LaTeX Mathematics Examples

tex4ht + mathjax-node-page sample

This is an example of LATEX to HTML conversion using tex4ht. The document was compiled to HTML + MathML first, then it has been post-processed using Mathjax-node-page to pure HTML + CSS. This method should work in all modern web browsers, as MathML has poor browser support. It should be also much faster than client side MathML rendering by MathJax.

I’ve used a sample document provided by Prof Toby Roberts.

The development version of make4ht, build system for tex4ht has been used. It has a support for Lua extensions, which can be used for modification of the generated files. It contains mathjaxnode extension for Mathjax-node-page (which needs to be installed on the system) support. The following command line invocation has been used for the compilation:

make4ht -uf html5+mathjaxnode -c stack.cfg maths.tex

The configuration file stack.cfg contains some commands to add some additional styling for the resulting HTML file:

\Preamble{xhtml,mathml,fn-in,html5}
\Css{body{
    margin:40px auto;
    max-width:36em;
    line-height:1.6;
    font-size:1em; 0.88em;
    color:\#222;
    background-color:\#FEFEF7;
    padding:0 10px;}}
\Css{h1,h2,h3{line-height:1.2}}
\Css{p.indent{text-indent:0;}}
\Css{@media (min-width: 29.63em){
  :root {
    font-size: 3vw;
  }
}}

\Css{@media (min-width: 40.74em){
  :root {
    font-size: 1.375em;
  }
}}
\Css{body {
font-family: STIXGeneral, "Linux Libertine O", Gentium, Georgia,
"Times New Roman",  "Iowan Old Style",
Palatino Linotype, Palatino, serif;
}}
\begin{document}
\EndPreamble

The original content follows:

LATEX Mathematics Examples

Prof Tony Roberts

February 19, 2018

1 Delimiters
2 Spacing
3 Arrays
4 Equation arrays
5 Functions
6 Accents
7 Command definition
8 Theorems et al.

1 Delimiters

See how the delimiters are of reasonable size in these examples

(a + b) [1 - \frac{b}{a + b}] = a,

\sqrt{| x y |} \leq ∣ ∣ ∣ \frac{x + y}{2} ∣ ∣ ∣,

even when there is no matching delimiter

\int_{a}^{b} u \frac{d^{2} v}{d x^{2}} d x = {u \frac{d v}{d x} ∣ ∣ ∣}_{a}^{b} - \int_{a}^{b} \frac{d u}{d x} \frac{d v}{d x} d x .

2 Spacing

Differentials often need a bit of help with their spacing as in

\iint x y^{2} d x d y = \frac{1}{6} x^{2} y^{3},

whereas vector problems often lead to statements such as

u = \frac{- y}{x^{2} + y^{2}}, v = \frac{x}{x^{2} + y^{2}}, and w = 0 .

Occasionally one gets horrible line breaks when using a list in mathematics such as listing the first twelve primes $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37$ . In such cases, perhaps include \mathcode‘\,="213B inside the inline maths environment so that the list breaks: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37$ . Be discerning about when to do this as the spacing is different.

3 Arrays

Arrays of mathematics are typeset using one of the matrix environments as in

[\begin{matrix} 1 & x & 0 0 & 1 & - 1 \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} 1 y 1 \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 + x y y - 1 \end{matrix}] .

Case statements use cases:

∣ ∣ ∣ x ∣ ∣ ∣ = {\begin{matrix} x, & if x \geq 0, - x, & if x < 0 . \end{matrix}

Many arrays have lots of dots all over the place as in

\begin{matrix} - 2 & 1 & 0 & 0 & \dots & 0 1 & - 2 & 1 & 0 & \dots & 0 0 & 1 & - 2 & 1 & \dots & 0 0 & 0 & 1 & - 2 & ⋱ & ⋮ ⋮ & ⋮ & ⋮ & ⋱ & ⋱ & 1 0 & 0 & 0 & \dots & 1 & - 2 \end{matrix}

4 Equation arrays

In the flow of a fluid film we may report

\begin{matrix} u_{α} & = & ϵ^{2} κ_{x x x} (y - \frac{1}{2} y^{2}), & (1) v & = & ϵ^{3} κ_{x x x} y, & (2) p & = & ϵ κ_{x x} . & (3) \end{matrix}

Alternatively, the curl of a vector field $(u, v, w)$ may be written with only one equation number:

\begin{matrix} ω_{1} & = & \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}, ω_{2} & = & \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}, & (4) ω_{3} & = & \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} . \end{matrix}

Whereas a derivation may look like

\begin{matrix} (p \land q) \lor (p \land \neg q) & = & p \land (q \lor \neg q) by distributive law = & p \land T by excluded middle = & p by identity \end{matrix}

5 Functions

Observe that trigonometric and other elementary functions are typeset properly, even to the extent of providing a thin space if followed by a single letter argument:

exp (i θ) = cos θ + i sin θ, sinh (log x) = \frac{1}{2} (x - \frac{1}{x}) .

With sub- and super-scripts placed properly on more complicated functions,

lim q \to \infty ∥ f (x) ∥_{q} = max x ∣ ∣ ∣ f (x) ∣ ∣ ∣,

and large operators, such as integrals and

\begin{matrix} e^{x} & = & \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} where n! = \prod_{i = 1}^{n} i, ¯ ¯¯¯¯ ¯ U_{α} & = & ⋂_{α} U_{α} . \end{matrix}

In inline mathematics the scripts are correctly placed to the side in order to conserve vertical space, as in $1 / (1 - x) = \sum_{n = 0}^{\infty} x^{n} .$

6 Accents

Mathematical accents are performed by a short command with one argument, such as

˜ f (ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} f (x) e^{- i ω x} d x,

˙ \to ω = \to r \times \to I .

7 Command definition

The Airy function, $Ai (x)$ , may be incorrectly defined as this integral

Ai (x) = \int exp (s^{3} + i s x) d s .

This vector identity serves nicely to illustrate two of the new commands:

\nabla \times q = i (\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}) + j (\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}) + k (\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}) .

8 Theorems et al.

Definition 1 (right-angled triangles) A right-angled triangle is a triangle whose sides of length $a$ , $b$ and $c$ ,in some permutation of order, satisfies $a^{2} + b^{2} = c^{2}$ .

Lemma 2 The triangle with sides of length $3$ , $4$ and $5$ is right-angled.

This lemma follows from the Definition 1 as $3^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}$ .

Theorem 3 (Pythagorean triplets) Triangles with sides of length $a = p^{2} - q^{2}$ , $b = 2 p q$ and $c = p^{2} + q^{2}$ are right-angled triangles.

Prove this Theorem 3 by the algebra $a^{2} + b^{2} = {(p^{2} - q^{2})}^{2} + {(2 p q)}^{2} = p^{4} - 2 p^{2} q^{2} + q^{4} + 4 p^{2} q^{2} = p^{4} + 2 p^{2} q^{2} + q^{4} = {(p^{2} + q^{2})}^{2} = c^{2}$ .