Equations

Overview

Teaching: 20 min
Exercises: 0 min

Questions

• How can we create equations in LaTeX?

• How can we make reference to these equations elsewhere in the document?

Objectives

• Learn to create both inline and display equations

• Understand when each should be used

• Learn how to use packages to extend LaTeX’s functionality

LaTeX’s equation tools are arguably the greatest reason for LaTeX’s popularity, and are widely imitated. They are implemented directly in MathJax, a tool used for typesetting equations on the web, and in that guise are used on thousands of websites, including large ones like Wikipedia. Many of the commands are also implemented in Microsoft Word’s equation tools as AutoCorrects. There are no other equation tools anywhere near as widely supported and used.

LaTeX divides equations into two categories:

• Inline equations: These appear within the flow of a line of text, and are squashed vertically to fit while causing minimal disruption to the surrounding paragraph. These are appropriate when inserting a short formula or equation, and when referring to individual mathematical symbols
• Display equations: These are given their own line, and so can take up more vertical space. They can have numbers. These are best suited to longer formulae and all but the shortest equations.

Inline equations

Let’s open a new blank article to use as a scratch area. Inline equations are placed in between $ signs; everything between two of these is seen to be in “math mode”, where a set of commands not previous accessible become usable. Adjust the new article so that it contains the following:

\documentclass[]{article}

%opening
\title{}
\author{}

\begin{document}

A mass $m$

\end{document}

Now build and view it. Notice that the $m$ appears in italic font; this is because LaTeX knows it is a mathematical symbol. Let’s build this up to instead have m =1+2. Build and view this, and you can see that LaTeX has once again ignored the spacing we placed in the source code, and chosen the most appropriate spacing for the equation entered.

Going back to the main document, look for the % FIX THIS comment, and adjust the lines so that the symbols referenced are in equation environments as we need them to be.

Greek, and other symbols

Greek letters we access with a \ and then the name of the letter. Both uppercase and lowercase letters are available, with the exception of those that look exactly like Latin letters. So \pi gives us a lowercase $\pi$, while \Pi gives us a capital $\Pi$. Similarly, $\infty$ is \infty, and $\hbar$ is \hbar.

Operators are also available like this; e.g. \pm$ is $$\pm$$, $$\ne$$ is \ne, and $$\Rightarrow$$ is \Rightarrow`. For a complete gallery, see this cheat sheet.

Some examples:

• $C = 2 \pi r$
• $c = f \lambda$
• $F = \mu R$

Superscripts and subscripts

As you might expect, superscripts and subscripts are added using ^ and _ respectively. However, these only apply to the next character; if you need more than one character to be superscripted (or subscripted), then you need to surround the group with curly braces. These can be nested, so you can have constructions like e^{x^2 + y^2}. You can attach both a superscript and a subscript to the same object, but not more than one of each.

Some examples:

• $ds^2 = dx^2 + dy^2 + dz^2 - dt^2$
• $r_i^2 = x_i^2 + y_i^2$
• $T_{\mu \nu} = T^{\alpha \beta} g_{\alpha\mu} g_{\beta \nu}$

Let’s try this now; find the % inline equation comment, and correct the equation to display properly.

Standard functions

If we try to use a standard function, e.g. sin, we see that it looks like we are trying to multiply the individual letters $s$,, $i$, and $n$. Since this isn’t what we want, we precede the sin with a \ to instead make it a command. This makes it display in the upright formatting that we expect.

Some examples:

• $\sin^2 \alpha + \cos^2 \alpha = 1$
• $\ln e = \log_10 10 = \exp(0) = 1$
• $\det A = 0$

Fractions and surds

We also have access to fractions, \frac{}{}, where the first set of curly braces contain the numerator, and the second the denominator. We also have \sqrt{}, which gives a square root above the contents of the curly braces.

Some examples:

• $\sec \omega = \sqrt{1 - \tan^2 \omega}$
• $\frac{\sin \beta}{\cos \beta} = \tan \beta$
• $e^w = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}$

$N$-ary operators

\sum, \int, and \prod should be used for summations, integrals, and products respectively. Add limits using superscripts and subscripts. These look like symbols for now; in the next section we will see how they differ. Limits work in the same way; use \lim with a subscript.

Some examples:

• $\sigma^2 = \sum_i \frac{(x_i - X)^2}{\varepsilon^2}$
• $V = \int_0^{2\pi} \int_0^\pi \int_0^r r^2 \sin \theta\ dr\ d\theta\ dphi$
• $\frac{d}{dx}f(x) = \lim_{\delta x \rightarow 0} \frac{f(x + \delta x) - f(x)}{\delta x}$

Upright text

SI units are generally displayed in upright text, to distinguish them from mathematical symbols. For example, while $\mathrm{nm}$ is nanometres, $nm$ could be the product of the two numbers $n$ and $m$, perhaps a number and a mass. Similarly, if you want to label a variable with a piece of text (e.g. radial and tangential components of velocity $v_{\mathrm{rad}}$ and $v_{\mathrm{tan}}$), these would want to be upright to show that they are text. If we need to get upright text in a math environment, we can use \mathrm to get it.

Display equations

We create numbered display equations using \begin{equation}...\end{equation}. Unnumbered display equations can be created using equation* instead of equation. You can choose to number every display equation, or only those you refer to elsewhere in the document; the important thing is to be consistent about this. It is somewhat more common to number every display equation; this makes life easier for anyone else referencing your report.

Make sure not to leave blank lines before and after the display equation, since it forms part of the flow of the paragraph. This is especially important where you have a comma before it, or a lowercase letter (e.g. “where”) after it.

The syntax for display equations is otherwise identical to inline equations. The differences appear in the output; fractions are no longer squashed to fit on the line, but rather the numerator and denominator each take up as much space as numbers not in a fraction. Similarly, integrals, sums, and products stand taller, and their limits shift to a more pleasing location.

Brackets

When we put brackets around a fraction, the result isn’t particularly pleasing—the fraction is significantly taller than the bracket that should enclose it. To fix this, add a \left before the opening bracket, and a \right before the closing bracket. Now these will grow as much as necessary to fit their contents.

These don’t have to be matched—you can open a round bracket and close a square one. If you want to have an opening but no closing bracket (or vice-versa), use \right. (or \left.).

In addition to round and square brackets, you can use | for magnitude, \{ and \} for curly braces, and \langle and \rangle for angle brackets (e.g. expectation values).

Let’s add a display equation to the document now. Place the following equation in place of the comment % ADD DISPLAY EQUATION HERE:

\begin{equation}
    \frac{\sum_i p_i O(C_i)}{\sum_i p_i} = \langle O \rangle = \lim_{N\rightarrow \infty} \frac{1}{N} \sum_{i=1}^{N} O(C_j)
\end{equation}

Multi-line equations

When an equation has multiple steps that need to be on separate lines, we use align instead of equation. This lets us use line breaks, \\, to start new lines within the equation. Then, we can use & to indicate the alignment point. Typically we will put & immediately before the = sign (or the inequality sign in the case of an inequality). If more than one & is present on a line, then LaTeX creates two or more columns of equations.

Let’s try and make this equation fit into the document nicely:

\begin{equation}
\langle O \rangle = \frac{\int O(x) e^{-\beta ' H(x)}dx}{e^{-\beta' H(x)} dx} = \frac{\int O(x) e^{-(\beta ' - \beta)H(x)} e^{-\beta H(x)}dx}{\int e^{-(\beta'-\beta)H(x)}e^{-\beta H(x)}dx} = \frac{\left\langle O(x)e^{-(\beta'-\beta)H(x)}\right\rangle}{\left\langle e^{-(\beta'-\beta)H(x)}\right\rangle}
\end{equation}

Cross-referencing equations

We normally place cross-references to equation numbers in round brackets. Doing this by hand can get tedious, so LaTeX gives us a separate command, \eqref, to do this. However, this is a command invented by the American Mathematical Society (AMS), and isn’t a built-in part of LaTeX. As such we need to import one of AMS’s packages to define the command for this.

We’re going to use the amsmath package, which we do using the \usepackage command. This is done in the preamble. Just like cross-referencing chapters, we place a \label in the equation, and then use \eqref when we want to use the number.

Practice

The only way to get better at writing equations in LaTeX is with practice. Get the following equations working in a sample document.

1. $$r = \sqrt{x^2 + y^2}$$
2. $$\partial_x = \frac{\partial}{\partial x}$$

   (Hint: \partial)

3. $$\frac{\delta (xy)}{xy} = \sqrt{\left(\frac{\delta x}{x}\right)^2 + \left( \frac{\delta y}{y}\right)^2}$$
4. $$r_n = \left(\sum_i x_i^2\right)^{\frac{1}{2}}$$
5. $$\mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}$$

   (Hint: \mathbf, \hat)

6. $$a_\mathrm{radial} = \frac{v_\mathrm{tangential}^2}{r}$$
7. $$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$
8. $$\begin{align} \nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf{B} &= 0 \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} & \nabla \times \mathbf{B} &= \mu_0 \left( \mathbf{B} + \varepsilon_0 \frac{\partial \mathbf E}{\partial t}\right) \end{align}$$

   (Hint: \cdot, \varepsilon)

Answers

1. $r = \sqrt{x^2 + y^2}$
2. $\partial_x = \frac{\partial}{\partial x}$
3. $\frac{\delta (xy)}{xy} = \sqrt{\left(\frac{\delta x}{x}\right)^2 + \left( \frac{\delta y}{y}\right)^2}$
4. $r_n = \left(\sum_i x_i^2\right)^{\frac{1}{2}}$
5. $\mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}$
6. $a_\mathrm{radial} = \frac{v_\mathrm{tangential}^2}{r}$
7. $R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$

\begin{align}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} &
\nabla \cdot \mathbf{B} &= 0 \\
\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} &
\nabla \times \mathbf{B} &= \mu_0 \left( \mathbf{B} + \varepsilon_0 \frac{\partial \mathbf E}{\partial t}\right)
\end{align}

Sequential indices in General Relativity

In General Relativity (and possibly some other fields), indices must be kept in order; you can’t have indices one above the other like LaTeX does. For example, $\Gamma^c_{ab}$ ($\Gamma^c_{ab}$) isn’t valid.

How can you make these display properly; i.e. as $\Gamma^c{}_{ab}$?

Answer

The trick is to add an empty {} to hang the successive indices off; i.e.

$\Gamma^c{}_{ab}$

(If you ever have very tall objects so this fails, then you’ll need to look up \vphantom.)

Key Points

• Equations can be created with the equation, equation*, and align environments, and with $ signs.

• Any time you refer to a mathematical symbol, it should be in an equation, most frequently inside $ signs.

• Use \eqref to refer to equation numbers within the text.