2 Math examples

Chapter 2
Math examples

2.1 Delimiters
2.2 Spacing
2.3 Arrays
2.4 Equation arrays
2.5 Functions
2.6 Accents
2.7 Command deﬁnition
2.8 Theorems et al.

These examples were copied from www.maths.adelaide.edu.au/. Math was converted to mathml, Mathjax is used to render it in browsers without mathml support.

2.1 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 .

2.2 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 .

2.3 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}

2.4 Equation arrays

In the ﬂow of a ﬂuid ﬁlm we may report

\begin{array}{rcl} u_{α} & = & 𝜖^{2} κ_{x x x} (y - \frac{1}{2} y^{2}), & (2.1) \\ v & = & 𝜖^{3} κ_{x x x} y, & (2.2) \\ p & = & 𝜖 κ_{x x} . & (2.3) \end{array}

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

\begin{array}{rcl} ω_{1} & = & \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}, \\ ω_{2} & = & \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}, & (2.4) \\ ω_{3} & = & \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} . \end{array}

Whereas a derivation may look like

\begin{array}{rcl} (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{array}

2.5 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{array}{rcl} e^{x} & = & \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} where n! = \prod_{i = 1}^{n} i, \\ \bar{U_{α}} & = & ⋂_{α} U_{α} . \end{array}

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

2.6 Accents

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

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

\dot{\vec{ω}} = \vec{r} \times \vec{I} .

2.7 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}) .

2.8 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 $a^{2} + b^{2} = {(p^{2} - q^{2})}^{2} + {(2 p q)}^{2} = p^{4} - 2 p^{2} q^{2} + q^{4} + 4 p^{2} q^{2} = p^{4} + 2 p^{2} q^{2} + q^{4} = {(p^{2} + q^{2})}^{2} = c^{2}$ .

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Chapter 2Math examples