Here is a collection of useful LaTex math commands. Good reference
Euler’s Identity:
\[e^{i \pi} + 1 = 0\] $e^{i \pi} + 1 = 0$
Fractions and brackets:
\[\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]$$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Commonly used:
\[\alpha = \beta + \mu \longrightarrow\] \[x^2 = y^{abc}de \times gh\] \[a = \frac{c}{def}\] \[f''(x)=\sum_{i=0}^3 \ddot{a} + \hat{b} + \bar{c} + \vec{v}\] \[\vec{u}=\boldsymbol{i}+\boldsymbol{j}+\boldsymbol{k}\] \[|a| = \underbrace{b + c}_{A} + \underbrace{\ldots}_{\mbox{some comment}}\] \[a < b \leq c \geq d > e \neq f \approx 3\] \[\nabla f = \frac{\partial f}{dx} + \frac{\partial f}{dy}+ \frac{\partial f}{dz} =\int_{0}^3dx\] \[\left[ \frac{-b\pm\sqrt{b^2-4ac}}{2a} \right]\] \[\mbox{spacing} \qquad ~~~\mbox{using quad or qquad or ~}\] \[v = 32 ~ \text{ft/s}^2\]$$\alpha = \beta + \mu \longrightarrow$$
$$x^2 = y^{abc}de \times gh$$
$$a = \frac{c}{def} $$
$$f''(x)=\sum_{i=0}^3 \ddot{a} + \hat{b} + \bar{c} + \vec{v} $$
$$\vec{u}=\boldsymbol{i}+\boldsymbol{j}+\boldsymbol{k} $$
$$|a| = \underbrace{b + c}_{A} + \underbrace{\ldots}_{\mbox{some comment}} $$
$$a < b \leq c \geq d > e \neq f \approx 3 $$
$$\nabla f = \frac{\partial f}{dx} + \frac{\partial f}{dy}+ \frac{\partial f}{dz} =\int_{0}^3dx $$
$$\left[ \frac{-b\pm\sqrt{b^2-4ac}}{2a} \right] $$
$$\mbox{spacing} \qquad ~~~\mbox{using quad or qquad or ~} $$
$$v = 32 ~ \text{ft/s}^2$$
Matrices and Arrays:
\[\pmatrix{ a&b&d \\ c&d&e}\] \[\left \{ \array{ a&b\\ b&b\\ c&b}\right\}\] \[\begin{bmatrix} a & b & c\\ d & e & f \end{bmatrix}\] \[\begin{Bmatrix} a & b & c\\ d & e & f \end{Bmatrix}\]$$
\pmatrix{
a&b&d \\
c&d&e}
$$
$$
\left \{ \array{
a&b\\
b&b\\
c&b}\right\}
$$
$$
\begin{bmatrix}
a & b & c\\
d & e & f
\end{bmatrix}
$$
$$
\begin{Bmatrix}
a & b & c\\
d & e & f
\end{Bmatrix}
$$
Aligning multiple lines:
\[\begin{align*} y = y(x,t) &= A e^{i\theta} \\ &= A (\cos \theta + i \sin \theta) \\ &= A (\cos(kx - \omega t) + i \sin(kx - \omega t)) \\ &= A\cos(kx - \omega t) + i A\sin(kx - \omega t) \\ &= A\cos \Big(\frac{2\pi}{\lambda}x - \frac{2\pi v}{\lambda} t \Big) + i A\sin \Big(\frac{2\pi}{\lambda}x - \frac{2\pi v}{\lambda} t \Big) \\ &= A\cos \frac{2\pi}{\lambda} (x - v t) + i A\sin \frac{2\pi}{\lambda} (x - v t) \end{align*}\]\begin{align*}
y = y(x,t) &= A e^{i\theta} \\
&= A (\cos \theta + i \sin \theta) \\
&= A (\cos(kx - \omega t) + i \sin(kx - \omega t)) \\
&= A\cos(kx - \omega t) + i A\sin(kx - \omega t) \\
&= A\cos \Big(\frac{2\pi}{\lambda}x - \frac{2\pi v}{\lambda} t \Big) +
i A\sin \Big(\frac{2\pi}{\lambda}x - \frac{2\pi v}{\lambda} t \Big) \\
&= A\cos \frac{2\pi}{\lambda} (x - v t) + i A\sin \frac{2\pi}{\lambda} (x - v t)
\end{align*}
Big Bracket. Stepwise functions:
\[sign(x) = \left\{ \begin{array}\\ 1 & \mbox{if } \ x \in \mathbf{N}^* \\ 0 & \mbox{if } \ x = 0 \\ -1 & \mbox{else.} \end{array} \right.\]sign(x) = \left\{
\begin{array}\\
1 & \mbox{if } \ x \in \mathbf{N}^* \\
0 & \mbox{if } \ x = 0 \\
-1 & \mbox{else.}
\end{array}
\right.
'''