = \displaystyle \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$ be a differentiable vector-valued function. Let $\map {\mathbf f} x$ be such that its magnitude is constant: :$\size {\map {\mathbf f} x} = c$ for some $c \in \R$. Then the dot product of $\mathbf f$ with its derivative is zero: :$\map {\mathbf f} x \cdot \dfrac {\d \map {\mathbf f} x} {\d x} = 0$
Expectation of Beta Distribution
Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname{Beta}$ denotes the beta distribution.  Then: :$\expect X = \dfrac \alpha {\alpha + \beta}$
Variance of Beta Distribution
Let $X \sim \map \Beta {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Beta$ is the Beta distribution.  Then: :$\var X = \dfrac {\alpha \beta} {\paren {\alpha + \beta}^2 \paren {\alpha + \beta + 1} }$
Moment Generating Function of Gamma Distribution
Let $X \sim \Gamma \left({\alpha, \beta}\right)$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution. Then the moment generating function of $X$, $M_X$, is given by:  :$\displaystyle M_X \left({t}\right) = \begin{cases} \left({1 - \frac t \beta}\right)^{-\alpha} & t < \beta \\ \text{does not exist} & t \ge \beta \end{cases}$
Gradient Operator Distributes over Addition
Let $\mathbf V$ be a vector space of $n$ dimensions. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis of $\mathbf V$. Let $f \left({x_1, x_2, \ldots, x_n}\right), g \left({x_1, x_2, \ldots, x_n}\right): \mathbf V \to \R$ be real-valued functions on $\mathbf V$. Let $\nabla f$ denote the gradient of $f$. Then: :$\nabla \left({f + g}\right) = \nabla f + \nabla g$
Divergence Operator Distributes over Addition
Let $\mathbf V \left({x_1, x_2, \ldots, x_n}\right)$ be a vector space of $n$ dimensions. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis of $\mathbf V$. Let $\mathbf f$ and $\mathbf g: \mathbf V \to \mathbf V$ be vector-valued functions on $\mathbf V$: :$\mathbf f := \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right)$ :$\mathbf g := \left({g_1 \left({\mathbf x}\right), g_2 \left({\mathbf x}\right), \ldots, g_n \left({\mathbf x}\right)}\right)$ Let $\nabla \cdot \mathbf f$ denote the divergence of $f$. Then: :$\nabla \cdot \left({\mathbf f + \mathbf g}\right) = \nabla \cdot \mathbf f + \nabla \cdot \mathbf g$
Curl Operator Distributes over Addition
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.. Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$. Let $\mathbf f$ and $\mathbf g: \R^3 \to \R^3$ be vector-valued functions on $\R^3$: :$\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }$ :$\mathbf g := \tuple {\map {g_x} {\mathbf x}, \map {g_y} {\mathbf x}, \map {g_z} {\mathbf x} }$ Let $\nabla \times \mathbf f$ denote the curl of $f$. Then: :$\nabla \times \paren {\mathbf f + \mathbf g} = \nabla \times \mathbf f + \nabla \times \mathbf g$
Product Rule for Divergence
Let $\mathbf V \left({x_1, x_2, \ldots, x_n}\right)$ be a vector space of $n$ dimensions. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis of $\mathbf V$. Let $\mathbf f := \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \mathbf V \to \mathbf V$ be a vector-valued function on $\mathbf V$. Let $g \left({x_1, x_2, \ldots, x_n}\right): \mathbf V \to \R$ be a real-valued function on $\mathbf V$. Let $\nabla \cdot \mathbf f$ denote the divergence of $f$. Then: :$\nabla \cdot \left({g \, \mathbf f}\right) = g \left({\nabla \cdot \mathbf f}\right) + \left({\nabla g}\right) \cdot \mathbf f$