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Heavy Duty Math Writing

Curtis Helms

October 14, 2011

1 Utilizing Large DelimitersLarger delimiters are sometimes required which have the appropriate heightto match the size of the subformula which they enclose. Consider, for in-stance, the problem of typesetting the following formula:

f (x,y,z) = 3y2z

3 +

7x + 6

1 + y2

.

One may also nest pairs of delimiters within one another: by typing

4x3 +

x + 421 + x4

.

Consider, for example, the problem of typesetting

du

dx

x=0

.

2 Multiline Formulae in LATEX

Consider the problem of typesetting the formula

cos2θ = cos2 θ − sin2θ

= 2 cos2 θ − 1.

The more complicated example

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If h

≤1

2

|ζ

−ξ

|then

|ζ − ξ − h| ≥ 1

2|ζ − ξ|

and hence

1

ζ − ξ − h− 1

ζ − ξ

=

(ζ − ξ) − (ζ − ξ − h)

(ζ − ξ − h)(ζ − ξ)

=

h

(ζ

−ξ

−h)(ζ

−ξ)

≤ 2|h||ζ − ξ|2 .

3 Matrices and other arrays in LATEX

Matrices and other arrays are produced in LATEXusing the array environ-ment. For example, suppose that we wish to typeset the following passage:

The characteristic polynominal χ(λ) of the 3 × 3 matrix a b c

d e f g h i

is given by the formula

χ(λ) =

λ − a −b −c−d λ − e −f

−g

−h λ

−i

.

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3.1 Creating Tables with Arrays

We can use the array construct to build tables that are primarily mathemat-ical in nature. Here is an example.

First number x 8Second number y 15Sum x − y 23Difference x − y −7Product xy 120

We can use the array environment to produce formulae such as

|x| =

x if x ≥ 0;

−x if x < 0.

3.2 Derivatives, Limits, Sums and Integrals

3.2.1 Derivatives

The expressionsdu

dtand

d2u

dx2

are obtained in LA

TEX by typing \frac{du}{dt}and \frac{d^2u}{dx^2} re-spectively. The mathematical symbol ∂ is produced using \partial. Thusthe Heat Equation

∂u

∂t=

∂ 2u

∂x2+

∂ 2u

∂y2+

∂ 2u

∂z2

obtained in LATEX by typing the appropriate expression.LATEX has the ability to display word based quantifiers. For example

expressions such aslim

x→+∞, inf x>s

and supK

are available to a document writer. In practical usage, a student wouldnormally embed an expression such as

limx→+∞

3x2 + 7

x2 + 1= 3.

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To obtain a summation sign such as

2ni=1

use the appropriate control sequences. You can create this expression

nk=1

=n(n + 1)

2.

via the correct LATEX codes.

3.2.2 Integrals

We now discuss how to obtain integrals in mathematical documents. Attypical integral is the following: b

af (x)dx

The integral sign

is typeset using the control sequence \int, and the limitsof integration . Most integrals occurring in mathematical documents beginwith an integral sign and contain one or more instances of d followed byanother (Latin or Greek) letter, as in dx, dy and dt. To obtain the correctappearance one should put extra space before the d, using \, . Thus

+∞

0

xne−x dx = n!.

cos θ dθ = sin θ.

x2+y2≤R2

f (x, y) dxdx = 2π

θ=0

Rr=0

f (r cos θ, r sin θ)r dr dθ

and R0

2x dx

1 + x2 = log(1 + R2

).

are obtained by typing the correct LATEX sequences. Here are some moreintegrals that use the control sequence \! to remove a thin strip of unwantedspace

1

0

1

0

x2y2 dxdy (1)

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Had we not used the space remover we would have obtained

1

0

1

0

x2y2 dxdy

A particularly noteworthy example comes when we are typesetting a multipleintegral such as

Df (x, y) dxdy. (2)

Had we not used \! three times we would have this D

f (x, y) dxdy

3.2.3 A Complex Set of Derivative and Integral Equations

In non-relativistic wave mechanics, the wave function ψ(r, t) of a particlesatisfies the Schr¨ odinger Wave Equation

ih∂ψ

∂t=−h2

2m

∂ 2

∂x2+

∂ 2

∂y2+

∂ 2

∂z2

ψ + V ψ.

It is customary to normalize the wave equation by demanding that

R3 |ψ(r, 0)|

2

dx dy dz = 1.

A simple calculation using the Schrodinger wave equation shows that

d

dt

R3|ψ(r, t)|2 dx dy dz = 0,

and hence R3|ψ(r, t)|2 dx dy dz = 1.

for all times t. If we normalize the wave function in this way then, for any(measurable) subset R3 and time t,

V |ψ(r, t)|2 dx dy dz

represents the probability that the particle is to be found within the regionV at time t.

This is a test of our macros.

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Macro #1 +∞−∞

f (x) dx.

Macro #2 +∞

−∞

f (x) dx.

Macro #3 +∞

−∞

f (y) dy

4 Calculus

4.1 Limits

4.1.1 Limits of a Function

If f is a function, then we say:

A is the limit of f(x) as x approaches a.

if the value if f (x) gets arbitrarily close to A as x approaches a. This iswritten in mathematical notation as:

limx→a

f (x) = A

The definition of limx→a f (x) = A was stated above in ordinary language.The definition can be stated in more precise mathematical language as fol-lows: limx→a f (x) = A if an only if, for any given positive number , howeversmall, there exists a positive number δ, such that, whenever 0 < |x− a| < δ,then |f (x) − A| < .

Notice the important fact that whether or not limx→a f (x) = A is truedoes not depend upon the value of f (x) when x = a. In fact, f (x) need noteven be defined when x = a.

4.1.2 Right and Left Limits

Next we want to talk about one-sided limits of f (x) as x approaches a fromthe right-hand side or from the left-hand side. By limx→a− f (x) = A we meanthat that f is defined in some open interval (c, a) and f (x) approaches A as

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x approaches a through values less than a, that is, as x approaches a from

the left . Similarly, limx→a+ f (x) = A means that f is defined in some openinterval (a, d) and f (x) approaches A as x approaches a from the right . If f is defined in an interval to the left of a and in an interval to the right of a,then the statement limx→a f (x) = A is equivalent to the conjunction of thetwo statements limx→a− f (x) = A and limx→a+ f (x) = A. We shall see byexamples below that the existence of the limit from the left does not implythe existence of the limit from the right, and conversely. When a functionis defined only on one side of a point a, then we shall identify limx→a f (x)with the one-sided limit, if it exists. For example, if f (x) =

√x, then f is

defined only at and to the right of 0. Hence, since limx→0+

√x = 0, we will

also write limx→0√x = 0. Of course, limx→0−

does not exist, since √x isnot defined when x < 0. This is an example where the existence of the limitfrom one side does not entail the existence of the limit from the other side.As another interesting example, consider the function g(x) =

1/x, which

is defined only for x > 0. In this case, limx→0+

1/x does not exist, since

1/x gets larger and larger without bound as x approaches 0 from the right.

Therefore, limx→0

1/x does not exist.

5 Further Features of LATEX

LATEX will automatically indent paragraphs (with the exception of the firstparagraph of a section). One can prevent LATEX from indenting a paragraphthrough by beginning the paragraph with the control sequence \noindent.Conversely, the control sequence \indent forces LATEX to indent the para-graph.

5.1 Lists

A metric space (X, d) consists of a set X on which is defined a distance function which assigns to each pair of points of X a distance between them,

and which satisfies the following four axioms:

1. d(x, y) ≥ 0 for all points x and y of X ;

2. d(x, y) = d(y, x) for all points x and y of X ;

3. d(x, y) ≤ d(x, y) + d(y, z) for all points x, y, z of X ;

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4. d(x, y) = 0 for all points x and y coincide.

• d(x, y) ≥ 0 for all points x and y of X ;

• d(x, y) = d(y, x) for all points x and y of X ;

• d(x, y) ≤ d(x, y) + d(y, z) for all points x, y, z of X ;

• d(x, y) = 0 for all points x and y coincide.

We now list the definitions of open ball, open set and closed set in a metricspace.

open ball The open ball of radius r about any point x is the set of all pointsof the metric space whose distance from x is strictly less than r;

open set A subset of a metric space is an open set if, given any point of the set, some open ball of sufficiently small radius about that point iscontained wholly within the set;

closed set A subset of a metric space is a closed set if its complement is an

open set.

5.2 Displayed Quotations

Issac Newton discovered the basis techniques of the differential and integralcalculus, and applied them in the study of many problems in mathematicalphysics. His main mathematical works are the Principia and the Optics. hesummed up his own estimate of his work as follows:

I do not know what I may appear to the world; but to myself Iseem to have been only like a boy, playing on the sea-shore, and

diverting myself, in now and then finding a smoother pebble, ora prettier shell than ordinary, whilst the great ocean truth lay allundiscovered before me

In later years newton became embroiled in a bitter priority dispute withLeibniz over the discover of the basic techniques of calculus.

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5.3 Tables

Tables can be produced in LATEX using the tabular environment. The\begin{tabular} command must be followed by a string of characters en-closed within braces which specifies the format of the table. In the aboveexample, the sting {lll} is a format specification for a table with threecolumns of left-justified text. Within the body of the table the ampersandcharacter &, and the double backslash \\ is used to separate the rows of thetable.

Table Example

Chicago U.S.A. 1893Zurich Switzerland 1897Paris Switzerland 1897Heidelberg Germany 1904Rome Italy 1908

The group of permutations of a set of n elements has order n!, wheren!, the factorial of n, is the product of all integers between 1 and n. Thefollowing table lists the values of the factorial of each integer n between 1and 10:

Table with Lines Example

n n!1 12 23 64 245 1206 7207 50408 40320

9 36288010 3628800

Note how rapidly the value of n! increases with n.

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