Mathematical Notation & CriticMarkup Demo

This page demonstrates mathematical notation with LaTeX and collaborative editing with CriticMarkup.

Mathematical Notation Examples

Basic Mathematics

Arithmetic and Algebra

Simple equations: - Addition: \(2 + 2 = 4\) - Multiplication: \(3 \times 4 = 12\) - Division: \(\frac{15}{3} = 5\) - Powers: \(2^8 = 256\)

Quadratic equation: \(ax^2 + bx + c = 0\)

Solution: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Trigonometry

Basic identities:

\[ \sin^2\theta + \cos^2\theta = 1 \]

\[ \tan\theta = \frac{\sin\theta}{\cos\theta} \]

Euler's formula: [ e^{itheta} = costheta + isintheta ]

Calculus

Derivatives

Definition of derivative: [ f'(x) = lim_{h to 0} frac{f(x + h) - f(x)}{h} ]

Common derivatives: - \(\frac{d}{dx}(x^n) = nx^{n-1}\) - \(\frac{d}{dx}(\sin x) = \cos x\) - \(\frac{d}{dx}(e^x) = e^x\) - \(\frac{d}{dx}(\ln x) = \frac{1}{x}\)

Integrals

Fundamental theorem of calculus: [ int_a^b f'(x),dx = f(b) - f(a) ]

Common integrals: [ begin{align} int x^n,dx &= frac{x^{n+1}}{n+1} + C quad (n neq -1) \ int frac{1}{x},dx &= ln|x| + C \ int e^x,dx &= e^x + C \ int sin x,dx &= -cos x + C end{align} ]

Linear Algebra

Matrices

Matrix multiplication: [ begin{bmatrix} a & b \ c & d end{bmatrix} begin{bmatrix} e & f \ g & h end{bmatrix} = begin{bmatrix} ae + bg & af + bh \ ce + dg & cf + dh end{bmatrix} ]

Determinant: [ det(A) = begin{vmatrix} a & b & c \ d & e & f \ g & h & i end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) ]

Eigenvalues

Characteristic equation: [ det(A - lambda I) = 0 ]

For a 2×2 matrix: [ lambda = frac{text{tr}(A) pm sqrt{text{tr}(A)^2 - 4det(A)}}{2} ]

Statistics

Probability Distributions

Normal distribution: [ f(x) = frac{1}{sigmasqrt{2pi}} expleft(-frac{(x-mu)^2}{2sigma2}right) ]

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

Statistical Measures

Sample mean: \(\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i\)

Sample variance: \(s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2\)

Correlation coefficient: [ r = frac{sum_{i=1}^n (x_i - bar{x})(y_i - bar{y})}{sqrt{sum_{i=1}^n (x_i - bar{x})^{2}sqrt{sum_{i=1}}n (y_i - bar{y})^2}} ]

CriticMarkup Examples

Document Revision

Mathematical Content with Edits

The quadratic formula was discoveredhas been known since ancient times and is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula only applies when \(a \neq 0\).Important constraint to mention

Updating Equations

Original equation: [ f(x) = x^2 + 2x + 1 ]

Updated equation: [ f(x) = (x + 1)^2 ]Factored form is clearer

Collaborative Math Review

Theorem Statement

Pythagorean TheoremNeeds formal proof

In a right triangle with legs of length \(a\) and \(b\), and hypotenuse of length \(c\):

\[ a^2 + b^2 = c^2c^2 = a^2 + b^2Standard form has c² on the left \]

Proof Corrections

The proof follows from the law of cosines.

Proof: Consider a right triangle with legs \(a\) and \(b\). Construct squares on each side. The area relationships give us:

\[ \text{Area of square on hypotenuse} = \text{Sum of areas on legs} \]

Therefore \(c^2 = a^2 + b^2\). \(\square\)

Statistical Analysis Updates

Data Summary

Our analysis showsdemonstrates the following results:

• Sample size: n = 50n = 75Updated after additional data collection
• Mean: \(\bar{x} = 12.313.7\)
• Standard deviation: (sigma = 2.12.4
• 95% Confidence interval: \([12.9, 14.5]\)

Statistical significancep < 0.05 using t-test

Physics Formula Corrections

Newton's Second Law

The force on an object is equal toproportional to its acceleration:

[ F = a ]

[ F = ma ]Don't forget mass!

where: - \(F\) = force (in Newtons) - \(m\) = mass (in kilograms) - \(a\) = acceleration (in m/s²)

Combined Examples

Calculus Textbook Revision

Chapter 3Section 3.1: Derivatives

The derivative of \(f(x) = x^n\) is:

[ f'(x) = nx^{n-1} quad text{for all } n ]

[ f'(x) = nx^{n-1} quad text{for } n neq 0 ]Special case needed for n=0

Important cases: - When \(n = 0\): \(f(x) = 1\), so \(f'(x) = 0\) - When \(n = 1\): \(f(x) = x\), so \(f'(x) = 1\) - When \(n = -1\): \(f(x) = \frac{1}{x}\), so \(f'(x) = -\frac{1}{x^2}\)

Research Paper Edits

Abstract

We studiedThis paper investigates the convergence of the series:

\[ \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6\frac{\pi^2}{6}Use fraction notation \]

Our results show that the series converges.Using the Weierstrass M-test, we prove uniform convergence on compact subsets.

Key findings:Expand in results section 1. Convergence rate: \(O(1/n^2)\) 2. Error bound: \(|S - S_n| < \frac{1}{n}\) 3. Applications in physicsApplications in quantum mechanics and statistical mechanics

Algorithm Documentation

Fast Fourier Transform

The DiscreteFast++} Fourier Transform {++efficiently++} computes:

\[ X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-i2\pi kn/N} \]

{--Time complexity: \(O(N^2)\)--} {++Time complexity: \(O(N \log N)\) using Cooley-Tukey algorithm++}{>>Major improvement!<<}

{Implementation notes:} - {++Requires \(N\) to be a power of 2++} - {~~Use complex arithmetic~>Can use real arithmetic for real inputs - Memory requirement: \(O(N)\)

Advanced Integration

Scientific Paper Review

Methodology Section

We usedemployed the following novel approach:

1. Data collection: 100 samples150 samplesIncreased for better statistics
2. Analysis method: Linear regressionMultiple regression with interaction terms
3. Validation: k-fold cross-validation with k=10

The model equation isbecomes:

[ y = beta_0 + beta_1 x + epsilon ]

[ y = beta_0 + beta_1 x_1 + beta_2 x_2 + beta_3 x_1 x_2 + epsilon ]

where the interaction term \(\beta_3 x_1 x_2\) captures non-linear effectsKey innovation.

Homework Solutions

Problem 1

Find the integral: \(\int x^2 e^x dx\)

Solution: Using substitutionUsing integration by partsBetter method for this type

Let \(u = x^2\) and \(dv = e^x dx\)

Then \(du = 2x dx\) and \(v = e^x\)

[ int x^2 e^x dx = x^2 e^x - int 2x e^x dx ]

[ begin{align} int x^2 e^x dx &= x^2 e^x - int 2x e^x dx \ &= x^2 e^x - 2(xe^x - e^x) + C \ &= e^x(x2 - 2x + 2) + C end{align} ]Complete solution shown

Best Practices Demo

Clear Mathematical Writing

Rule 1: Define all variables before using them.

The equation \(E = mc^2\) shows energy-mass equivalence.

In Einstein's mass-energy equation \(E = mc^2\): - \(E\) represents energy (in joules) - \(m\) represents mass (in kilograms)
- \(c\) represents the speed of light (≈ \(3 \times 10^8\) m/s)

Collaborative Formula Development

Original proposal: [ text{Efficiency} = frac{text{Output}}{text{Input}} ]

Refined formula: [ eta = frac{W_{text{out}}}{Q_{text{in}}} times 100% ]

Where: - \(\eta\) = efficiency (percentage) - \(W_{\text{out}}\) = useful work output (J) - \(Q_{\text{in}}\) = total energy input (J)

Industry standard notation

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Note: This page demonstrates the integration of mathematical notation with collaborative editing features. Both systems work seamlessly togetherPerfect for scientific and technical documentation in Material for MkDocs.