Typesetting Mathematics with LaTeX - Day 2

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Recall Symbols AdEqn Greek Letters

Typesetting mathematics with LATEX

Suddhasheel Ghosh

Department of Civil Engineering,Jawaharlal Nehru Engineering College

Aurangabad, Maharashtra431003

LATEX for Technical and Scientific DocumentsJanuary 12 - 16, 2015

Day II

shudh@JNEC LATEX@ JNEC 1 / 30

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Outline

1 What did we learn about typesetting Math

2 Mathematics Symbology

3 Advanced equations

4 Greek Letters


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In this workshop:

Day 1: Introduction to LATEX

Day 2: Typesetting Mathematics

Day 3: Writing your thesis / technical report

Day 4: Make your presentations

Day 5: Basics of science communication


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Outline




4 Greek Letters


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Playing with mathematics textInline and display math

There are two kinds of math used in LATEX.

Inline math (surrounded by $ signs)

Display math (surrounded by square brackets \[ and \] or $$

About any point x in a metric space M we define the open ball ofradius r > 0 about x as the set

B(x;r) = y ∈ M : d(x,y) < r.

About any point $x$ in a metric space $M$ we define theopen ball of radius $r > 0$ about $x$ as the set\[B(x;r) = \y \in M : d(x,y) < r\.\]


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Typesetting equationsNumbered equations

Numbered equations can be typeset by using

\beginequationR(x,y) = \fracAx^3+Bx^2+Cx+DEy^3 + Fy + G\endequation

R(x,y) = Ax3 +Bx2 +Cx+D

Ey3 +Fy+G(1)


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Outline




4 Greek Letters


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Definition symbolsAn example of a Cauchy sequence

A sequence of numbers ⟨xi⟩ni=1, is called a Cauchy sequence, if

∃ε> 0, ∀N ∈N, such that for all m,n ∈N,

|xm −xn| < ε

A sequence of numbers $\left\langle x_i\right\rangle_i=1^n$,

is called a Cauchy sequence, if $\exists\,\epsilon > 0$,$\forall N\in\mathbbN$, such that for all$m,n\in\mathbbN$,

\[\left\vert x_m - x_n \right\vert < \epsilon\]


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Definition symbolsContinuous function example

We can say thatlimx→c

f (x) = L

if ∀ε> 0 , ∃δ> 0 such that ∀x ∈ D that satisfy 0 < |x− c| < δ , theinequality |f (x)−L| < ε holds.

We can say that\[\lim\limits_x\rightarrow c f(x) = L\]if $\forall\,\varepsilon > 0$ , $\exists\, \delta > 0$such that $\forall x\in D$ that satisfy $0 < | x - c | <

\delta$ ,the inequality $|f(x) - L| < \varepsilon$ holds.


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Outline




4 Greek Letters


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Multiline equations... and alignment

If h ≤ 12 |ζ−z| then

|ζ−z−h| ≥ 1

2|ζ−z|

and hence∣∣∣∣ 1

ζ−z−h− 1

ζ−z

∣∣∣∣ =∣∣∣∣ (ζ−z)− (ζ−z−h)

(ζ−z−h)(ζ−z)

∣∣∣∣=

∣∣∣∣ h

(ζ−z−h)(ζ−z)

∣∣∣∣≤ 2|h|

|ζ−z|2 . (2)

If $h \leq \frac12 |\zeta - z|$ then\[ |\zeta - z - h| \geq \frac12

|\zeta - z|\]and hence\begineqnarray\left| \frac1\zeta - z - h -\frac1\zeta - z \right|& = & \left|\frac(\zeta - z) -(\zeta - z - h)(\zeta - z - h)(\zeta -

z)\right| \nonumber \\ & = &\left| \frach(\zeta - z - h)(\zeta -

z)\right| \nonumber \\& \leq & \frac2 |h||\zeta - z|^2.

\endeqnarray


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Case based definitions

An example of a continuous, nowhere differential function is givenas

f (x) =

0 x is irrational

1 x is rational

An example of a continuous, nowhere differential functionis given as

\[f(x) =\begincases0 &x\ \mathrm\ is\ irrational \\1 &x\ \mathrm\ is\ rational\endcases\]


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Case based definitions... a more professional one

An example of a continuous, nowhere differential function is givenas

f (x) =

1 x ∈Q0 x ∈R−Q

An example of a continuous, nowhere differential functionis given as\[f(x) =\begincases1 &x\in\mathbbQ \\0 &x\in\mathbbR-\mathbbQ\endcases\]


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Fractions and binomials

To typeset:n!

k!(n−k)!=

(n

k

)Use

\[\fracn!k!(n-k)! = \binomnk\]

For the computation of a permutation nPk, we would use thefollowing formula:

nPk = (n−1)(n−2) . . . (n−k+1)︸︷︷︸Exactly k factors

\[^n P_k = \underbrace(n-1)(n-2)

\dotsc(n-k+1)_\mboxExactly $k$ factors\]


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What about a continued fraction?Any idea?

To get,

x = a0 +1

a1 +1

a2 +1

a3 +1

a4

\[x = a_0 + \cfrac1a_1

+ \cfrac1a_2+ \cfrac1a_3 + \cfrac1a_4

\]


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Square roots... and other roots too!

A =√

4x3 +3x2 −5x+1

B = 3√

4x3 +3x2 −5x+1

\[A = \sqrt4x^3 + 3x^2 - 5x + 1\]\[B = \sqrt[3]4x^3 + 3x^2 - 5x + 1\]


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Square roots...use in mathematical expressions

Given a quadratic equation ax2 +bx+ c = 0, the roots of theequation are given by the following formula:

x = −b±p

b2 −4ac

2a

Given a quadratic equation $ax^2 + bx + c = 0$, the rootsof the equation are given by the following formula:\[x = \frac-b \pm \sqrtb^2 - 4ac 2a\]


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Sums and Products

n∑i=0

ai = a1 +a2 +a3 +·· ·+an

\[\sum_i=0^n a_i = a_1 + a_2 + a_3 + \dots + a_n\]

n∏i=0

ai = a1 ·a2 ·a3 · · · · ·an

\[\prod_i=0^n a_i = a_1 \cdot a_2 \cdot a_3\cdot \dots \cdot a_n\]


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Integrals and Differentials

∫ ∞

0e−x2

dx =pπ

2

\[\int_0^\infty e^-x^2 dx=\frac\sqrt\pi2\]

If f (x,y) = x2 +y2, then:∂f

∂x= 2x

If $f(x,y) = x^2 + y^2$, then:\[\frac\partial f\partial x = 2x\]


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Mathematical operatorsIntegration examples

In the field of integral calculus, it is known that∫sinx =−cosx

and ∫cosx = sinx

In the field of integral calculus, it is known that\[\int\sin x = -\cos x\]and\[\int\cos x = \sin x\]


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Mathematical operators and notationFourier transform

The Fourier transform, for a function f (x), is given as

f (ξ) =∫ ∞

−∞f (x) e−2πixξdx, ∀ξ ∈R

and the inverse Fourier transform is given as,

f (x) =∫ ∞

−∞f (ξ) e2πixξdξ ∀x ∈R

The Fourier transform, for a function $f(x)$, is given as\[\hatf(\xi) = \int_-\infty^\infty f(x)\ e^-2 \pi i

x \xi\, dx, \, \qquad \forall \xi\in\mathbbR\]and the inverse Fourier transform is given as,\[f(x) = \int_-\infty^\infty \hatf(\xi)\ e^2 \pi i x

\xi\, d\xi \, \qquad \forall x \in\mathbbR\]


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Mathematical OperatorsContinuity equation ... and vector notation

Mathematically, the integral form of the continuity equation is:

dq

dt+

ÓS

j ·dS =Σ

which in vector notation can also be written as

dq

dt+

ÓS

~j ·d~S =Σ

Mathematically, the integral form of the continuityequation is:

\[\fracd qd t + \oiint_S\mathbfj \cdotd\mathbfS = \Sigma

\]which in vector notation can also be written as\[

\fracd qd t + \oiint_S\vecj \cdot d\vecS =\Sigma

\]


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Typesetting Matrices... and determinants

The determinant of the matrix A, where

A =∣∣∣∣∣∣1 2 34 5 67 8 9

∣∣∣∣∣∣is zero.

\[A =\beginvmatrix1 &2 &3 \\4 &5 &6 \\7 &8 &9\endvmatrix\]


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Typesetting MatricesTypically some long ones

The Gaussian Elimination Method can be used to find the solutionof the following system of equations, represented in the matrixform:

a11 a12 a13 . . . a1n

a21 a22 a23 . . . a2n...

......

. . ....

an1 an2 an3 . . . ann

x1

x2...

xn

=

b1

b2...

bn

\[\beginbmatrixa_11 & a_12 &a_13 &\dots &a_1n \\ a_21 & a_22

&a_23 &\dots &a_2n \\\vdots &\vdots &\vdots &\ddots &\vdots \\ a_n1 & a_n2

&a_n3 &\dots &a_nn \\\endbmatrix\beginbmatrixx_1 \\ x_2 \\ \vdots \\ x_n\endbmatrix=\beginbmatrixb_1 \\ b_2 \\ \vdots \\b_n\endbmatrix\]


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Array based typesetting of equationsLeast squares example

The least squares method is used to find the solution of a system ofn equations, with m variables where m > n. The system ofequations is given as follows:

a11x1 +a12x2 + . . . +a1mxm = b1

a21x1 +a22x2 + . . . +a2mxm = b2

. . . . . . . . . . . . . . . . . .an1x1 +an2x2 + . . . +anmxm = bm

(3)

The least squares method is used to find the solution ofa system of $n$ equations, with $m$ variables where$m > n$. The system of equations is given as follows:\beginequation\beginarraylllllla_11 x_1 &+a_12 x_2 &+\dotsc &+a_1m x_m &= &b_1 \\a_21 x_1 &+a_22 x_2 &+\dotsc &+a_2m x_m &= &b_2 \\\dots &\dots &\dots &\dots &\dots &\dots \\a_n1 x_1 &+a_n2 x_2 &+\dotsc &+a_nm x_m &= &b_m \\\endarray\endequation


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Array based typesetting of equationsLeast squares example ... with a twist

The least squares method is used to find the solution of a system ofn equations, with m variables where m > n. The system ofequations is given as follows:

Solv

e

a11x1 +a12x2 + . . . +a1mxm = b1

a21x1 +a22x2 + . . . +a2mxm = b2

. . . . . . . . . . . . . . . . . .an1x1 +an2x2 + . . . +anmxm = bm

(4)

The least squares method is used to find the solution ofa system of $n$ equations, with $m$ variables where$m > n$. The system of equations is given as follows:\beginequation\rotatebox90Solve \left\ \beginarraylllllla_11 x_1 &+a_12 x_2 &+\dotsc &+a_1m x_m &= &b_1 \\a_21 x_1 &+a_22 x_2 &+\dotsc &+a_2m x_m &= &b_2 \\\dots &\dots &\dots &\dots &\dots &\dots \\a_n1 x_1 &+a_n2 x_2 &+\dotsc &+a_nm x_m &= &b_m \\\endarray\right.\endequation


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Outline




4 Greek Letters


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Greek letters

The small greek letters areα,β,γ,δ,ε,ε,ζ,η,θ,ϑ,κ,Å,λ,µ,ν,ξ,π,$,ρ,%,σ,ς,τ,υ,φ,ϕ,χ,ψ,ω,and the capital greek letters are given asA,B,Γ,∆,E,Z ,H ,Θ,K ,Λ,M ,N ,Ξ,Π,P,Σ,T ,Υ,Φ,X ,Ψ,Ω

The small greek letters are $\alpha, \beta, \gamma,\delta, \epsilon, \varepsilon, \zeta, \eta, \theta,\vartheta, \kappa, \varkappa, \lambda, \mu, \nu, \xi,\pi, \varpi, \rho, \varrho, \sigma, \varsigma, \tau,\upsilon, \phi, \varphi, \chi, \psi, \omega$, andthe capital greek letters are given as $ A, B, \Gamma,\Delta, E, Z, H, \Theta, K, \Lambda, M, N, \Xi, \Pi, P,\Sigma, T, \Upsilon, \Phi, X, \Psi, \Omega $


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Example with greek letters... and something more

If Z1, . . . ,Zk are independent, standard normalrandom variables, then the sum of their squares,

Q =k∑

i=1Z2

i ,

is distributed according to the ξ2 distributionwith k degrees of freedom. This is usuallydenoted as

Q ∼ χ2(k) or Q ∼ χ2k.

The chi-squared distribution has oneparameter: k – a positive integer that specifiesthe number of degrees of freedom (i.e. thenumber of Zi’s)

If $Z_1, \dots, Z_k$ are independent,standard normal random variables,then the sum of their squares,\[Q\ = \sum_i=1^k Z_i^2 ,\]is distributed according to the $\xi^2$distribution with $k$ degrees of freedom.This is usually denoted as\[Q\ \sim\ \chi^2(k)\ \ \textor\ \ Q\

\sim\ \chi^2_k .\]The chi-squared distribution has oneparameter: $k$ -- a positive integer thatspecifies the number of degreesof freedom (i.e. the number of $Z_i$’s)


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In the upcoming sessions

Typing a thesis / report / dissertation

Table of contents / figure / tables

Managing Bibliography

Cross referencing

Paper referencing

Style files for theses and bibliography


Typesetting Mathematics with LaTeX - Day 2

Education

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