LaTeX Math Demo#

This post demonstrates various LaTeX math formulas rendered with KaTeX.

Calculus#

Derivatives#

$$\frac{d}{dx} \left( \int_{a}^{x} f(t)\,dt \right) = f(x)$$

The product rule:

$$(fg)' = f'g + fg'$$

Integrals#

Double integral:

$$\iint_{D} f(x,y)\,dx\,dy$$

Line integral:

$$\oint_{C} \vec{F} \cdot d\vec{r}$$

Linear Algebra#

Matrix:

$$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$

Determinant:

$$\det(A) = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \prod_{i=1}^{n} a_{i,\sigma(i)}$$

Physics#

Maxwell’s equations:

$$\begin{aligned} \nabla \cdot \vec{E} &= \frac{\rho}{\varepsilon_0} \\ \nabla \cdot \vec{B} &= 0 \\ \nabla \times \vec{E} &= -\frac{\partial \vec{B}}{\partial t} \\ \nabla \times \vec{B} &= \mu_0\vec{J} + \mu_0\varepsilon_0\frac{\partial \vec{E}}{\partial t} \end{aligned}$$

Schrödinger equation:

$$i\hbar\frac{\partial}{\partial t}|\Psi\rangle = \hat{H}|\Psi\rangle$$

Statistics#

Normal distribution:

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$

Bayes’ theorem:

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

Greek Alphabet#

$$\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \iota, \kappa, \lambda, \mu, \nu, \xi, \omicron, \pi, \rho, \sigma, \tau, \upsilon, \phi, \chi, \psi, \omega$$ $$\Gamma, \Delta, \Theta, \Lambda, \Xi, \Pi, \Sigma, \Phi, \Psi, \Omega$$