Formula Sheet

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\fancyhead[C]{\footnotesize\textbf{Mathematics Formula Sheet --- Calculus, Linear Algebra \& Probability}}
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\begin{multicols}{4}

% ============================================================
\section{Single-Variable Calculus}

\subsection{Limits}
$\lim_{x\to 0}\frac{\sin x}{x}=1$\quad
$\lim_{x\to 0}\frac{1-\cos x}{x^2}=\frac{1}{2}$

$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e$\quad
$\lim_{x\to 0}\frac{e^x-1}{x}=1$

\subsection{Derivatives}
$\frac{d}{dx}x^n = nx^{n-1}$\quad
$\frac{d}{dx}e^x = e^x$\quad
$\frac{d}{dx}\ln x = \frac{1}{x}$

$\frac{d}{dx}\sin x = \cos x$\quad
$\frac{d}{dx}\cos x = -\sin x$

$\frac{d}{dx}\tan x = \sec^2 x$\quad
$\frac{d}{dx}\arctan x = \frac{1}{1+x^2}$

$\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}$\quad
$\frac{d}{dx}\arccos x = \frac{-1}{\sqrt{1-x^2}}$

\textbf{Chain:} $(f\circ g)' = f'(g(x))\cdot g'(x)$

\textbf{Product:} $(fg)' = f'g + fg'$

\textbf{Quotient:} $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

\subsection{Integrals}
$\int x^n\dd x = \frac{x^{n+1}}{n+1}+C$\quad ($n\neq -1$)

$\int \frac{1}{x}\dd x = \ln|x|+C$\quad
$\int e^x\dd x = e^x+C$

$\int \sin x\dd x = -\cos x+C$\quad
$\int \cos x\dd x = \sin x+C$

$\int \sec^2 x\dd x = \tan x+C$

$\int \frac{1}{1+x^2}\dd x = \arctan x+C$

$\int \frac{1}{\sqrt{1-x^2}}\dd x = \arcsin x + C$

\textbf{IBP:} $\int u\dd v = uv - \int v\dd u$

\textbf{FTC:} $\int_a^b f'(x)\dd x = f(b)-f(a)$

\subsection{Series}
$e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$\quad
$\sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}$

$\cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}$\quad
$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$

$\ln(1+x)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^n}{n}$,\; $|x|\le 1$

% ============================================================
\section{Multivariable Calculus}

\subsection{Gradient \& Directional Deriv.}
$\nabla f = \left(\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n}\right)$

$D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}$\quad ($\|\mathbf{u}\|=1$)

\subsection{Vector Calculus}
$\nabla\cdot\mathbf{F}=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}$

$\nabla\times\mathbf{F}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\\partial_x&\partial_y&\partial_z\\F_1&F_2&F_3\end{vmatrix}$

$\nabla\cdot(\nabla\times\mathbf{F})=0$\quad
$\nabla\times(\nabla f)=\mathbf{0}$

\subsection{Integral Theorems}
\textbf{Green:} $\oint_C P\dd x+Q\dd y = \iint_D\!\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\dd A$

\textbf{Stokes:} $\oint_C\mathbf{F}\cdot\dd\mathbf{r}=\iint_S(\nabla\times\mathbf{F})\cdot\dd\mathbf{S}$

\textbf{Divergence:} $\oiint_S\mathbf{F}\cdot\dd\mathbf{S}=\iiint_V\nabla\cdot\mathbf{F}\dd V$

\subsection{Coordinate Transforms}
\textbf{Polar:} $x=r\cos\theta$, $y=r\sin\theta$, $\dd A=r\dd r\dd\theta$

\textbf{Cylindrical:} $\dd V=r\dd r\dd\theta\dd z$

\textbf{Spherical:} $x=\rho\sin\phi\cos\theta$, $y=\rho\sin\phi\sin\theta$,
$z=\rho\cos\phi$, $\dd V=\rho^2\sin\phi\dd\rho\dd\phi\dd\theta$

% ============================================================
\section{Linear Algebra}

\subsection{Matrix Operations}
$(AB)^T = B^T A^T$\quad
$(AB)^{-1}=B^{-1}A^{-1}$

$A^{-1}=\frac{1}{\det A}\operatorname{adj}(A)$

For $2\times 2$: $\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$

\subsection{Determinants}
$\det(AB)=\det A\cdot\det B$\quad
$\det(A^T)=\det A$

$\det(cA)=c^n\det A$\quad ($A$ is $n\times n$)

$\det A=\prod_{i}\lambda_i$\quad
$\tr A = \sum_i \lambda_i$

\subsection{Eigenvalues}
$A\mathbf{v}=\lambda\mathbf{v}$\quad
$\det(A-\lambda I)=0$

If $A=PDP^{-1}$, then $A^k=PD^kP^{-1}$

$e^{At}=Pe^{Dt}P^{-1}$ where $e^{Dt}=\diag(e^{\lambda_1 t},\ldots)$

\subsection{SVD \& Norms}
$A = U\Sigma V^T$\quad (Singular Value Decomposition)

$\|A\|_F = \sqrt{\tr(A^TA)} = \sqrt{\sum\sigma_i^2}$

$\|A\|_2 = \sigma_{\max}(A)$

\subsection{Key Identities}
$\rank(A) = \rank(A^T) = \rank(A^TA)$

Rank-Nullity: $\rank(A)+\dim\ker(A)=n$

$\langle A\mathbf{x},\mathbf{y}\rangle = \langle\mathbf{x},A^T\mathbf{y}\rangle$

\subsection{Projections}
$\text{proj}_\mathbf{u}\mathbf{v}=\frac{\mathbf{v}\cdot\mathbf{u}}{\mathbf{u}\cdot\mathbf{u}}\mathbf{u}$

Projection onto col($A$): $P=A(A^TA)^{-1}A^T$

Least squares: $\hat{\mathbf{x}}=(A^TA)^{-1}A^T\mathbf{b}$

% ============================================================
\section{Probability \& Statistics}

\subsection{Combinatorics}
$\binom{n}{k}=\frac{n!}{k!(n-k)!}$\quad
$P(n,k)=\frac{n!}{(n-k)!}$

$(x+y)^n=\sum_{k=0}^n\binom{n}{k}x^ky^{n-k}$

\subsection{Basic Probability}
$P(A\cup B) = P(A)+P(B)-P(A\cap B)$

$P(A|B) = \frac{P(A\cap B)}{P(B)}$

\textbf{Bayes:} $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

$P(B) = \sum_i P(B|A_i)P(A_i)$\quad (Law of Total Prob.)

\subsection{Expectation \& Variance}
$\E[X]=\sum_x xP(X\!=\!x)$ or $\int xf(x)\dd x$

$\Var(X)=\E[X^2]-(\E[X])^2$

$\E[aX+b]=a\E[X]+b$

$\Var(aX+b)=a^2\Var(X)$

$\Cov(X,Y)=\E[XY]-\E[X]\E[Y]$

$\Var(X+Y)=\Var(X)+\Var(Y)+2\Cov(X,Y)$

\subsection{Common Distributions}
\textbf{Bernoulli}$(p)$: $\E=p$, $\Var=p(1\!-\!p)$

\textbf{Binomial}$(n,p)$: $P(k)=\binom{n}{k}p^k(1\!-\!p)^{n-k}$,
$\E=np$, $\Var=np(1\!-\!p)$

\textbf{Poisson}$(\lambda)$: $P(k)=\frac{\lambda^k e^{-\lambda}}{k!}$,
$\E=\Var=\lambda$

\textbf{Geometric}$(p)$: $P(k)=(1\!-\!p)^{k-1}p$,
$\E=\frac{1}{p}$, $\Var=\frac{1-p}{p^2}$

\textbf{Uniform}$(a,b)$: $f=\frac{1}{b-a}$,
$\E=\frac{a+b}{2}$, $\Var=\frac{(b-a)^2}{12}$

\textbf{Normal}$(\mu,\sigma^2)$: $f=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

\textbf{Exponential}$(\lambda)$: $f=\lambda e^{-\lambda x}$,
$\E=\frac{1}{\lambda}$, $\Var=\frac{1}{\lambda^2}$

\subsection{Limit Theorems}
\textbf{LLN:} $\bar{X}_n \xrightarrow{P} \mu$ as $n\to\infty$

\textbf{CLT:} $\frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0,1)$

\textbf{Markov:} $P(X\ge a)\le\frac{\E[X]}{a}$

\textbf{Chebyshev:} $P(|X-\mu|\ge k\sigma)\le\frac{1}{k^2}$

\subsection{MGF \& Transforms}
$M_X(t)=\E[e^{tX}]$\quad
$\E[X^n]=M_X^{(n)}(0)$

$M_{aX+b}(t)=e^{bt}M_X(at)$

\end{multicols}

\end{document}