# Markdown Formatting Guide

## Headers
### Basic Header Syntax
# An H1 markdown
--------------
--------------

# This is an H1, too

## An H2 markdown
--------------

## This is an H2, too
### H3
###### H6, you get the idea...

## Text Formatting
### Basic Text Styles
*italic*, **bold**, ~~Strikethrough~~
### Blockquotes
> The answer to everything…Life, the Universe, and Everything...is...Forty-two.
## Lists
### Unordered Lists
List of animals:
* Dog
* Cat
* Lion
* Elephant
### Ordered Lists
Ordered lists:
1. Cow
2. Tiger
3. Bird
4. Ant

# Special Elements
### Keyboard Input
Press <kbd>Winkbd> + <kbd>Skbd> (Start menu)
### Links
[Google](https://www.google.com/)
## Escaping Techniques
**Preventing Automatic Numbering**. \
Using a Backslash: \
1\. Autonomous Nature. \
Using a Non-Breaking Space: \
1 . Autonomous Nature.

# Emoji Support
## Common Emoji Categories
I :heart: Math!
Think about it: :thinking: :bulb:
- Basic operations: :heavy_plus_sign: :heavy_minus_sign: :heavy_multiplication_x::heavy_division_sign:
- Symbols: :infinity: :pi:
- Geometry: :triangular_ruler: :black_square_button: :red_circle:
- Tools: :abacus: :straight_ruler:
- Academic: :mortar_board: :books:
- Thinking/Problem-solving: :thinking: :bulb:
- Miscellaneous: :atom_symbol: :bar_chart: :sunglasses: :eight_pointed_black_star: :cry: :wink: :laughing: :yum:

# Code Blocks
## Python Example
Ensure that there are three backticks (```) before (``` language, e.g., ``` python) and after the code block to properly format it in Markdown.
def my_function():
   print("Hello from a function")

my_function()
# Tables
| Basic Syntax | Description                             |
| ------------ | --------------------------------------- |
| Heading      | # H1, ## H2, ### H3                     |
| Format       | **bold**, *italic*, ~~strike~~          |
| Blockquote   | > blockquote                            |
| List         | - unordered, 1. ordered                 |
| Code         | \`inline\` or \`\`\`block\`\`\`         |
| Link         | [text](url)                             |
| Image        | ![alt](/maths/images/PeterCuclillasFindPotentialFunction.jpg)                         |
| Table        | pipes \| and hyphens ---                |

def my_function():
    print("Hello from a function")

# Mathematical Notation Guide

## Basic Symbols
### Greek Letters
$\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \iota, \kappa, \lambda, \mu, \nu, \xi, \pi, \rho, \sigma, \tau, \upsilon, \phi, \chi, \psi, \omega$

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

### Operators
- Arithmetic: $\times \div \pm \mp \cdot$
- Comparison: $\lt \gt \le \ge \neq \not\geq \approx$
- Set Operations: $\cup \cap \setminus \subset \subseteq \subsetneq \supset \in \notin$
- $\circ,\cdot, \ast, \times, \oplus, \ominus, \otimes, \odot, \bullet, \cup, \cap, \vee, \wedge$
- Special Symbols: $\forall \exists \infty \emptyset \varnothing$

## Functions & Notation
### Roots and Limits
- Square root: $\sqrt{2}$
- n-th root: $\sqrt[n]{2}$
- Limits: $\lim_{x \to 0} f(x)$, $\lim_{\Delta x \to 0} \frac{f(x_0+\Delta x) -f(x_0)}{\Delta x}$

### Piecewise Functions
$f(x) =
\begin{cases}
  x + 1, &x > 0  \\\\
  -2x + 2, &x < 0
\end{cases}$

### Special Functions
* Fraction: $\frac{2}{3}$
* Binomial Coefficient: ${n+1 \choose 2k}$
* Vector Symbol: $\hat{\mathbf{T}}$
* Tilde Notation: $\tilde{x}$
* Blackboard Font: $\mathbb{R}$ (real numbers)
* Periodical Numbers: $0.\widehat{3}$ (repeating decimal)

## Calculus
### Sums and Integrals
* Summation: $\sum_{i=0}^\infty i$
* Integral: $\int x^{n} dx$
* Definite Integral: $\int_{a}^{b} f(x)dx = F(b)-F(a) = F(x) \bigg|_{a}^{b}$
* Contour Integral: $\oint_C \vec{F} \cdot d\vec{r}$
### Derivatives
* Derivative Operator: $\frac{d}{dx}$
* Evaluation: $e^{u}|_{u=0}$
* Gradient: $\nabla w = \langle 2x, 2y \rangle$
### Other
* LaTeX/Boxes: $\boxed{f(z) = C + e^{y + ix}}$
* The proof is left as an exercise to the reader: $\blacksquare$ or QED (Quod Erat Demonstrandum); Proof Completion: $\square$; contradiction: ⊥, $\bot$.
* Multi-Line Math: $$
\begin{aligned}
\int_{2\pi}^{4\pi} f(t)\,dt
  &=\int_{2\pi}^{4\pi}t\cos t\,dt + i\int_{2\pi}^{4\pi} t\sin t\,dt \\[2pt]
  &=t\sin t\Big|_{2\pi}^{4\pi}-\int_{2\pi}^{4\pi}\sin t\,dt
     +i\Bigl[-t\cos t\Big|_{2\pi}^{4\pi}-\int_{2\pi}^{4\pi}1·(-cos t)dt\Bigr]\\[2pt]
  &=0+\cos t\Big|_{2\pi}^{4\pi}+i\bigl[-4\pi+2\pi+\sin t\big|_{2\pi}^{4\pi}\bigr]\\[2pt]
  &=0+i(-2\pi)=-2\pi i.
\end{aligned}
$$

## Linear Algebra
### Matrices and Determinants
$(\begin{smallmatrix}a & 0\\\ 0 & b\end{smallmatrix})$
$\Bigl \vert\begin{smallmatrix}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}}\\\ -1 & 1 & 0\\\ -1 & 0 & 1\end{smallmatrix}\Bigr \vert$

## Linear Systems
### General form
$\begin{cases}
a_{11}x_1 +a_{12}x_2 + \cdots + a_{1n}x_n = b_1 \\\\
a_{21}x_1 +a_{22}x_2 + \cdots + a_{2n}x_n = b_2 \\\\
\vdots \\\\
a_{n1}x_1 +a_{n2}x_2 + \cdots + a_{nn}x_n = b_n
\end{cases}$
### Transformation example
$T(x)=0 \leftrightarrow \left( \begin{smallmatrix}x_1 + 2x_2\\\ x_3-3x_2\end{smallmatrix} \right) = \left( \begin{smallmatrix}0\\\ 0\end{smallmatrix} \right)$
$\begin{cases} x_1 + 2x_2 = 0 \\\ x_3 -3x_2 = 0 \end{cases}$

## Advanced Notation
### Sequences and Sets
* Sequence: $\\{a_n\\}_{n=1}^\infty$
* Intersection: $\bigcap^{\infty}_{i=1} {I_i} \neq \emptyset$
### Text Embedding
$\text{In a different notation}$

### Laplace Transforms
$\mathcal{L}(af+bg) = a\mathcal{L}(f)+b\mathcal{L}(g), \quad \mathcal{L}^{-1}(F(s)) = f(t)u(t)$

### Spacing Control
* Thick space: $;$
* Medium space: $:$
* Thin space: $,$
* Negative space: $!$