LaTeX/MathJax Demo Notebook¶

This Jupyter notebook demonstrates mathematical rendering with our enhanced MathJax configuration.

Basic Mathematical Expressions¶

Inline math examples:

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
• Circle area: $A = \pi r^2$
• Einstein equation: $E = mc^2$

Display Equations¶

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

$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

In [ ]:

import matplotlib.pyplot as plt
import numpy as np
from IPython.display import Latex, Math, display

# Test Python mathematical computation
x = np.linspace(-5, 5, 100)
y = x**2

plt.figure(figsize=(8, 6))
plt.plot(x, y, "b-", linewidth=2, label=r"$f(x) = x^2$")
plt.xlabel(r"$x$")
plt.ylabel(r"$f(x)$")
plt.title(r"Graph of $f(x) = x^2$")
plt.grid(True, alpha=0.3)
plt.legend()
plt.show()

print(f"The derivative of f(x) = x² is f'(x) = 2x")

import matplotlib.pyplot as plt import numpy as np from IPython.display import Latex, Math, display # Test Python mathematical computation x = np.linspace(-5, 5, 100) y = x**2 plt.figure(figsize=(8, 6)) plt.plot(x, y, "b-", linewidth=2, label=r"$f(x) = x^2$") plt.xlabel(r"$x$") plt.ylabel(r"$f(x)$") plt.title(r"Graph of $f(x) = x^2$") plt.grid(True, alpha=0.3) plt.legend() plt.show() print(f"The derivative of f(x) = x² is f'(x) = 2x")

Using Custom Macros¶

Our MathJax configuration includes custom macros:

Mathematical Sets¶

• Real numbers: $\R$
• Complex numbers: $\C$
• Natural numbers: $\N$
• Integers: $\Z$
• Rational numbers: $\Q$

Vector and Matrix Notation¶

Vector: $\vec{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$

Matrix: $\mat{A} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

In [ ]:

# Display LaTeX using IPython's display capabilities
display(
    Latex(
        r"\begin{align}"
        r"\vec{v} \cdot \vec{w} &= v_x w_x + v_y w_y + v_z w_z \\"
        r"&= |\vec{v}| |\vec{w}| \cos \theta"
        r"\end{align}"
    )
)

print("Vector dot product formula displayed above")

# Display LaTeX using IPython's display capabilities display( Latex( r"\begin{align}" r"\vec{v} \cdot \vec{w} &= v_x w_x + v_y w_y + v_z w_z \\" r"&= |\vec{v}| |\vec{w}| \cos \theta" r"\end{align}" ) ) print("Vector dot product formula displayed above")

Machine Learning Mathematics¶

Loss Function¶

Mean squared error: $\loss(\params) = \frac{1}{n} \sum_{i=1}^n (y_i - \hat{y}_i)^2$

Gradient Descent¶

Parameter update: $\params_{t+1} = \params_t - \alpha \nabla_{\params} \loss(\params_t)$

In [ ]:

# Demonstrate gradient descent


def f(x):
    return x**2 + 2 * x + 1


def df_dx(x):
    return 2 * x + 2


# Gradient descent parameters
x = 5.0  # starting point
alpha = 0.1  # learning rate
iterations = 20

x_history = [x]
for i in range(iterations):
    grad = df_dx(x)
    x = x - alpha * grad
    x_history.append(x)

print(f"Starting point: {x_history[0]:.3f}")
print(f"Final point: {x_history[-1]:.3f}")
print(f"Minimum of f(x) = x² + 2x + 1 is at x = -1")

# Demonstrate gradient descent def f(x): return x**2 + 2 * x + 1 def df_dx(x): return 2 * x + 2 # Gradient descent parameters x = 5.0 # starting point alpha = 0.1 # learning rate iterations = 20 x_history = [x] for i in range(iterations): grad = df_dx(x) x = x - alpha * grad x_history.append(x) print(f"Starting point: {x_history[0]:.3f}") print(f"Final point: {x_history[-1]:.3f}") print(f"Minimum of f(x) = x² + 2x + 1 is at x = -1")

Probability and Statistics¶

Normal Distribution¶

$$f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

Bayes' Theorem¶

$$\Pr(A|B) = \frac{\Pr(B|A) \Pr(A)}{\Pr(B)}$$

In [ ]:

# Plot normal distribution
from scipy import stats

x = np.linspace(-4, 4, 100)
y1 = stats.norm.pdf(x, 0, 1)  # μ=0, σ=1
y2 = stats.norm.pdf(x, 0, 0.5)  # μ=0, σ=0.5
y3 = stats.norm.pdf(x, 1, 1)  # μ=1, σ=1

plt.figure(figsize=(10, 6))
plt.plot(x, y1, "b-", label=r"$\mu=0, \sigma=1$")
plt.plot(x, y2, "r-", label=r"$\mu=0, \sigma=0.5$")
plt.plot(x, y3, "g-", label=r"$\mu=1, \sigma=1$")
plt.xlabel(r"$x$")
plt.ylabel(r"$f(x)$")
plt.title(
    r"Normal Distribution: $f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$"
)
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

# Plot normal distribution from scipy import stats x = np.linspace(-4, 4, 100) y1 = stats.norm.pdf(x, 0, 1) # μ=0, σ=1 y2 = stats.norm.pdf(x, 0, 0.5) # μ=0, σ=0.5 y3 = stats.norm.pdf(x, 1, 1) # μ=1, σ=1 plt.figure(figsize=(10, 6)) plt.plot(x, y1, "b-", label=r"$\mu=0, \sigma=1$") plt.plot(x, y2, "r-", label=r"$\mu=0, \sigma=0.5$") plt.plot(x, y3, "g-", label=r"$\mu=1, \sigma=1$") plt.xlabel(r"$x$") plt.ylabel(r"$f(x)$") plt.title( r"Normal Distribution: $f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$" ) plt.legend() plt.grid(True, alpha=0.3) plt.show()

Advanced Mathematics¶

Fourier Transform¶

$$\mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$$

Maxwell's Equations¶

\begin{align} \nabla \times \vec{E} &= -\frac{\partial \vec{B}}{\partial t} \\ \nabla \times \vec{B} &= \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} \\ \nabla \cdot \vec{E} &= \frac{\rho}{\epsilon_0} \\ \nabla \cdot \vec{B} &= 0 \end{align}

Chemical Equations¶

Using the mhchem extension:

$$\ce{H2O + NaCl -> Na+ + Cl- + H2O}$$

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

$$\ce{CaCO3 + 2HCl -> CaCl2 + H2O + CO2}$$

Testing Output Formats¶

This notebook tests various MathJax features:

1. ✅ Inline math with $ delimiters
2. ✅ Display math with $$ delimiters
3. ✅ LaTeX environments (align, equation)
4. ✅ Custom macros for sets, vectors, operators
5. ✅ Chemical equations with mhchem
6. ✅ Complex mathematical expressions
7. ✅ Integration with Python code and matplotlib

When this notebook is rendered in MkDocs, all mathematical expressions should display correctly with proper formatting and typography.

In [ ]:

# Final test: Display some mathematical constants
import math

print("Mathematical Constants:")
print(f"π (pi) = {math.pi:.10f}")
print(f"e (Euler's number) = {math.e:.10f}")
print(f"φ (Golden ratio) = {(1 + math.sqrt(5))/2:.10f}")

# Display as LaTeX
display(
    Latex(
        r"""\begin{align}
\pi &\approx 3.14159 \\
e &\approx 2.71828 \\
\phi &= \frac{1 + \sqrt{5}}{2} \approx 1.61803
\end{align}"""
    )
)

# Final test: Display some mathematical constants import math print("Mathematical Constants:") print(f"π (pi) = {math.pi:.10f}") print(f"e (Euler's number) = {math.e:.10f}") print(f"φ (Golden ratio) = {(1 + math.sqrt(5))/2:.10f}") # Display as LaTeX display( Latex( r"""\begin{align} \pi &\approx 3.14159 \\ e &\approx 2.71828 \\ \phi &= \frac{1 + \sqrt{5}}{2} \approx 1.61803 \end{align}""" ) )