Advanced Mathematical Expressions with LaTeX

This document demonstrates the power of LaTeX for representing mathematical expressions in technical documentation, from basic notation to complex equations and proofs.

Basic Mathematical Notation

Arithmetic Operations

$a + b - c \times d \div e = f$

$7 \times 8 = 56$

$\frac{20}{5} = 4$

Exponents and Subscripts

$x^2 + y^2 = z^2$

$a_1 + a_2 + a_3 + \ldots + a_n = \sum_{i=1}^{n} a_i$

Chemical formulas: $H_2O$, $CO_2$

Greek Letters

Lowercase: $\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \iota, \kappa, \lambda, \mu, \nu, \xi, \pi, \rho, \sigma, \tau, \upsilon, \phi, \chi, \psi, \omega$

Uppercase: $\Gamma, \Delta, \Theta, \Lambda, \Xi, \Pi, \Sigma, \Upsilon, \Phi, \Psi, \Omega$

Variants: $\varepsilon, \vartheta, \varkappa, \varpi, \varrho, \varsigma, \varphi$

Set Theory and Logic

Set Notation

$A \cup B$ - Union of sets A and B

$A \cap B$ - Intersection of sets A and B

$A \setminus B$ - Set difference (elements in A but not in B)

$A \subset B$ - A is a subset of B

$A \subseteq B$ - A is a subset of or equal to B

$x \in A$ - x is an element of set A

$\emptyset$ - Empty set

$\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$ - Number sets (natural, integer, rational, real, complex)

Logic Operators

$P \land Q$ - Logical AND

$P \lor Q$ - Logical OR

$\lnot P$ - Logical NOT

$P \implies Q$ - Implication (if P then Q)

$P \iff Q$ - Logical equivalence (P if and only if Q)

$\forall x$ - Universal quantifier (for all x)

$\exists x$ - Existential quantifier (there exists x)

Calculus

Limits

$\lim_{x \to a} f(x) = L$

$\lim_{x \to \infty} \frac{1}{x} = 0$

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

Derivatives

$\frac{d}{dx}f(x) = f'(x)$

$\frac{d^2y}{dx^2} = f''(x)$

$\frac{\partial f}{\partial x}$ - Partial derivative

$\nabla f$ - Gradient of f

Integrals

Indefinite integral: $\int f(x) \, dx = F(x) + C$

Definite integral: $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$

Double integral: $\iint_D f(x,y) \, dx \, dy$

Line integral: $\oint_C f(x,y) \, ds$

Linear Algebra

Vectors and Matrices

Column vector: $\vec{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}$

Row vector: $\vec{v} = \begin{pmatrix} v_1 & v_2 & \cdots & v_n \end{pmatrix}$

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

Identity matrix: $I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Matrix Operations

Matrix multiplication: $C = AB = \begin{pmatrix} \sum_{k=1}^{n} a_{1k}b_{k1} & \sum_{k=1}^{n} a_{1k}b_{k2} & \cdots & \sum_{k=1}^{n} a_{1k}b_{kp} \\ \sum_{k=1}^{n} a_{2k}b_{k1} & \sum_{k=1}^{n} a_{2k}b_{k2} & \cdots & \sum_{k=1}^{n} a_{2k}b_{kp} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{k=1}^{n} a_{mk}b_{k1} & \sum_{k=1}^{n} a_{mk}b_{k2} & \cdots & \sum_{k=1}^{n} a_{mk}b_{kp} \end{pmatrix}$

Determinant: $\det(A)$ or $|A|$

Transpose: $A^T$

Inverse: $A^{-1}$

Trace: $\text{tr}(A) = \sum_{i=1}^{n} a_{ii}$

Equations and Systems

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for $ax^2 + bx + c = 0$

Systems of Linear Equations

$\begin{cases} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = b_m \end{cases}$

In matrix form: $A\vec{x} = \vec{b}$

Advanced Calculus

Taylor Series

$f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$

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

Fourier Series

$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right]$

Where: $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx$

$b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx$

Differential Equations

Ordinary Differential Equations

First-order linear ODE: $\frac{dy}{dx} + P(x)y = Q(x)$

Second-order linear ODE: $\frac{d^2y}{dx^2} + p(x)\frac{dy}{dx} + q(x)y = g(x)$

Partial Differential Equations

Wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

Heat equation: $\frac{\partial u}{\partial t} = \alpha \nabla^2 u$

Laplace equation: $\nabla^2 \phi = 0$ or $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} = 0$

Probability and Statistics

Probability

Probability of event A: $P(A)$

Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

Bayes’ theorem: $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Distributions

Binomial distribution: $P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$

Normal distribution probability density function: $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$

Poisson distribution: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$

Statistical Measures

Expected value: $E[X] = \sum_{i} x_i p_i$ or $E[X] = \int_{-\infty}^{\infty} x f(x) \, dx$

Variance: $\text{Var}(X) = E[(X - \mu)^2] = E[X^2] - E[X]^2$

Covariance: $\text{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]$

Correlation coefficient: $\rho_{X,Y} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}$

Number Theory

Divisibility and Congruences

$a \mid b$ means a divides b

$a \equiv b \pmod{m}$ means a is congruent to b modulo m

Euler’s totient function: $\phi(n) = n \prod_{p \mid n} \left(1 - \frac{1}{p}\right)$

Prime Numbers

Prime factorization: $n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$

The prime number theorem: $\pi(n) \sim \frac{n}{\ln n}$

Complex Analysis

Complex Numbers

$z = a + bi$ where $i^2 = -1$

$|z| = \sqrt{a^2 + b^2}$ - Modulus of z

$\arg(z) = \text{arctan}\frac{b}{a}$ - Argument of z

Complex Functions

Cauchy-Riemann equations: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Cauchy’s integral formula: $f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - z_0} \, dz$

Residue theorem: $\oint_C f(z) \, dz = 2\pi i \sum_{k=1}^{n} \text{Res}(f, z_k)$

Physics Equations

Mechanics

Newton’s second law: $\vec{F} = m\vec{a}$

Kinetic energy: $K = \frac{1}{2}mv^2$

Potential energy (gravitational): $U = mgh$

Electromagnetism

Maxwell’s Equations:

$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$ (Gauss’s law)

$\nabla \cdot \vec{B} = 0$ (Gauss’s law for magnetism)

$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$ (Faraday’s law)

$\nabla \times \vec{B} = \mu_0\left(\vec{J} + \epsilon_0 \frac{\partial \vec{E}}{\partial t}\right)$ (Ampère’s law with Maxwell’s addition)

Thermodynamics

First law: $\Delta U = Q - W$

Entropy change: $\Delta S = \int \frac{dQ}{T}$

Quantum Mechanics

Schrödinger equation: $i\hbar \frac{\partial}{\partial t}\Psi(\vec{r},t) = \hat{H}\Psi(\vec{r},t)$

Heisenberg uncertainty principle: $\Delta x \Delta p \geq \frac{\hbar}{2}$

Mathematical Proofs and Theorems

Proof by Induction

Base case: P(1) is true.

Inductive step: Assume P(k) is true, show P(k+1) is true.

Therefore, P(n) is true for all positive integers n.

Mathematical Theorems

The Fundamental Theorem of Calculus:

$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$

Where F is an antiderivative of f.

Pythagorean Theorem:

For a right triangle with sides a, b and hypotenuse c:

$a^2 + b^2 = c^2$

Commutative Diagrams

This diagram commutes if $h \circ f = j \circ g$

Advanced Mathematical Structures

Groups, Rings, and Fields

Group axioms for $(G, \cdot)$:

1. Closure: $\forall a,b \in G, a \cdot b \in G$
2. Associativity: $\forall a,b,c \in G, (a \cdot b) \cdot c = a \cdot (b \cdot c)$
3. Identity: $\exists e \in G, \forall a \in G, e \cdot a = a \cdot e = a$
4. Inverse: $\forall a \in G, \exists a^{-1} \in G, a \cdot a^{-1} = a^{-1} \cdot a = e$

Transformations and Mappings

Linear transformation: $T(\alpha \vec{v} + \beta \vec{w}) = \alpha T(\vec{v}) + \beta T(\vec{w})$

Eigenvalue equation: $T(\vec{v}) = \lambda \vec{v}$ where $\lambda$ is the eigenvalue

Notable Equations in Science

Einstein’s Mass-Energy Equivalence

$E = mc^2$

Euler’s Identity

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

Boltzmann’s Entropy Formula

$S = k_B \ln W$

The Standard Model Lagrangian (Simplified)

$\mathcal{L}_{SM} = \mathcal{L}_{gauge} + \mathcal{L}_{fermion} + \mathcal{L}_{Higgs} + \mathcal{L}_{Yukawa}$

Combinatorial Mathematics

Permutations and Combinations

Number of permutations: $P(n,r) = \frac{n!}{(n-r)!}$

Number of combinations: $C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$

Binomial Theorem

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

Optimization Problems

Lagrange Multipliers

To maximize/minimize f(x,y) subject to g(x,y) = c:

$\nabla f(x,y) = \lambda \nabla g(x,y)$

Multi-Dimensional Calculus

Multiple Integrals

Triple integral in rectangular coordinates: $\iiint_E f(x,y,z) \, dx \, dy \, dz$

Triple integral in spherical coordinates: $\iiint_E f(\rho,\theta,\phi) \, \rho^2\sin\phi \, d\rho \, d\theta \, d\phi$

Stokes’ Theorem

$\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$

Divergence Theorem

$\iiint_V (\nabla \cdot \vec{F}) \, dV = \oiint_S \vec{F} \cdot d\vec{S}$

Advanced Theoretical Mathematics

The Riemann Hypothesis

One of the most famous unsolved problems in mathematics, the Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function have real part $\frac{1}{2}$.

The Riemann zeta function is defined as:

$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \quad \text{for } \Re(s) > 1$

And it can be analytically continued to the entire complex plane except for a simple pole at $s=1$:

$\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)$

The Riemann Hypothesis conjectures that:

$\zeta(\sigma + it) = 0 \Rightarrow \sigma = \frac{1}{2}$

The zeros of the zeta function are closely related to the distribution of prime numbers through the explicit formula:

$\psi(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \log(2\pi) - \frac{1}{2}\log\left(1-\frac{1}{x^2}\right)$

Where $\rho$ runs through the non-trivial zeros of the zeta function.

Mandelbrot Set and Complex Dynamics

The Mandelbrot set $M$ is defined as the set of complex numbers $c$ for which the sequence defined by the recurrence relation:

$z_{n+1} = z_n^2 + c \quad \text{with } z_0 = 0$

remains bounded. Mathematically:

$M = \{c \in \mathbb{C} : \lim_{n \to \infty} |z_n| \not\to \infty \}$

The boundary of the Mandelbrot set exhibits infinite complexity with self-similar structures at all scales. The “bulb” regions correspond to different periods in the orbit dynamics:

$\text{Period-}p \text{ bulb: } z^p + c_p = z \text{ has an attracting fixed point}$

The Mandelbrot set can also be characterized using the potential function:

$G(c) = \lim_{n \to \infty} \frac{1}{2^n} \log|z_n|$

where $G(c) = 0$ if $c \in M$ and $G(c) > 0$ otherwise.

Euler’s Remarkable Identities

The Most Beautiful Equation in Mathematics

Euler’s identity elegantly connects five fundamental mathematical constants:

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

This can be derived from the more general Euler’s formula:

$e^{ix} = \cos x + i \sin x$

Setting $x = \pi$ yields:

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

Extended Euler’s Formula Applications

For any complex number $z = x + iy$:

$e^z = e^{x+iy} = e^x \cdot e^{iy} = e^x(\cos y + i\sin y)$

This leads to the hyperbolic functions:

$\cosh z = \frac{e^z + e^{-z}}{2} \quad \sinh z = \frac{e^z - e^{-z}}{2}$

And the remarkable identity:

$\cos z = \cosh(iz) \quad \sin z = -i\sinh(iz)$

Euler’s Infinite Series

$e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!} = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots$

$\sin z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots$

$\cos z = \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \cdots$

Advanced Trigonometry

Trigonometric Identities

The double angle formulas:

$\sin 2\theta = 2\sin \theta \cos \theta$

$\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$

$\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$

The sum and difference formulas:

$\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$

$\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$

$\tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}$

Hyperbolic Trigonometry

The hyperbolic functions and their relationship to the exponential function:

$\sinh x = \frac{e^x - e^{-x}}{2} \quad \cosh x = \frac{e^x + e^{-x}}{2} \quad \tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$

Fundamental hyperbolic identity:

$\cosh^2 x - \sinh^2 x = 1$

Complex Trigonometry

Extended complex trigonometric identities:

$\sin(x+iy) = \sin x \cosh y + i\cos x \sinh y$

$\cos(x+iy) = \cos x \cosh y - i\sin x \sinh y$

$\tan(x+iy) = \frac{\sin(x+iy)}{\cos(x+iy)} = \frac{\sin x \cosh y + i\cos x \sinh y}{\cos x \cosh y - i\sin x \sinh y}$

Mathematical Visualizations (TikZ-Style)

Vector Field Representation

The visualization of a vector field $\vec{F}(x,y) = (y, -x)$ representing rotational motion:

$\vec{F}(x,y) = \begin{pmatrix} y \\ -x \end{pmatrix}$

This field has curl:

$\nabla \times \vec{F} = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = -1 - 1 = -2$

And divergence:

$\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} = 0 + 0 = 0$

Parametric Surface Plotting

A torus with major radius $R$ and minor radius $r$ can be parametrized as:

$\vec{r}(u,v) = \begin{pmatrix} (R + r\cos v)\cos u \\ (R + r\cos v)\sin u \\ r\sin v \end{pmatrix}$

where $u, v \in [0, 2\pi]$.

The surface area of this torus is:

$A = \iint_D \left| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \right| \,du\,dv = \iint_D (R + r\cos v) \cdot r \,du\,dv = 4\pi^2 Rr$

Complex Function Visualization

For a complex function $f(z) = z^2$, the conformal mapping of a grid in the complex plane:

If $z = x + iy$, then $f(z) = (x^2-y^2) + i(2xy)$

The Jacobian of this transformation is:

$J = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix} = \begin{vmatrix} 2x & -2y \\ 2y & 2x \end{vmatrix} = 4x^2 + 4y^2 = 4|z|^2$

Quantum Mechanics and Advanced Physics

Schrödinger’s Wave Equation (Time-Dependent)

$i\hbar \frac{\partial}{\partial t} \Psi(\vec{r},t) = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\vec{r},t) \right] \Psi(\vec{r},t)$

For a particle in a 1D infinite potential well of width $L$, the energy eigenfunctions are:

$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$

with energy eigenvalues:

$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$

Dirac Equation

The Dirac equation for a relativistic electron:

$i\hbar \frac{\partial}{\partial t}\psi = \left(c\vec{\alpha} \cdot \vec{p} + \beta mc^2\right) \psi$

Where $\vec{\alpha}$ and $\beta$ are the Dirac matrices:

$\alpha_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix}, \quad \beta = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}$

And $\sigma_i$ are the Pauli matrices:

$\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Path Integrals in Quantum Field Theory

Feynman path integral formulation of quantum mechanics:

$\langle x_b, t_b | x_a, t_a \rangle = \int_{x(t_a)=x_a}^{x(t_b)=x_b} \mathcal{D}[x(t)] \, e^{\frac{i}{\hbar}S[x(t)]}$

Where $S[x(t)]$ is the action:

$S[x(t)] = \int_{t_a}^{t_b} L(x(t), \dot{x}(t), t) \, dt$

For a non-relativistic free particle:

$S[x(t)] = \int_{t_a}^{t_b} \frac{1}{2}m\dot{x}^2(t) \, dt$

Einstein Field Equations

$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$

The Schwarzschild solution for a non-rotating spherical mass:

$ds^2 = -\left(1-\frac{2GM}{c^2r}\right)c^2dt^2 + \left(1-\frac{2GM}{c^2r}\right)^{-1}dr^2 + r^2(d\theta^2 + \sin^2\theta \, d\phi^2)$

Fractals and Iterative Systems

Julia Sets

For a complex function $f_c(z) = z^2 + c$, the Julia set $J_c$ is the boundary of the set of points that escape to infinity under iteration:

$J_c = \partial\{z \in \mathbb{C} : \lim_{n\to\infty} |f_c^n(z)| = \infty\}$

The dynamics near fixed points $z_0$ where $f_c(z_0) = z_0$ are determined by the multiplier:

$\lambda = f'_c(z_0)$

• If $|\lambda| < 1$, the fixed point is attracting
• If $|\lambda| > 1$, the fixed point is repelling
• If $|\lambda| = 1$, the fixed point is indifferent

Iterated Function Systems (IFS)

For a set of contractive mappings $\{f_1, f_2, \ldots, f_n\}$ in a metric space, the attractor $A$ satisfies:

$A = \bigcup_{i=1}^n f_i(A)$

The Hausdorff dimension $d$ of a self-similar fractal created by $n$ copies scaled by factor $r$ satisfies:

$n \cdot r^d = 1$

For example, the Sierpinski triangle has $n=3$ and $r=\frac{1}{2}$, so:

$3 \cdot \left(\frac{1}{2}\right)^d = 1 \implies d = \frac{\log 3}{\log 2} \approx 1.585$

The Logistic Map and Chaos

The logistic map is defined by the recurrence relation:

$x_{n+1} = rx_n(1-x_n)$

The orbit diagram shows period doubling bifurcations leading to chaos. The Feigenbaum constant:

$\delta = \lim_{n\to\infty} \frac{r_n - r_{n-1}}{r_{n+1} - r_n} \approx 4.669201609102990671853203821578$

Where $r_n$ is the parameter value at which the period-$2^n$ orbit appears.

Non-Euclidean Geometry

Hyperbolic Geometry

In the Poincaré disk model, the hyperbolic distance between points $z_1$ and $z_2$ is:

$d_H(z_1, z_2) = 2\tanh^{-1}\left|\frac{z_1-z_2}{1-\bar{z_1}z_2}\right|$

The area of a hyperbolic triangle with angles $\alpha, \beta, \gamma$ is:

$A = \pi - (\alpha + \beta + \gamma)$

Riemannian Geometry

The Riemann curvature tensor measures how the geometry deviates from Euclidean space:

$R^{\rho}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}_{\nu\sigma} - \partial_{\nu}\Gamma^{\rho}_{\mu\sigma} + \Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma} - \Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma}$

Where $\Gamma^{\rho}_{\mu\nu}$ are the Christoffel symbols:

$\Gamma^{\rho}_{\mu\nu} = \frac{1}{2}g^{\rho\sigma}\left(\partial_{\mu}g_{\nu\sigma} + \partial_{\nu}g_{\mu\sigma} - \partial_{\sigma}g_{\mu\nu}\right)$

Category Theory

Functors and Natural Transformations

For categories $\mathcal{C}$ and $\mathcal{D}$, a functor $F: \mathcal{C} \rightarrow \mathcal{D}$ maps:

• Objects $X$ in $\mathcal{C}$ to objects $F(X)$ in $\mathcal{D}$
• Morphisms $f: X \rightarrow Y$ in $\mathcal{C}$ to morphisms $F(f): F(X) \rightarrow F(Y)$ in $\mathcal{D}$

Preserving composition: $F(g \circ f) = F(g) \circ F(f)$

A natural transformation $\eta: F \Rightarrow G$ between functors $F, G: \mathcal{C} \rightarrow \mathcal{D}$ consists of a family of morphisms $\eta_X: F(X) \rightarrow G(X)$ such that for any $f: X \rightarrow Y$ in $\mathcal{C}$:

$\eta_Y \circ F(f) = G(f) \circ \eta_X$

This is captured by the commutative diagram:

Yoneda Lemma

For a locally small category $\mathcal{C}$ and an object $A$ in $\mathcal{C}$, the Yoneda embedding:

$y_A = \text{Hom}_{\mathcal{C}}(A, -): \mathcal{C} \rightarrow \text{Set}$

The Yoneda lemma states that for any functor $F: \mathcal{C} \rightarrow \text{Set}$:

$\text{Nat}(y_A, F) \cong F(A)$

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