Advanced Mathematics¶

Topic: LaTeX Lesson: 5 of 16 Prerequisites: Basic Math Typesetting, Packages & Document Classes Objective: Master advanced mathematical typesetting including multi-line equations, matrices, theorem environments, and specialized notation for physics and computer science.

Introduction¶

While basic math mode covers inline equations and simple displays, professional mathematical writing requires sophisticated tools for multi-line derivations, aligned equations, matrices, theorem statements, and domain-specific notation. This lesson explores the powerful amsmath package ecosystem and specialized packages that make LaTeX the gold standard for mathematical typesetting.

The amsmath Package¶

The amsmath package is essential for advanced mathematics. Load it in your preamble:

\usepackage{amsmath}

It provides numerous environments and commands that improve upon LaTeX's basic math capabilities.

Display Math Environments¶

equation and equation*¶

The equation environment creates a numbered display equation:

\begin{equation}
  E = mc^2
\end{equation}

The starred version equation* suppresses numbering:

\begin{equation*}
  \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
\end{equation*}

align and align*¶

The align environment is for multiple equations aligned at specific points (usually = or \leq):

\begin{align}
  x^2 + y^2 &= 1 \\
  x &= \cos\theta \\
  y &= \sin\theta
\end{align}

The & symbol marks the alignment point. Each line gets its own equation number. Use align* to suppress all numbering:

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

gather and gather*¶

The gather environment centers multiple equations without alignment:

\begin{gather}
  a = b + c \\
  x = y + z \\
  p = q \cdot r
\end{gather}

multline and multline*¶

For a single long equation that needs to break across lines:

\begin{multline}
  p(x) = 3x^6 + 14x^5y + 590x^4y^2 + 19x^3y^3 \\
  - 12x^2y^4 - 12xy^5 + 2y^6 - a^3b^3
\end{multline}

The first line is left-aligned, the last is right-aligned, and middle lines are centered.

Equation Numbering Control¶

Custom Tags¶

Override automatic numbering with \tag{}:

\begin{equation}
  E = mc^2 \tag{Einstein}
\end{equation}

Suppressing Individual Numbers¶

In multi-line environments, suppress a specific line's number with \notag:

\begin{align}
  x &= a + b \\
  y &= c + d \notag \\
  z &= e + f
\end{align}

Only the first and third equations are numbered.

Labels and References¶

Label equations for cross-referencing:

\begin{equation}
  \label{eq:pythagorean}
  a^2 + b^2 = c^2
\end{equation}

By the Pythagorean theorem (Equation~\ref{eq:pythagorean}), we have...

The \eqref{} command adds parentheses automatically:

As shown in \eqref{eq:pythagorean}, the relationship holds.

This produces: "As shown in (1), the relationship holds."

Matrices¶

The amsmath package provides several matrix environments:

pmatrix (Parentheses)¶

\[
  A = \begin{pmatrix}
    a_{11} & a_{12} & a_{13} \\
    a_{21} & a_{22} & a_{23} \\
    a_{31} & a_{32} & a_{33}
  \end{pmatrix}
\]

bmatrix (Brackets)¶

\[
  B = \begin{bmatrix}
    1 & 0 & 0 \\
    0 & 1 & 0 \\
    0 & 0 & 1
  \end{bmatrix}
\]

vmatrix and Vmatrix (Determinants)¶

\[
  \det(A) = \begin{vmatrix}
    a & b \\
    c & d
  \end{vmatrix} = ad - bc
\]

Vmatrix uses double vertical bars:

\[
  \|A\| = \begin{Vmatrix}
    1 & 2 \\
    3 & 4
  \end{Vmatrix}
\]

smallmatrix (Inline)¶

For inline matrices, use smallmatrix:

The transformation matrix $\bigl(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\bigr)$ maps...

Note: smallmatrix doesn't add delimiters, so use \bigl( and \bigr) manually.

Matrix Examples¶

\begin{align*}
  \mathbf{A} &= \begin{bmatrix}
    1 & 2 & 3 \\
    4 & 5 & 6 \\
    7 & 8 & 9
  \end{bmatrix} \\
  \mathbf{I}_3 &= \begin{pmatrix}
    1 & 0 & 0 \\
    0 & 1 & 0 \\
    0 & 0 & 1
  \end{pmatrix}
\end{align*}

Piecewise Functions with cases¶

\[
  f(x) = \begin{cases}
    x^2 & \text{if } x \geq 0 \\
    -x^2 & \text{if } x < 0
  \end{cases}
\]

More complex example:

\begin{equation}
  |x| = \begin{cases}
    x & \text{if } x > 0 \\
    0 & \text{if } x = 0 \\
    -x & \text{if } x < 0
  \end{cases}
\end{equation}

Theorem Environments¶

The amsthm Package¶

Load the amsthm package:

\usepackage{amsthm}

Define theorem-like environments:

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

The optional [theorem] argument makes these environments share the same counter.

Using Theorem Environments¶

\begin{theorem}[Pythagorean Theorem]
  \label{thm:pythagoras}
  In a right triangle with legs of length $a$ and $b$ and hypotenuse of length $c$,
  \[
    a^2 + b^2 = c^2
  \]
\end{theorem}

\begin{proof}
  Consider a square of side length $a+b$...

  Thus, we have shown that $a^2 + b^2 = c^2$.
\end{proof}

The proof environment automatically adds "Proof" at the start and a QED symbol (□) at the end.

Custom Theorem Styles¶

Define custom styles for definitions, remarks, etc.:

\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}

\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}

The three built-in styles are: - plain: italicized text (for theorems, lemmas) - definition: upright text (for definitions, examples) - remark: upright text with different spacing (for remarks, notes)

Complete Example¶

\documentclass{article}
\usepackage{amsmath,amsthm}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\begin{document}

\section{Fundamental Concepts}

\begin{definition}[Continuity]
  A function $f: \mathbb{R} \to \mathbb{R}$ is continuous at $x = a$ if
  \[
    \lim_{x \to a} f(x) = f(a)
  \]
\end{definition}

\begin{theorem}[Intermediate Value Theorem]
  If $f$ is continuous on $[a,b]$ and $f(a) < 0 < f(b)$, then there exists
  $c \in (a,b)$ such that $f(c) = 0$.
\end{theorem}

\begin{remark}
  This theorem does not hold for discontinuous functions.
\end{remark}

\end{document}

Customizing the QED Symbol¶

\renewcommand{\qedsymbol}{$\blacksquare$}

Custom Operators¶

Use \DeclareMathOperator for custom operators that should be typeset in roman (upright) font:

\DeclareMathOperator{\argmax}{arg\,max}
\DeclareMathOperator{\argmin}{arg\,min}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\diag}{diag}

Usage:

\[
  \theta^* = \argmax_\theta \mathcal{L}(\theta)
\]

\[
  \tr(AB) = \tr(BA)
\]

For operators with limits (like \max and \min), use the starred version:

\DeclareMathOperator*{\argmax}{arg\,max}

\[
  x^* = \argmax_{x \in \mathbb{R}^n} f(x)
\]

Multiline Equations¶

split Environment¶

Use split inside equation for multi-line derivations with a single number:

\begin{equation}
  \begin{split}
    (a + b)^2 &= (a + b)(a + b) \\
    &= a^2 + ab + ba + b^2 \\
    &= a^2 + 2ab + b^2
  \end{split}
\end{equation}

Aligned Equations with Annotations¶

\begin{align}
  f(x) &= x^2 + 2x + 1 \\
  &= (x + 1)^2 && \text{(completing the square)} \\
  &\geq 0 && \text{(squares are non-negative)}
\end{align}

The && creates a second alignment point for annotations.

Stacked Symbols¶

overset and underset¶

\[
  A \overset{\text{def}}{=} B
\]

\[
  \lim_{n \to \infty} a_n \overset{?}{=} L
\]

\[
  X \underset{\text{i.i.d.}}{\sim} \mathcal{N}(0, 1)
\]

stackrel¶

\[
  f(x) \stackrel{x \to 0}{\longrightarrow} L
\]

Multiple Stacks¶

\[
  A \underset{\text{below}}{\overset{\text{above}}{=}} B
\]

Advanced Examples¶

Complex Integral¶

\begin{equation}
  \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}
\end{equation}

Summation with Conditions¶

\[
  \sum_{\substack{1 \leq i \leq n \\ i \text{ odd}}} i^2
\]

Continued Fraction¶

\[
  x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cdots}}}
\]

Note: Use \cfrac (continued fraction) instead of \frac for better spacing.

System of Equations¶

\[
  \left\{
    \begin{aligned}
      x + y + z &= 6 \\
      2x - y + 3z &= 14 \\
      -x + 3y - 2z &= -8
    \end{aligned}
  \right.
\]

Commutative Diagrams¶

For category theory and algebra, use the tikz-cd package:

\usepackage{tikz-cd}

\begin{tikzcd}
  A \arrow[r, "f"] \arrow[d, "g"] & B \arrow[d, "h"] \\
  C \arrow[r, "k"] & D
\end{tikzcd}

A simple commutative square:

\[
  \begin{tikzcd}
    X \times Y \arrow[r, "\pi_1"] \arrow[d, "\pi_2"] & X \arrow[d, "f"] \\
    Y \arrow[r, "g"] & Z
  \end{tikzcd}
\]

Diagonal arrows:

\[
  \begin{tikzcd}
    A \arrow[r] \arrow[dr] & B \arrow[d] \\
    & C
  \end{tikzcd}
\]

Physics Package¶

The physics package provides shortcuts for quantum mechanics and calculus notation:

\usepackage{physics}

Derivatives¶

% Ordinary derivatives
\dv{x}  % d/dx
\dv{f}{x}  % df/dx
\dv[2]{f}{x}  % d²f/dx²

% Partial derivatives
\pdv{x}  % ∂/∂x
\pdv{f}{x}  % ∂f/∂x
\pdv{f}{x}{y}  % ∂²f/∂x∂y
\pdv[2]{f}{x}  % ∂²f/∂x²

Example:

\begin{equation}
  \pdv{u}{t} = \alpha \pdv[2]{u}{x}
\end{equation}

Quantum Mechanics Notation¶

% Bra-ket notation
\bra{\psi}  % ⟨ψ|
\ket{\phi}  % |φ⟩
\braket{\psi|\phi}  % ⟨ψ|φ⟩
\braket{\psi}  % ⟨ψ|ψ⟩
\ketbra{\psi}{\phi}  % |ψ⟩⟨φ|

% Expectation value
\expval{A}  % ⟨A⟩
\expval{A}{\psi}  % ⟨ψ|A|ψ⟩

Example:

\begin{equation}
  \expval{\hat{H}}{\psi} = \int_{-\infty}^{\infty} \psi^*(x) \hat{H} \psi(x) \, dx
\end{equation}

Vector Notation¶

\vb{v}  % bold vector
\vb*{v}  % arrow vector
\grad  % gradient ∇
\div  % divergence
\curl  % curl
\laplacian  % Laplacian ∇²

Matrix Operations¶

\tr{A}  % trace
\Tr{A}  % trace (capital)
\rank{A}  % rank
\det{A}  % determinant

Complete Advanced Example¶

\documentclass{article}
\usepackage{amsmath,amsthm,amssymb}
\usepackage{physics}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}

\DeclareMathOperator*{\argmin}{arg\,min}

\begin{document}

\section{Optimization Theory}

\begin{definition}[Convex Function]
  A function $f: \mathbb{R}^n \to \mathbb{R}$ is convex if for all $x, y \in \mathbb{R}^n$
  and $\lambda \in [0,1]$,
  \begin{equation}
    f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)
  \end{equation}
\end{definition}

\begin{theorem}[First-Order Condition]
  \label{thm:first-order}
  Let $f: \mathbb{R}^n \to \mathbb{R}$ be differentiable. If $x^*$ is a local minimum, then
  \begin{equation}
    \nabla f(x^*) = \mathbf{0}
  \end{equation}
\end{theorem}

\begin{proof}
  Suppose $\nabla f(x^*) \neq \mathbf{0}$. Then we can find a direction $d$ such that
  \begin{align}
    \nabla f(x^*)^\top d &< 0 \\
    f(x^* + \epsilon d) &< f(x^*) && \text{for sufficiently small } \epsilon > 0
  \end{align}
  This contradicts the assumption that $x^*$ is a local minimum.
\end{proof}

\begin{lemma}[Gradient Descent Update]
  The gradient descent iteration
  \begin{equation}
    x_{k+1} = x_k - \alpha_k \nabla f(x_k)
  \end{equation}
  decreases the objective value when $\alpha_k$ is sufficiently small.
\end{lemma}

Consider the quadratic optimization problem:
\begin{equation}
  \begin{split}
    \min_{x \in \mathbb{R}^n} \quad & \frac{1}{2} x^\top Q x - b^\top x \\
    \text{subject to} \quad & Ax = c
  \end{split}
\end{equation}

The Lagrangian is:
\begin{align}
  \mathcal{L}(x, \lambda) &= \frac{1}{2} x^\top Q x - b^\top x + \lambda^\top (Ax - c)
\end{align}

The optimality conditions are:
\begin{align}
  \nabla_x \mathcal{L} &= Qx - b + A^\top \lambda = 0 \\
  \nabla_\lambda \mathcal{L} &= Ax - c = 0
\end{align}

In matrix form:
\begin{equation}
  \begin{bmatrix}
    Q & A^\top \\
    A & 0
  \end{bmatrix}
  \begin{bmatrix}
    x^* \\
    \lambda^*
  \end{bmatrix}
  =
  \begin{bmatrix}
    b \\
    c
  \end{bmatrix}
\end{equation}

\end{document}

Typography Best Practices¶

Use the right environment: align for aligned equations, gather for centered, multline for single long equations
Don't overuse numbering: Use starred versions (align*, equation*) when references aren't needed
Alignment consistency: Align at relation symbols (=, <, \leq)
Spacing in matrices: LaTeX handles this automatically, don't force spacing
Text in math mode: Use \text{} for annotations
Punctuation: Display equations are part of sentences, include punctuation
Consistent notation: Define custom operators for repeated use

Common Mistakes¶

Using eqnarray: This environment is obsolete, use align instead
Manual spacing: Let LaTeX handle spacing, avoid \,, \! unless necessary
Breaking alignment: Every line in align needs exactly one & before continuing
Forgetting \\: Multi-line environments need \\ to break lines (except the last)
Nested equation environments: Don't put equation inside align

Exercises¶

Exercise 1: Maxwell's Equations¶

Typeset Maxwell's equations in both differential and integral form using the align environment. Add equation numbers and labels.

Exercise 2: Matrix Proof¶

Typeset this theorem and proof:

Theorem: If $A$ and $B$ are $n \times n$ matrices, then $\det(AB) = \det(A)\det(B)$.

Proof: Use the fact that determinants are multiplicative...

Exercise 3: Piecewise Function¶

Create the Heaviside step function:

H(x) = { 0  if x < 0
       { 1  if x ≥ 0

Exercise 4: Custom Theorem¶

Create a document with: - Three theorem styles (theorem, definition, remark) - At least one theorem with proof - Numbered definitions - Cross-references between theorems

Exercise 5: Optimization Problem¶

Typeset the following constrained optimization problem with Lagrangian and KKT conditions:

minimize    f(x)
subject to  g_i(x) ≤ 0, i = 1,...,m
            h_j(x) = 0, j = 1,...,p

Exercise 6: Quantum Mechanics¶

Using the physics package, typeset the time-dependent Schrödinger equation and show that the expectation value of the Hamiltonian is conserved.

Exercise 7: Continued Fractions¶

Typeset the golden ratio as a continued fraction:

φ = 1 + 1/(1 + 1/(1 + 1/(1 + ...)))

Exercise 8: Commutative Diagram¶

Create a commutative diagram showing a pullback or pushout in category theory.

Summary¶

This lesson covered: - amsmath environments: equation, align, gather, multline - Equation numbering with \tag, \notag, \label, \eqref - Matrix environments: pmatrix, bmatrix, vmatrix - Piecewise functions with cases - Theorem environments with amsthm - Custom operators with \DeclareMathOperator - Multiline equation techniques - Stacked symbols and annotations - Commutative diagrams with tikz-cd - Physics notation with the physics package

With these tools, you can typeset virtually any mathematical content at a professional level.

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