Mathematics

One of the hallmarks of academic writing is beautiful mathematical typography. This theme uses KaTeX to render LaTeX-style equations directly in your browser.

Inline Mathematics

Mathematical expressions can appear inline with text. For example, the famous equation $E = mc^2$ shows the relationship between energy and mass. The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves any quadratic equation.

Greek letters like $\alpha$, $\beta$, $\gamma$, and $\theta$ are commonly used in mathematics and physics. Subscripts and superscripts work naturally: $x_1, x_2, \ldots, x_n$ and $2^{10} = 1024$.

Display Equations

For important equations, display mode centers them on their own line:

$$ \int_{-\infty}^{\infty} e^{-x^2} , dx = \sqrt{\pi} $$

This is the Gaussian integral, one of the most important integrals in mathematics.

The Euler Identity

Often called the most beautiful equation in mathematics:

$$ e^{i\pi} + 1 = 0 $$

It connects five fundamental constants: $e$, $i$, $\pi$, $1$, and $0$.

Calculus

Derivatives and integrals are the foundation of calculus:

$$ \frac{d}{dx}\left[ \int_a^x f(t) , dt \right] = f(x) $$

The chain rule for composite functions:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$

Taylor series expansion:

$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n $$

Linear Algebra

Matrix notation is fully supported:

$$ \mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} $$

The determinant of a 2×2 matrix:

$$ \det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc $$

Probability and Statistics

Bayes' theorem:

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

The normal distribution probability density function:

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

Physics

Maxwell's equations in differential form:

$$ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} $$

$$ \nabla \cdot \mathbf{B} = 0 $$

$$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$

$$ \nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t} $$

Schrödinger's equation:

$$ i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r}, t) = \hat{H}\Psi(\mathbf{r}, t) $$

Aligned Equations

Multi-line derivations can be aligned:

$$ \begin{aligned} (x + y)^2 &= (x + y)(x + y) \\ &= x^2 + xy + yx + y^2 \\ &= x^2 + 2xy + y^2 \end{aligned} $$

Cases and Piecewise Functions

$$ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} $$

Enabling Math

To enable KaTeX rendering on a page, add math = true to your page's frontmatter:

[extra]
math = true