Mathematical Notation with KaTeX

1. Inline Mathematics

Mathematical notation should flow naturally within the text. For example, the Pythagorean theorem is defined as $a^2 + b^2 = c^2$.

We can also discuss Euler's identity, $e^{i\pi} + 1 = 0$, which is often cited as one of the most beautiful equations in mathematics. When $x$ approaches infinity ($x \to \infty$), the function $f(x) = \frac{1}{x}$ approaches zero.

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2. Display (Block) Mathematics

Equations that need emphasis should be rendered in their own block.

The Quadratic Formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The Gaussian Integral:

This is a famous integral in statistics and physics.

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

Limit Definition of the Derivative:

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

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3. Matrices and Arrays

Testing grid alignment and bracket scaling.

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

A system of equations:

$$\begin{cases} 3x + 5y - z = 10 \\ 2x - y + 2z = 5 \\ x + y + z = 4 \end{cases}$$

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4. Complex Notations & Summation

Riemann Sum:

$$S = \sum_{i=1}^{n} f(x_i^*) \Delta x_i$$

Maxwell's Equations (Differential Form):

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

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5. Logic and Set Theory

• Union: $A \cup B$
• Intersection: $A \cap B$
• Subset: $A \subseteq B$
• For All: $\forall x \in \mathbb{R}$
• Exists: $\exists n \in \mathbb{N}$ such that $n > 10$.

$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$

6. Chemistry Test (mhchem support check)

Note: Standard KaTeX supports basic chemistry, but mhchem extension is required for complex reactions. If this renders as plain text variables, you may need to enable the extension in your config.

$$\ce{2H2 + O2 -> 2H2O}$$

(If the above looks like "ce 2H2..." it means the mhchem extension is not enabled, which is the default for standard KaTeX).