About This Page
This page introduces basic mathematical notation common to all departments.
For more advanced mathematics (vector analysis, Fourier transforms, etc.), see the Advanced Science/Engineering page.
Table of Contents
Limits and Continuity
Epsilon-Delta Definition of Limit
Code:
$\displaystyle\lim_{x \to a} f(x) = L \Leftrightarrow \forall \varepsilon > 0, \exists \delta > 0 : 0 < |x - a| < \delta \Rightarrow |f(x) - L| < \varepsilon$
Display:
$\displaystyle\lim_{x \to a} f(x) = L \Leftrightarrow \forall \varepsilon > 0, \exists \delta > 0 : 0 < |x - a| < \delta \Rightarrow |f(x) - L| < \varepsilon$
Epsilon-N Definition of Sequence Limit
Code:
$\displaystyle\lim_{n \to \infty} a_n = L \Leftrightarrow \forall \varepsilon > 0, \exists N \in \mathbb{N} : n > N \Rightarrow |a_n - L| < \varepsilon$
Display:
$\displaystyle\lim_{n \to \infty} a_n = L \Leftrightarrow \forall \varepsilon > 0, \exists N \in \mathbb{N} : n > N \Rightarrow |a_n - L| < \varepsilon$
Definition of Continuity
Code:
$f$ is continuous at $a$ $\Leftrightarrow \displaystyle\lim_{x \to a} f(x) = f(a)$
Display:
$f$ is continuous at $a$ $\Leftrightarrow \displaystyle\lim_{x \to a} f(x) = f(a)$
Uniform Continuity
Code:
$\forall \varepsilon > 0, \exists \delta > 0 : |x - y| < \delta \Rightarrow |f(x) - f(y)| < \varepsilon$
Display:
$\forall \varepsilon > 0, \exists \delta > 0 : |x - y| < \delta \Rightarrow |f(x) - f(y)| < \varepsilon$
Two-Variable Function Limit
Code:
$\displaystyle\lim_{(x,y) \to (a,b)} f(x,y) = L$
Display:
$\displaystyle\lim_{(x,y) \to (a,b)} f(x,y) = L$
Limit Superior and Limit Inferior
Code:
$\displaystyle\limsup_{n \to \infty} a_n = \lim_{n \to \infty} \sup_{k \geq n} a_k$, $\displaystyle\liminf_{n \to \infty} a_n = \lim_{n \to \infty} \inf_{k \geq n} a_k$
Display:
$\displaystyle\limsup_{n \to \infty} a_n = \lim_{n \to \infty} \sup_{k \geq n} a_k$, $\displaystyle\liminf_{n \to \infty} a_n = \lim_{n \to \infty} \inf_{k \geq n} a_k$
Partial Derivatives
Partial Derivative Notation
Code:
$\displaystyle\frac{\partial f}{\partial x}$
Display:
$\displaystyle\frac{\partial f}{\partial x}$
Second-Order Partial Derivative
Code:
$\displaystyle\frac{\partial^2 f}{\partial x^2}$
Display:
$\displaystyle\frac{\partial^2 f}{\partial x^2}$
Mixed Partial Derivative
Code:
$\displaystyle\frac{\partial^2 f}{\partial x \partial y}$
Display:
$\displaystyle\frac{\partial^2 f}{\partial x \partial y}$
Jacobian
Code:
$\displaystyle J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$
Display:
$\displaystyle J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$
Multivariable Calculus
Total Differential
Code:
$\displaystyle df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$
Display:
$\displaystyle df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$
Directional Derivative
Code:
$\displaystyle D_{\vec{u}}f = \nabla f \cdot \vec{u} = \frac{\partial f}{\partial x}u_1 + \frac{\partial f}{\partial y}u_2$
Display:
$\displaystyle D_{\vec{u}}f = \nabla f \cdot \vec{u} = \frac{\partial f}{\partial x}u_1 + \frac{\partial f}{\partial y}u_2$
Chain Rule
Code:
$\displaystyle\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$
Display:
$\displaystyle\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$
Implicit Function Theorem
Code:
If $F(x,y) = 0$, then $\displaystyle\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y} = -\frac{F_x}{F_y}$
Display:
If $F(x,y) = 0$, then $\displaystyle\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y} = -\frac{F_x}{F_y}$
Hessian Matrix
Code:
$\displaystyle H = \begin{pmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\ \frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} \end{pmatrix}$
Display:
$\displaystyle H = \begin{pmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\ \frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} \end{pmatrix}$
Lagrange Multipliers
Code:
$\nabla f = \lambda \nabla g$ (to extremize $f(x,y)$ subject to constraint $g(x,y) = c$)
Display:
$\nabla f = \lambda \nabla g$ (to extremize $f(x,y)$ subject to constraint $g(x,y) = c$)
Taylor Expansion (Two Variables)
Code:
$\displaystyle f(a+h, b+k) = f(a,b) + \left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right)f + \frac{1}{2!}\left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right)^2 f + \cdots$
Display:
$\displaystyle f(a+h, b+k) = f(a,b) + \left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right)f + \frac{1}{2!}\left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right)^2 f + \cdots$
Various Types of Integration
Double Integral
Code:
$\displaystyle\iint_D f(x,y) \, dx \, dy$
Display:
$\displaystyle\iint_D f(x,y) \, dx \, dy$
Triple Integral
Code:
$\displaystyle\iiint_V f(x,y,z) \, dx \, dy \, dz$
Display:
$\displaystyle\iiint_V f(x,y,z) \, dx \, dy \, dz$
Line Integral
Code:
$\displaystyle\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt$
Display:
$\displaystyle\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt$
Surface Integral
Code:
$\displaystyle\iint_S \vec{F} \cdot d\vec{S} = \iint_D \vec{F} \cdot \vec{n} \, dS$
Display:
$\displaystyle\iint_S \vec{F} \cdot d\vec{S} = \iint_D \vec{F} \cdot \vec{n} \, dS$
Linear Algebra
Matrix Multiplication
Code:
$AB$
Display:
$AB$
Identity Matrix
Code:
$I$ or $E$
Display:
$I$ or $E$
Eigenvalue
Code:
$\lambda$
Display:
$\lambda$
Characteristic Equation
Code:
$\det(A - \lambda I) = 0$
Display:
$\det(A - \lambda I) = 0$
Trace
Code:
$\text{tr}(A)$
Display:
$\text{tr}(A)$
Rank
Code:
$\text{rank}(A)$
Display:
$\text{rank}(A)$
Condition for Inverse Matrix
Code:
$A^{-1}$ exists $\Leftrightarrow \det(A) \neq 0$
Display:
$A^{-1}$ exists $\Leftrightarrow \det(A) \neq 0$
2×2 Inverse Matrix
Code:
$\displaystyle\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
Display:
$\displaystyle\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
Properties of Determinants
Code:
$\det(AB) = \det(A)\det(B)$, $\det(A^{-1}) = \frac{1}{\det(A)}$, $\det(A^T) = \det(A)$
Display:
$\det(AB) = \det(A)\det(B)$, $\det(A^{-1}) = \frac{1}{\det(A)}$, $\det(A^T) = \det(A)$
Eigenvector
Code:
$A\vec{v} = \lambda\vec{v}$ ($\vec{v} \neq \vec{0}$)
Display:
$A\vec{v} = \lambda\vec{v}$ ($\vec{v} \neq \vec{0}$)
Diagonalization
Code:
$A = PDP^{-1}$ ($D$ is diagonal, $P$ is matrix of eigenvectors)
Display:
$A = PDP^{-1}$ ($D$ is diagonal, $P$ is matrix of eigenvectors)
Cayley-Hamilton Theorem
Code:
If $p(\lambda)$ is the characteristic polynomial of $\det(A - \lambda I) = 0$, then $p(A) = O$
Display:
If $p(\lambda)$ is the characteristic polynomial of $\det(A - \lambda I) = 0$, then $p(A) = O$
Orthogonal Matrix
Code:
$Q^TQ = QQ^T = I$ (i.e., $Q^{-1} = Q^T$)
Display:
$Q^TQ = QQ^T = I$ (i.e., $Q^{-1} = Q^T$)
Inner Product (Dot Product)
Code:
$\langle \vec{u}, \vec{v} \rangle = \vec{u} \cdot \vec{v} = \sum_{i=1}^{n} u_i v_i = \vec{u}^T \vec{v}$
Display:
$\langle \vec{u}, \vec{v} \rangle = \vec{u} \cdot \vec{v} = \sum_{i=1}^{n} u_i v_i = \vec{u}^T \vec{v}$
Gram-Schmidt Orthonormalization
Code:
$\displaystyle \vec{u}_k = \vec{v}_k - \sum_{j=1}^{k-1} \frac{\langle \vec{v}_k, \vec{u}_j \rangle}{\langle \vec{u}_j, \vec{u}_j \rangle} \vec{u}_j$
Display:
$\displaystyle \vec{u}_k = \vec{v}_k - \sum_{j=1}^{k-1} \frac{\langle \vec{v}_k, \vec{u}_j \rangle}{\langle \vec{u}_j, \vec{u}_j \rangle} \vec{u}_j$
Rank-Nullity Theorem
Code:
$\dim(\text{Ker}(A)) + \dim(\text{Im}(A)) = n$ (for $m \times n$ matrix $A$)
Display:
$\dim(\text{Ker}(A)) + \dim(\text{Im}(A)) = n$ (for $m \times n$ matrix $A$)
Differential Equations
First-Order Linear ODE
Code:
$\displaystyle\frac{dy}{dx} + P(x)y = Q(x)$
Display:
$\displaystyle\frac{dy}{dx} + P(x)y = Q(x)$
First-Order Linear General Solution
Code:
$\displaystyle y = e^{-\int P dx}\left(\int Q e^{\int P dx} dx + C\right)$
Display:
$\displaystyle y = e^{-\int P dx}\left(\int Q e^{\int P dx} dx + C\right)$
Separable Equation
Code:
$\displaystyle\frac{dy}{dx} = f(x)g(y) \Rightarrow \int \frac{dy}{g(y)} = \int f(x)dx$
Display:
$\displaystyle\frac{dy}{dx} = f(x)g(y) \Rightarrow \int \frac{dy}{g(y)} = \int f(x)dx$
Second-Order Linear Homogeneous ODE
Code:
$\displaystyle\frac{d^2y}{dx^2} + p\frac{dy}{dx} + qy = 0$
Display:
$\displaystyle\frac{d^2y}{dx^2} + p\frac{dy}{dx} + qy = 0$
Characteristic Equation
Code:
$\lambda^2 + p\lambda + q = 0$ (substituting $y = e^{\lambda x}$)
Display:
$\lambda^2 + p\lambda + q = 0$ (substituting $y = e^{\lambda x}$)
General Solution (Distinct Real Roots)
Code:
$y = C_1 e^{\lambda_1 x} + C_2 e^{\lambda_2 x}$
Display:
$y = C_1 e^{\lambda_1 x} + C_2 e^{\lambda_2 x}$
General Solution (Repeated Root)
Code:
$y = (C_1 + C_2 x)e^{\lambda x}$
Display:
$y = (C_1 + C_2 x)e^{\lambda x}$
General Solution (Complex Roots)
Code:
For $\lambda = \alpha \pm \beta i$: $y = e^{\alpha x}(C_1 \cos\beta x + C_2 \sin\beta x)$
Display:
For $\lambda = \alpha \pm \beta i$: $y = e^{\alpha x}(C_1 \cos\beta x + C_2 \sin\beta x)$
Laplace Transform
Code:
$\displaystyle\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) dt$
Display:
$\displaystyle\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) dt$
Inverse Laplace Transform
Code:
$\displaystyle\mathcal{L}^{-1}\{F(s)\} = f(t) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} e^{st} F(s) ds$
Display:
$\displaystyle\mathcal{L}^{-1}\{F(s)\} = f(t) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} e^{st} F(s) ds$
Series Expansions
Taylor Series
Code:
$\displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$
Display:
$\displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$
Maclaurin Series
Code:
$\displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$
Display:
$\displaystyle f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$
Maclaurin Series of $e^x$
Code:
$\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$
Display:
$\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$
Maclaurin Series of $\sin x$
Code:
$\displaystyle \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$
Display:
$\displaystyle \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$
Maclaurin Series of $\cos x$
Code:
$\displaystyle \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$
Display:
$\displaystyle \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$
Maclaurin Series of $\ln(1+x)$
Code:
$\displaystyle \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ $(-1 < x \leq 1)$
Display:
$\displaystyle \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ $(-1 < x \leq 1)$
Maclaurin Series of $(1+x)^\alpha$ (Binomial Series)
Code:
$\displaystyle (1+x)^\alpha = \sum_{n=0}^{\infty} \binom{\alpha}{n} x^n = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \cdots$
Display:
$\displaystyle (1+x)^\alpha = \sum_{n=0}^{\infty} \binom{\alpha}{n} x^n = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \cdots$
Radius of Convergence
Code:
$\displaystyle R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|$ or $\displaystyle R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}}$
Display:
$\displaystyle R = \lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right|$ or $\displaystyle R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}}$
Complex Analysis
Complex Number Representation
Code:
$z = x + iy = r(\cos\theta + i\sin\theta) = re^{i\theta}$
Display:
$z = x + iy = r(\cos\theta + i\sin\theta) = re^{i\theta}$
Euler's Formula
Code:
$e^{i\theta} = \cos\theta + i\sin\theta$, $e^{i\pi} + 1 = 0$
Display:
$e^{i\theta} = \cos\theta + i\sin\theta$, $e^{i\pi} + 1 = 0$
Complex Conjugate and Modulus
Code:
$\bar{z} = x - iy$, $|z| = \sqrt{x^2 + y^2} = \sqrt{z\bar{z}}$
Display:
$\bar{z} = x - iy$, $|z| = \sqrt{x^2 + y^2} = \sqrt{z\bar{z}}$
Cauchy-Riemann Equations
Code:
$\displaystyle\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$, $\displaystyle\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ (condition for $f(z) = u + iv$ to be holomorphic)
Display:
$\displaystyle\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$, $\displaystyle\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ (condition for $f(z) = u + iv$ to be holomorphic)
Cauchy's Integral Theorem
Code:
$\displaystyle\oint_C f(z) dz = 0$ (when $f$ is holomorphic in a simply connected region)
Display:
$\displaystyle\oint_C f(z) dz = 0$ (when $f$ is holomorphic in a simply connected region)
Cauchy's Integral Formula
Code:
$\displaystyle f(a) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a} dz$
Display:
$\displaystyle f(a) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a} dz$
Laurent Series
Code:
$\displaystyle f(z) = \sum_{n=-\infty}^{\infty} a_n (z-a)^n$ (about $a$)
Display:
$\displaystyle f(z) = \sum_{n=-\infty}^{\infty} a_n (z-a)^n$ (about $a$)
Residue Theorem
Code:
$\displaystyle\oint_C f(z) dz = 2\pi i \sum_{k} \text{Res}(f, a_k)$
Display:
$\displaystyle\oint_C f(z) dz = 2\pi i \sum_{k} \text{Res}(f, a_k)$
Residue at a Simple Pole
Code:
$\displaystyle\text{Res}(f, a) = \lim_{z \to a}(z-a)f(z)$
Display:
$\displaystyle\text{Res}(f, a) = \lim_{z \to a}(z-a)f(z)$
Probability and Statistics
Conditional Probability
Code:
$\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)}$
Display:
$\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)}$
Bayes' Theorem
Code:
$\displaystyle P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
Display:
$\displaystyle P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
Covariance
Code:
$\text{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)]$
Display:
$\text{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)]$
Correlation Coefficient
Code:
$\displaystyle\rho = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$
Display:
$\displaystyle\rho = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$
Normal Distribution PDF
Code:
$\displaystyle f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
Display:
$\displaystyle f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
Central Limit Theorem
Code:
$\displaystyle \frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0, 1)$
Display:
$\displaystyle \frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} N(0, 1)$
Moment Generating Function
Code:
$M_X(t) = E[e^{tX}]$
Display:
$M_X(t) = E[e^{tX}]$
Binomial Distribution
Code:
$\displaystyle P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$, $E[X] = np$, $V[X] = np(1-p)$
Display:
$\displaystyle P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$, $E[X] = np$, $V[X] = np(1-p)$
Poisson Distribution
Code:
$\displaystyle P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$, $E[X] = V[X] = \lambda$
Display:
$\displaystyle P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$, $E[X] = V[X] = \lambda$
Exponential Distribution
Code:
$f(x) = \lambda e^{-\lambda x}$ $(x \geq 0)$, $E[X] = \frac{1}{\lambda}$, $V[X] = \frac{1}{\lambda^2}$
Display:
$f(x) = \lambda e^{-\lambda x}$ $(x \geq 0)$, $E[X] = \frac{1}{\lambda}$, $V[X] = \frac{1}{\lambda^2}$
Chi-squared Distribution
Code:
$\chi^2_n = \sum_{i=1}^{n} Z_i^2$ ($Z_i \sim N(0,1)$ independent), $E[\chi^2_n] = n$, $V[\chi^2_n] = 2n$
Display:
$\chi^2_n = \sum_{i=1}^{n} Z_i^2$ ($Z_i \sim N(0,1)$ independent), $E[\chi^2_n] = n$, $V[\chi^2_n] = 2n$
t-Distribution
Code:
$\displaystyle t_n = \frac{Z}{\sqrt{\chi^2_n/n}}$ ($Z \sim N(0,1)$, $\chi^2_n$ independent)
Display:
$\displaystyle t_n = \frac{Z}{\sqrt{\chi^2_n/n}}$ ($Z \sim N(0,1)$, $\chi^2_n$ independent)
F-Distribution
Code:
$\displaystyle F_{m,n} = \frac{\chi^2_m/m}{\chi^2_n/n}$ ($\chi^2_m$, $\chi^2_n$ independent)
Display:
$\displaystyle F_{m,n} = \frac{\chi^2_m/m}{\chi^2_n/n}$ ($\chi^2_m$, $\chi^2_n$ independent)
Law of Large Numbers
Code:
$\displaystyle\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n}X_i \xrightarrow{P} \mu$ (as $n \to \infty$)
Display:
$\displaystyle\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n}X_i \xrightarrow{P} \mu$ (as $n \to \infty$)
Unbiased Estimator (Sample Variance)
Code:
$\displaystyle s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^2$ ($E[s^2] = \sigma^2$)
Display:
$\displaystyle s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^2$ ($E[s^2] = \sigma^2$)
Logical Symbols and Set Theory
Universal Quantifier
Code:
$\forall x \in A$ (for all x)
Display:
$\forall x \in A$ (for all x)
Existential Quantifier
Code:
$\exists x \in A$ (there exists an x)
Display:
$\exists x \in A$ (there exists an x)
Unique Existence
Code:
$\exists! x$ (there exists exactly one x)
Display:
$\exists! x$ (there exists exactly one x)
Implication and Equivalence
Code:
$P \Rightarrow Q$ (if P then Q), $P \Leftrightarrow Q$ (P is equivalent to Q)
Display:
$P \Rightarrow Q$ (if P then Q), $P \Leftrightarrow Q$ (P is equivalent to Q)
Negation, Conjunction, Disjunction
Code:
$\neg P$, $P \land Q$, $P \lor Q$
Display:
$\neg P$, $P \land Q$, $P \lor Q$
Number Sets
Code:
$\mathbb{N}$ (naturals), $\mathbb{Z}$ (integers), $\mathbb{Q}$ (rationals), $\mathbb{R}$ (reals), $\mathbb{C}$ (complex)
Display:
$\mathbb{N}$ (naturals), $\mathbb{Z}$ (integers), $\mathbb{Q}$ (rationals), $\mathbb{R}$ (reals), $\mathbb{C}$ (complex)
Set Operations
Code:
$A \cup B$ (union), $A \cap B$ (intersection), $A \setminus B$ (difference), $A^c$ (complement)
Display:
$A \cup B$ (union), $A \cap B$ (intersection), $A \setminus B$ (difference), $A^c$ (complement)
Subset and Proper Subset
Code:
$A \subseteq B$, $A \subset B$, $A \subsetneq B$
Display:
$A \subseteq B$, $A \subset B$, $A \subsetneq B$
Empty Set
Code:
$\emptyset$ or $\varnothing$
Display:
$\emptyset$ or $\varnothing$
Mapping/Function
Code:
$f: A \to B$, $f: x \mapsto f(x)$
Display:
$f: A \to B$, $f: x \mapsto f(x)$
Definition
Code:
$A := B$ or $A \equiv B$ (A is defined as B)
Display:
$A := B$ or $A \equiv B$ (A is defined as B)
Vector Calculus
Gradient (grad)
Code:
$\displaystyle\nabla f = \mathrm{grad}\, f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$
Display:
$\displaystyle\nabla f = \mathrm{grad}\, f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$
Divergence (div)
Code:
$\displaystyle\nabla \cdot \vec{F} = \mathrm{div}\, \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$
Display:
$\displaystyle\nabla \cdot \vec{F} = \mathrm{div}\, \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$
Curl (rot)
Code:
$\displaystyle\nabla \times \vec{F} = \mathrm{rot}\, \vec{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)$
Display:
$\displaystyle\nabla \times \vec{F} = \mathrm{rot}\, \vec{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)$
Laplacian
Code:
$\displaystyle\nabla^2 f = \Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$
Display:
$\displaystyle\nabla^2 f = \Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$
Divergence Theorem (Gauss's Theorem)
Code:
$\displaystyle\iiint_V (\nabla \cdot \vec{F})\, dV = \iint_S \vec{F} \cdot d\vec{S}$
Display:
$\displaystyle\iiint_V (\nabla \cdot \vec{F})\, dV = \iint_S \vec{F} \cdot d\vec{S}$
Stokes' Theorem
Code:
$\displaystyle\iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = \oint_C \vec{F} \cdot d\vec{r}$
Display:
$\displaystyle\iint_S (\nabla \times \vec{F}) \cdot d\vec{S} = \oint_C \vec{F} \cdot d\vec{r}$
Green's Theorem
Code:
$\displaystyle\oint_C (P dx + Q dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$
Display:
$\displaystyle\oint_C (P dx + Q dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$
Usage Examples in Reading Notes
Example 1: Partial Derivative Problem
Code:
Problem on p.56: For $f(x,y) = x^2 + 3xy + y^2$, $\displaystyle\frac{\partial f}{\partial x} = 2x + 3y$, $\displaystyle\frac{\partial f}{\partial y} = 3x + 2y$
Display:
Problem on p.56: For $f(x,y) = x^2 + 3xy + y^2$, $\displaystyle\frac{\partial f}{\partial x} = 2x + 3y$, $\displaystyle\frac{\partial f}{\partial y} = 3x + 2y$
Example 2: Eigenvalue Calculation
Code:
Eigenvalues of matrix $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$: From $\det(A - \lambda I) = 0$, $\lambda = 1, 3$
Display:
Eigenvalues of matrix $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$: From $\det(A - \lambda I) = 0$, $\lambda = 1, 3$
Example 3: Maximum Likelihood Estimation
Code:
Maximize likelihood $L(\theta) = \displaystyle\prod_{i=1}^{n} f(x_i; \theta)$. From log-likelihood $\ell(\theta) = \displaystyle\sum_{i=1}^{n} \log f(x_i; \theta)$, solve $\displaystyle\frac{\partial \ell}{\partial \theta} = 0$
Display:
Maximize likelihood $L(\theta) = \displaystyle\prod_{i=1}^{n} f(x_i; \theta)$. From log-likelihood $\ell(\theta) = \displaystyle\sum_{i=1}^{n} \log f(x_i; \theta)$, solve $\displaystyle\frac{\partial \ell}{\partial \theta} = 0$
Common Symbol Reference
\partial→ Partial derivative symbol ∂\iint, \iiint→ Multiple integrals\forall, \exists→ Universal ∀, Existential ∃\det→ Determinant det\text{}→ Text within formulas