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Inline Math
Einstein’s famous equation \(E = mc^2\) changed physics forever.
The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) solves any quadratic equation.
Euler’s identity \(e^{i\pi} + 1 = 0\) is often called the most beautiful equation in mathematics.
Display Math
Fractions and Roots
\[f(x) = \frac{x^2 + 2x + 1}{x - 1}\]
\[\sqrt[n]{x} = x^{\frac{1}{n}}, \quad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]
Greek Letters
\[\alpha + \beta = \gamma, \quad \Delta = \Sigma \cdot \Pi\]
\[\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618, \quad \lambda \in \mathbb{R}\]
\[\theta, \vartheta, \Theta, \quad \epsilon, \varepsilon, \quad \rho, \varrho\]
Integrals
\[\int_0^\infty e^{-x^2}\, dx = \frac{\sqrt{\pi}}{2}\]
\[\int_a^b f(x)\, dx = F(b) - F(a)\]
\[\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}\]
Summations and Products
\[\sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x\]
\[\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\]
\[\prod_{k=1}^{n} k = n!\]
Limits and Derivatives
\[\lim_{x \to 0} \frac{\sin x}{x} = 1\]
\[\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e\]
\[\frac{d}{dx}\left[\int_a^x f(t)\, dt\right] = f(x)\]
Vectors and Operators
\[\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)\]
\[\mathbf{F} = m\mathbf{a} = m\frac{d^2\mathbf{r}}{dt^2}\]
\[\hat{H}\psi = E\psi\]
Matrices
\[\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
ax + by \\
cx + dy
\end{pmatrix}\]
\[A = \begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{bmatrix}, \quad
\det(A) = \begin{vmatrix}
a & b \\
c & d
\end{vmatrix} = ad - bc\]
Cases
\[f(x) = \begin{cases}
x^2 & \text{if } x \geq 0 \\
-x & \text{if } x < 0
\end{cases}\]
\[|x| = \begin{cases}
x & \text{if } x \geq 0 \\
-x & \text{if } x < 0
\end{cases}\]
Aligned Equations
\[\begin{aligned}
(a + b)^2 &= a^2 + 2ab + b^2 \\
(a - b)^2 &= a^2 - 2ab + b^2 \\
(a + b)(a - b) &= a^2 - b^2
\end{aligned}\]
Maxwell’s Equations
\[\begin{aligned}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} \\
\nabla \cdot \mathbf{B} &= 0 \\
\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\
\nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\end{aligned}\]
Binomial and Combinatorics
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]
Number Sets
\[\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}\]
\[\forall x \in \mathbb{R},\; \exists y \in \mathbb{R} : y > x\]
Logic and Sets
\[A \cup B = \{x \mid x \in A \lor x \in B\}\]
\[A \cap B \subseteq A \subseteq A \cup B\]
\[P \Rightarrow Q \iff \neg P \lor Q\]
