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Typesetting Mathematics with LaTeX - Day 2

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LaTeX for Technical and Scientific Documents - Day 2 (January 12 - 16, 2015)

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Typesetting Mathematics with LaTeX - Day 2

  1. 1. SU D D H ASH EEL.CO M Recall Symbols AdEqn Greek Letters Typesetting mathematics with LATEX Suddhasheel Ghosh Department of Civil Engineering, Jawaharlal Nehru Engineering College Aurangabad, Maharashtra 431003 LATEX for Technical and Scientific Documents January 12 - 16, 2015 Day II shudh@JNEC LATEX@ JNEC 1 / 30
  2. 2. Recall Symbols AdEqn Greek Letters Outline 1 What did we learn about typesetting Math 2 Mathematics Symbology 3 Advanced equations 4 Greek Letters shudh@JNEC LATEX@ JNEC 2 / 30
  3. 3. Recall Symbols AdEqn Greek Letters 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 shudh@JNEC LATEX@ JNEC 3 / 30
  4. 4. Recall Symbols AdEqn Greek Letters Outline 1 What did we learn about typesetting Math 2 Mathematics Symbology 3 Advanced equations 4 Greek Letters shudh@JNEC LATEX@ JNEC 4 / 30
  5. 5. Recall Symbols AdEqn Greek Letters Playing with mathematics text Inline 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 of radius 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 the open ball of radius $r > 0$ about $x$ as the set [ B(x;r) = {y in M : d(x,y) < r}. ] shudh@JNEC LATEX@ JNEC 5 / 30
  6. 6. SU D D H ASH EEL.CO M Recall Symbols AdEqn Greek Letters Typesetting equations Numbered equations Numbered equations can be typeset by using begin{equation} R(x,y) = frac{Ax^3+Bx^2+Cx+D}{Ey^3 + Fy + G} end{equation} R(x,y) = Ax3 +Bx2 +Cx+D Ey3 +Fy +G (1) shudh@JNEC LATEX@ JNEC 6 / 30
  7. 7. Recall Symbols AdEqn Greek Letters Outline 1 What did we learn about typesetting Math 2 Mathematics Symbology 3 Advanced equations 4 Greek Letters shudh@JNEC LATEX@ JNEC 7 / 30
  8. 8. Recall Symbols AdEqn Greek Letters Definition symbols An example of a Cauchy sequence A sequence of numbers 〈xi〉n i=1 , is called a Cauchy sequence, if ∃ > 0, ∀N ∈ N, such that for all m,n ∈ N, |xm −xn| < A sequence of numbers $leftlangle x_i rightrangle_{i=1}^n$, is called a Cauchy sequence, if $exists,epsilon > 0$, $forall Ninmathbb{N}$, such that for all $m,ninmathbb{N}$, [ leftvert x_m - x_n rightvert < epsilon ] shudh@JNEC LATEX@ JNEC 8 / 30
  9. 9. Recall Symbols AdEqn Greek Letters Definition symbols Continuous function example We can say that lim x→c f (x) = L if ∀ε > 0 , ∃δ > 0 such that ∀x ∈ D that satisfy 0 < |x−c| < δ , the inequality |f (x)−L| < ε holds. We can say that [ limlimits_{xrightarrow c} f(x) = L ] if $forall,varepsilon > 0$ , $exists, delta > 0$ such that $forall xin D$ that satisfy $0 < | x - c | < delta$ , the inequality $|f(x) - L| < varepsilon$ holds. shudh@JNEC LATEX@ JNEC 9 / 30
  10. 10. Recall Symbols AdEqn Greek Letters Outline 1 What did we learn about typesetting Math 2 Mathematics Symbology 3 Advanced equations 4 Greek Letters shudh@JNEC LATEX@ JNEC 10 / 30
  11. 11. Recall Symbols AdEqn Greek Letters Multiline equations ... and alignment If h ≤ 1 2 |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 $h leq frac{1}{2} |zeta - z|$ then [ |zeta - z - h| geq frac{1}{2} |zeta - z|] and hence begin{eqnarray} left| frac{1}{zeta - z - h} - frac{1}{zeta - z} right| & = & left| frac{(zeta - z) - (zeta - z - h)}{(zeta - z - h)(zeta - z)} right| nonumber & = & left| frac{h}{(zeta - z - h)(zeta - z)} right| nonumber & leq & frac{2 |h|}{|zeta - z|^2}. end{eqnarray} shudh@JNEC LATEX@ JNEC 11 / 30
  12. 12. Recall Symbols AdEqn Greek Letters Case based definitions An example of a continuous, nowhere differential function is given as f (x) = 0 x is irrational 1 x is rational An example of a continuous, nowhere differential function is given as [ f(x) = begin{cases} 0 &{x mathrm{ is irrational}} 1 &{x mathrm{ is rational}} end{cases} ] shudh@JNEC LATEX@ JNEC 12 / 30
  13. 13. Recall Symbols AdEqn Greek Letters Case based definitions ... a more professional one An example of a continuous, nowhere differential function is given as f (x) = 1 x ∈ Q 0 x ∈ R−Q An example of a continuous, nowhere differential function is given as [ f(x) = begin{cases} 1 &{xinmathbb{Q}} 0 &{xinmathbb{R}-mathbb{Q}} end{cases} ] shudh@JNEC LATEX@ JNEC 13 / 30
  14. 14. Recall Symbols AdEqn Greek Letters Fractions and binomials To typeset: n! k!(n−k)! = n k Use [ frac{n!}{k!(n-k)!} = binom{n}{k} ] For the computation of a permutation n Pk, we would use the following formula: n Pk = (n−1)(n−2)...(n−k +1) Exactly k factors [ {}^n P_k = underbrace{(n-1)(n-2) dotsc(n-k+1)}_{mbox{Exactly $k$ factors}} ] shudh@JNEC LATEX@ JNEC 14 / 30
  15. 15. Recall Symbols AdEqn Greek Letters What about a continued fraction? Any idea? To get, x = a0 + 1 a1 + 1 a2 + 1 a3 + 1 a4 [ x = a_0 + cfrac{1}{a_1 + cfrac{1}{a_2 + cfrac{1}{a_3 + cfrac{1}{a_4} } } } ] shudh@JNEC LATEX@ JNEC 15 / 30
  16. 16. Recall Symbols AdEqn Greek Letters What about a continued fraction? Any idea? To get, x = a0 + 1 a1 + 1 a2 + 1 a3 + 1 a4 [ x = a_0 + cfrac{1}{a_1 + cfrac{1}{a_2 + cfrac{1}{a_3 + cfrac{1}{a_4} } } } ] shudh@JNEC LATEX@ JNEC 15 / 30
  17. 17. Recall Symbols AdEqn Greek Letters Square roots ... and other roots too! A = 4x3 +3x2 −5x+1 B = 3 4x3 +3x2 −5x+1 [ A = sqrt{4x^3 + 3x^2 - 5x + 1} ] [ B = sqrt[3]{4x^3 + 3x^2 - 5x + 1} ] shudh@JNEC LATEX@ JNEC 16 / 30
  18. 18. Recall Symbols AdEqn Greek Letters Square roots ...use in mathematical expressions Given a quadratic equation ax2 +bx+c = 0, the roots of the equation are given by the following formula: x = −b± b2 −4ac 2a Given a quadratic equation $ax^2 + bx + c = 0$, the roots of the equation are given by the following formula: [ x = frac{-b pm sqrt{b^2 - 4ac} }{2a} ] shudh@JNEC LATEX@ JNEC 17 / 30
  19. 19. Recall Symbols AdEqn Greek Letters 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 ] shudh@JNEC LATEX@ JNEC 18 / 30
  20. 20. Recall Symbols AdEqn Greek Letters Integrals and Differentials ∞ 0 e−x2 dx = π 2 [ int_0^infty e^{-x^2} dx=frac{sqrt{pi}}{2} ] 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 ] shudh@JNEC LATEX@ JNEC 19 / 30
  21. 21. Recall Symbols AdEqn Greek Letters Mathematical operators Integration 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 [ intsin x = -cos x ] and [ intcos x = sin x ] shudh@JNEC LATEX@ JNEC 20 / 30
  22. 22. Recall Symbols AdEqn Greek Letters Mathematical operators and notation Fourier transform The Fourier transform, for a function f (x), is given as ˆf (ξ) = ∞ −∞ f (x) e−2πixξ dx, ∀ξ ∈ R and the inverse Fourier transform is given as, f (x) = ∞ −∞ ˆf (ξ) e2πixξ dξ ∀x ∈ R The Fourier transform, for a function $f(x)$, is given as [ hat{f}(xi) = int_{-infty}^{infty} f(x) e^{-2 pi i x xi}, dx, , qquad forall xiinmathbb{R} ] and the inverse Fourier transform is given as, [ f(x) = int_{-infty}^{infty} hat{f}(xi) e^{2 pi i x xi}, dxi , qquad forall x inmathbb{R} ] shudh@JNEC LATEX@ JNEC 21 / 30
  23. 23. Recall Symbols AdEqn Greek Letters Mathematical Operators Continuity equation ... and vector notation Mathematically, the integral form of the continuity equation is: dq dt + S j·dS = Σ which in vector notation can also be written as dq dt + S j ·dS = Σ Mathematically, the integral form of the continuity equation is: [ frac{d q}{d t} + oiint_Smathbf{j} cdot dmathbf{S} = Sigma ] which in vector notation can also be written as [ frac{d q}{d t} + oiint_Svec{j} cdot dvec{S} = Sigma ] shudh@JNEC LATEX@ JNEC 22 / 30
  24. 24. Recall Symbols AdEqn Greek Letters Mathematical Operators Continuity equation ... and vector notation Mathematically, the integral form of the continuity equation is: dq dt + S j·dS = Σ which in vector notation can also be written as dq dt + S j ·dS = Σ Mathematically, the integral form of the continuity equation is: [ frac{d q}{d t} + oiint_Smathbf{j} cdot dmathbf{S} = Sigma ] which in vector notation can also be written as [ frac{d q}{d t} + oiint_Svec{j} cdot dvec{S} = Sigma ] shudh@JNEC LATEX@ JNEC 22 / 30
  25. 25. Recall Symbols AdEqn Greek Letters Typesetting Matrices ... and determinants The determinant of the matrix A, where A = 1 2 3 4 5 6 7 8 9 is zero. [ A = begin{vmatrix} 1 &2 &3 4 &5 &6 7 &8 &9 end{vmatrix} ] shudh@JNEC LATEX@ JNEC 23 / 30
  26. 26. Recall Symbols AdEqn Greek Letters Typesetting Matrices ... and determinants The determinant of the matrix A, where A = 1 2 3 4 5 6 7 8 9 is zero. [ A = begin{vmatrix} 1 &2 &3 4 &5 &6 7 &8 &9 end{vmatrix} ] shudh@JNEC LATEX@ JNEC 23 / 30
  27. 27. Recall Symbols AdEqn Greek Letters Typesetting Matrices Typically some long ones The Gaussian Elimination Method can be used to find the solution of the following system of equations, represented in the matrix form:       a11 a12 a13 ... a1n a21 a22 a23 ... a2n ... ... ... ... ... an1 an2 an3 ... ann             x1 x2 ... xn       =       b1 b2 ... bn       [ begin{bmatrix} a_{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} end{bmatrix} begin{bmatrix} x_1 x_2 vdots x_n end{bmatrix} = begin{bmatrix} b_1 b_2 vdots b_n end{bmatrix} ] shudh@JNEC LATEX@ JNEC 24 / 30
  28. 28. Recall Symbols AdEqn Greek Letters Typesetting Matrices Typically some long ones The Gaussian Elimination Method can be used to find the solution of the following system of equations, represented in the matrix form:       a11 a12 a13 ... a1n a21 a22 a23 ... a2n ... ... ... ... ... an1 an2 an3 ... ann             x1 x2 ... xn       =       b1 b2 ... bn       [ begin{bmatrix} a_{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} end{bmatrix} begin{bmatrix} x_1 x_2 vdots x_n end{bmatrix} = begin{bmatrix} b_1 b_2 vdots b_n end{bmatrix} ] shudh@JNEC LATEX@ JNEC 24 / 30
  29. 29. Recall Symbols AdEqn Greek Letters Array based typesetting of equations Least squares example The least squares method is used to find the solution of a system of n equations, with m variables where m > n. The system of equations 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 of a system of $n$ equations, with $m$ variables where $m > n$. The system of equations is given as follows: begin{equation} begin{array}{llllll} a_{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 end{array} end{equation} shudh@JNEC LATEX@ JNEC 25 / 30
  30. 30. Recall Symbols AdEqn Greek Letters Array based typesetting of equations Least squares example The least squares method is used to find the solution of a system of n equations, with m variables where m > n. The system of equations 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 of a system of $n$ equations, with $m$ variables where $m > n$. The system of equations is given as follows: begin{equation} begin{array}{llllll} a_{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 end{array} end{equation} shudh@JNEC LATEX@ JNEC 25 / 30
  31. 31. Recall Symbols AdEqn Greek Letters Array based typesetting of equations Least squares example ... with a twist The least squares method is used to find the solution of a system of n equations, with m variables where m > n. The system of equations is given as follows: Solve    a11x1 +a12x2 +... +a1mxm = b1 a21x1 +a22x2 +... +a2mxm = b2 ... ... ... ... ... ... an1x1 +an2x2 +... +anmxm = bm (4) The least squares method is used to find the solution of a system of $n$ equations, with $m$ variables where $m > n$. The system of equations is given as follows: begin{equation} rotatebox{90}{Solve} left{ begin{array}{llllll} a_{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 end{array} right. end{equation} shudh@JNEC LATEX@ JNEC 26 / 30
  32. 32. Recall Symbols AdEqn Greek Letters Array based typesetting of equations Least squares example ... with a twist The least squares method is used to find the solution of a system of n equations, with m variables where m > n. The system of equations is given as follows: Solve    a11x1 +a12x2 +... +a1mxm = b1 a21x1 +a22x2 +... +a2mxm = b2 ... ... ... ... ... ... an1x1 +an2x2 +... +anmxm = bm (4) The least squares method is used to find the solution of a system of $n$ equations, with $m$ variables where $m > n$. The system of equations is given as follows: begin{equation} rotatebox{90}{Solve} left{ begin{array}{llllll} a_{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 end{array} right. end{equation} shudh@JNEC LATEX@ JNEC 26 / 30
  33. 33. Recall Symbols AdEqn Greek Letters Outline 1 What did we learn about typesetting Math 2 Mathematics Symbology 3 Advanced equations 4 Greek Letters shudh@JNEC LATEX@ JNEC 27 / 30
  34. 34. SU D D H ASH EEL.CO M Recall Symbols AdEqn Greek Letters Greek letters The small greek letters are α,β,γ,δ, ,ε,ζ,η,θ,ϑ,κ, ,λ,µ,ν,ξ,π, ,ρ, ,σ,ς,τ,υ,φ,ϕ,χ,ψ,ω, and the capital greek letters are given as A,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 $alpha, beta, gamma, delta, epsilon, varepsilon, zeta, eta, theta, vartheta, kappa, varkappa, lambda, mu, nu, xi, pi, varpi, rho, varrho, sigma, varsigma, tau, upsilon, phi, varphi, chi, psi, omega$, and the 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 $ shudh@JNEC LATEX@ JNEC 28 / 30
  35. 35. Recall Symbols AdEqn Greek Letters Example with greek letters ... and something more If Z1,...,Zk are independent, standard normal random variables, then the sum of their squares, Q = k i=1 Z2 i , is distributed according to the ξ2 distribution with k degrees of freedom. This is usually denoted as Q ∼ χ2 (k) or Q ∼ χ2 k. The chi-squared distribution has one parameter: k – a positive integer that specifies the number of degrees of freedom (i.e. the number 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) text{or} Q sim chi^2_k . ] The chi-squared distribution has one parameter: $k$ -- a positive integer that specifies the number of degrees of freedom (i.e. the number of $Z_i$’s) shudh@JNEC LATEX@ JNEC 29 / 30
  36. 36. Recall Symbols AdEqn Greek Letters Example with greek letters ... and something more If Z1,...,Zk are independent, standard normal random variables, then the sum of their squares, Q = k i=1 Z2 i , is distributed according to the ξ2 distribution with k degrees of freedom. This is usually denoted as Q ∼ χ2 (k) or Q ∼ χ2 k. The chi-squared distribution has one parameter: k – a positive integer that specifies the number of degrees of freedom (i.e. the number 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) text{or} Q sim chi^2_k . ] The chi-squared distribution has one parameter: $k$ -- a positive integer that specifies the number of degrees of freedom (i.e. the number of $Z_i$’s) shudh@JNEC LATEX@ JNEC 29 / 30
  37. 37. Recall Symbols AdEqn Greek Letters 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 shudh@JNEC LATEX@ JNEC 30 / 30

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