222 lines
6.6 KiB
TypeScript
222 lines
6.6 KiB
TypeScript
/* eslint-disable no-irregular-whitespace */
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// Math Formulas Content
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export const MATH_FORMULAS_MD = String.raw`
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# Mathematical Formulas and Expressions
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This document demonstrates various mathematical notation and formulas that can be rendered using LaTeX syntax in markdown.
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## Basic Arithmetic
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### Addition and Summation
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$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$
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## Algebra
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### Quadratic Formula
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The solutions to $ax^2 + bx + c = 0$ are:
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$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
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### Binomial Theorem
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$$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$
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## Calculus
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### Derivatives
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The derivative of $f(x) = x^n$ is:
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$$f'(x) = nx^{n-1}$$
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### Integration
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$$\int_a^b f(x) \, dx = F(b) - F(a)$$
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### Fundamental Theorem of Calculus
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$$\frac{d}{dx} \int_a^x f(t) \, dt = f(x)$$
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## Linear Algebra
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### Matrix Multiplication
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If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then:
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$$C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$$
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### Eigenvalues and Eigenvectors
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For a square matrix $A$, if $Av = \lambda v$ for some non-zero vector $v$, then:
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- $\lambda$ is an eigenvalue
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- $v$ is an eigenvector
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## Statistics and Probability
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### Normal Distribution
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The probability density function is:
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$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$
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### Bayes' Theorem
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$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$
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### Central Limit Theorem
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For large $n$, the sample mean $\bar{X}$ is approximately:
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$$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$$
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## Trigonometry
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### Pythagorean Identity
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$$\sin^2\theta + \cos^2\theta = 1$$
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### Euler's Formula
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$$e^{i\theta} = \cos\theta + i\sin\theta$$
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### Taylor Series for Sine
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$$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
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## Complex Analysis
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### Complex Numbers
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A complex number can be written as:
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$$z = a + bi = r e^{i\theta}$$
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where $r = |z| = \sqrt{a^2 + b^2}$ and $\theta = \arg(z)$
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### Cauchy-Riemann Equations
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For a function $f(z) = u(x,y) + iv(x,y)$ to be analytic:
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$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$
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## Differential Equations
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### First-order Linear ODE
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$$\frac{dy}{dx} + P(x)y = Q(x)$$
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Solution: $y = e^{-\int P(x)dx}\left[\int Q(x)e^{\int P(x)dx}dx + C\right]$
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### Heat Equation
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$$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$$
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## Number Theory
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### Prime Number Theorem
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$$\pi(x) \sim \frac{x}{\ln x}$$
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where $\pi(x)$ is the number of primes less than or equal to $x$.
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### Fermat's Last Theorem
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For $n > 2$, there are no positive integers $a$, $b$, and $c$ such that:
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$$a^n + b^n = c^n$$
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## Set Theory
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### De Morgan's Laws
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$$\overline{A \cup B} = \overline{A} \cap \overline{B}$$
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$$\overline{A \cap B} = \overline{A} \cup \overline{B}$$
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## Advanced Topics
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### Riemann Zeta Function
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$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}$$
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### Maxwell's Equations
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$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$
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$$\nabla \cdot \mathbf{B} = 0$$
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$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$
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$$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}$$
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### Schrödinger Equation
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$$i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)$$
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## Inline Math Examples
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Here are some inline mathematical expressions:
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- The golden ratio: $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$
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- Euler's number: $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$
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- Pi: $\pi = 4 \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$
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- Square root of 2: $\sqrt{2} = 1.41421356...$
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## Fractions and Radicals
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Complex fraction: $\frac{\frac{a}{b} + \frac{c}{d}}{\frac{e}{f} - \frac{g}{h}}$
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Nested radicals: $\sqrt{2 + \sqrt{3 + \sqrt{4 + \sqrt{5}}}}$
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## Summations and Products
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### Geometric Series
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$$\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \quad \text{for } |r| < 1$$
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### Product Notation
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$$n! = \prod_{k=1}^{n} k$$
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### Double Summation
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$$\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij}$$
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## Limits
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$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$
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$$\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x$$
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## Further Bracket Styles and Amounts
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- \( \mathrm{GL}_2(\mathbb{F}_7) \): Group of invertible matrices with entries in \(\mathbb{F}_7\).
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- Some kernel of \(\mathrm{SL}_2(\mathbb{F}_7)\):
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\[
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\left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \right\} = \{\pm I\}
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\]
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- Algebra:
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\[
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x = \frac{-b \pm \sqrt{\,b^{2}-4ac\,}}{2a}
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\]
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- $100 and $12.99 are amounts, not LaTeX.
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- I have $10, $3.99 and $x + y$ and $100x$. The amount is $2,000.
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- Emma buys 2 cupcakes for $3 each and 1 cookie for $1.50. How much money does she spend in total?
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- Maria has $20. She buys a notebook for $4.75 and a pack of pencils for $3.25. How much change does she receive?
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- 1 kg の質量は
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\[
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E = (1\ \text{kg}) \times (3.0 \times 10^8\ \text{m/s})^2 \approx 9.0 \times 10^{16}\ \text{J}
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\]
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というエネルギーに相当します。これは約 21 百万トンの TNT が爆発したときのエネルギーに匹敵します。
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- Algebra: \[
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x = \frac{-b \pm \sqrt{\,b^{2}-4ac\,}}{2a}
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\]
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- Algebraic topology, Homotopy Groups of $\mathbb{S}^3$:
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$$\pi_n(\mathbb{S}^3) = \begin{cases}
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\mathbb{Z} & n = 3 \\
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0 & n > 3, n \neq 4 \\
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\mathbb{Z}_2 & n = 4 \\
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\end{cases}$$
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- Spacer preceded by backslash:
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\[
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\boxed{
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\begin{aligned}
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N_{\text{att}}^{\text{(MHA)}} &=
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h \bigl[\, d_{\text{model}}\;d_{k} + d_{\text{model}}\;d_{v}\, \bigr] && (\text{Q,K,V の重み})\\
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&\quad+ h(d_{k}+d_{k}+d_{v}) && (\text{バイアス Q,K,V）}\\[4pt]
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&\quad+ (h d_{v})\, d_{\text{model}} && (\text{出力射影 }W^{O})\\
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&\quad+ d_{\text{model}} && (\text{バイアス }b^{O})
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\end{aligned}}
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\]
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## Formulas in a Table
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| Area | Expression | Comment |
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|------|------------|---------|
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| **Algebra** | \[
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x = \frac{-b \pm \sqrt{\,b^{2}-4ac\,}}{2a}
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\] | Quadratic formula |
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| | \[
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(a+b)^{n} = \sum_{k=0}^{n}\binom{n}{k}\,a^{\,n-k}\,b^{\,k}
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\] | Binomial theorem |
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| | \(\displaystyle \prod_{k=1}^{n}k = n! \) | Factorial definition |
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| **Geometry** | \( \mathbf{a}\cdot \mathbf{b} = \|\mathbf{a}\|\,\|\mathbf{b}\|\,\cos\theta \) | Dot product & angle |
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## No math (but chemical)
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Balanced chemical reaction with states:
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\[
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\ce{2H2(g) + O2(g) -> 2H2O(l)}
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\]
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The standard enthalpy change for the reaction is: $\Delta H^\circ = \pu{-572 kJ mol^{-1}}$.
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---
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*This document showcases various mathematical notation and formulas that can be rendered in markdown using LaTeX syntax.*
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`;
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