LaTeX Math I often use

Vectors

Info

use \vec{} to denote vectors, as opposed to scalars. Useful for vector fields vs. scalar fields too.

$$\begin{array}{rlrl}\vec{F}& =k\vec{x}& & \text{(Hooke’s law)}\end{array}$$

Logical operators

Info

Logical operators in LaTeX typically are l[name], like \lnot, \land, etc.

$$
\lnot (A \land B) \iff \lnot B \lor \lnot A \quad \text{(De Morgan’s law)}
$$

$$\neg (A\wedge B)⟺\neg B\vee \neg A\text{(De Morgan’s law)}$$

Mid for probabilities and sets

$$
Pr(x \mid y) = \frac{Pr(y \mid x) Pr(x)}{Pr(y)} \quad \text{(Bayes theorem)}
$$

$$Pr(x\mid y)=\frac{Pr(y\mid x)Pr(x)}{Pr(y)}\text{(Bayes theorem)}$$

$$
X = \{(y,z) \mid y \in Y, z \in Z \} \quad \text{(Cartesian Product)}
$$

$$X=\{(y,z)\mid y\in Y,z\in Z\}\text{(Cartesian Product)}$$

Norm of matrix or vector

$$
\| x \| = ()
$$

$$\begin{array}{rlr}\Vert x\Vert :=\sqrt{(x\cdot x)}& & \text{(Euclidean norm)}\end{array}$$

Multi-line equations

Info

use the \begin{aligned} and \end{aligned} to make multi-line equations, where each line is separated by \\. use & before operator (e.g., =) to make equations line up at them, and \\ for new lines. You can then use && to make \text{} or other math notes line up on the right side.

$$
\begin{aligned}
\vec{E}(\vec{x}) &= \int_{V} \frac{\rho(\vec{r})\widehat{(\vec{x} - \vec{r})}}{4 \pi \epsilon_0\mid \vec{x} - \vec{r} \mid ^2} d\tau && \text{(Electric field from charge distribution)} \\
\vec{E} &= - \nabla V && \text{(Electric field from electric potential )} \\
\end{aligned} 
$$

$$\begin{array}{rlrl}\vec{E}(\vec{x})& ={\int }_V\frac{\rho (\vec{r})\widehat{(\vec{x}-\vec{r})}}{4\pi {ϵ}_0\mid \vec{x}-\vec{r}{\mid }^2}d\tau & & \text{(Electric field from charge distribution)}\\ \vec{E}& =-\nabla V& & \text{(Electric field from electric potential )}\end{array}$$

Making column vectors and row vectors

Info

Use vdots and ldots for 3 dots to be rendered vertically and horizontally, respectively.

$$
\begin{bmatrix}
x_1 \\
\vdots \\
x_n \\
\end{bmatrix}
$$

$$\left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right]$$

$$
[x_1, \ldots, x_n]
$$

$$[x_1,\dots ,x_n]$$