A Handbook of Essential Mathematical Formulae

MathsPrelims308

6/3/08

12:04 pm

Page 1

A handbook of essential mathematical formulae Alan Davies and Diane Crann

University of Hertfordshire Press

MathsPrelims308

6/3/08

12:04 pm

Page 2

First published in Great Britain in 2004 by University of Hertfordshire Press Learning and Information Services University of Hertfordshire College Lane Hatfield Hertfordshire AL10 9AB Reprinted in 2008 © University of Hertfordshire Higher Education Corporation

All rights reserved. No part of this book may be reproduced or utilised in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from the author.

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library

ISBN 978-1-902806-41-9

Design by Diane Crann Cover design by John Robertshaw, AL5 2JB Printed in Great Britain by Antony Rowe Ltd, Chippenham SN14 6LH

Contents 1 Algebra, Trigonometry and Geometry 1.1 Indices and logarithms . . . . . . . . . 1.2 Factors and roots of equations . . . . . 1.3 Partial fractions . . . . . . . . . . . . . 1.4 Permutations and combinations . . . . 1.5 Trigonometric functions . . . . . . . . 1.6 Hyperbolic functions . . . . . . . . . . 1.7 Geometry . . . . . . . . . . . . . . . . 1.8 Conic sections . . . . . . . . . . . . . . 1.9 Mensuration . . . . . . . . . . . . . . . 1.10 Complex numbers . . . . . . . . . . . 1.11 Inequalities . . . . . . . . . . . . . . . 2 Determinants and Matrices 2.1 Determinants . . . . . . . . . 2.2 Matrices . . . . . . . . . . . . 2.3 Systems of equations . . . . . 2.4 Eigenvalues and eigenvectors

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1 1 2 2 3 3 5 5 6 7 7 9

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10 10 11 12 13

3 Vector Algebra 14 3.1 Vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Vector products . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Polar coordinates in two and three dimensions . . . . . . . . 15 4 Calculus 4.1 Rules for manipulation of derivatives 4.2 Standard derivatives and integrals . 4.3 Definite integrals . . . . . . . . . . . 4.4 Radius of curvature of a curve . . . . 4.5 Stationary points . . . . . . . . . . . 4.6 Limits and series . . . . . . . . . . . 4.7 Multiple integration . . . . . . . . . 4.8 Applications of integration . . . . . .

iii

and integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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17 17 19 21 23 23 24 25 27

4.9

Kinematics and dynamics . . . . . . . . . . . . . . . . . . . .

31

5 Ordinary Differential Equations 32 5.1 First order equations . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Second order equations . . . . . . . . . . . . . . . . . . . . . . 33 5.3 Higher order equations . . . . . . . . . . . . . . . . . . . . . . 34 6 Fourier Series

35

7 Vector Calculus 37 7.1 grad, div and curl . . . . . . . . . . . . . . . . . . . . . . . . 37 7.2 Integral theorems of the vector calculus . . . . . . . . . . . . 40 8 Tables of Transforms 8.1 The Laplace transform . . . . . . . . . . . . . . . . . . . 8.2 The Z-transform . . . . . . . . . . . . . . . . . . . . . . 8.3 The Fourier transform . . . . . . . . . . . . . . . . . . . 8.4 Fourier sine and cosine transforms . . . . . . . . . . . . 8.5 Some periodic functions: Laplace transforms and Fourier 9 Numerical Methods 9.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . 9.2 Finite difference operators . . . . . . . . . . . . . . . 9.3 Non-linear algebraic equations . . . . . . . . . . . . . 9.4 Numerical integration . . . . . . . . . . . . . . . . . 9.5 Numerical solution of ordinary differential equations 9.6 Systems of linear equations, n × n . . . . . . . . . . 9.7 Chebyshev polynomials . . . . . . . . . . . . . . . . 9.8 Numerical eigenvalues and eigenvectors . . . . . . . . 9.9 Least squares approximation . . . . . . . . . . . . .

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50 50 51 52 52 54 55 57 57 59

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10 Statistics 60 10.1 Sample statistics . . . . . . . . . . . . . . . . . . . . . . . . . 60 10.2 Regression and correlation . . . . . . . . . . . . . . . . . . . . 61 10.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 11 S. I. Units (Syst` eme International d’Unit´ es) 11.1 Fundamental units . . . . . . . . . . . . . . . 11.2 S. I. Prefixes and multiplication factors . . . . 11.3 Basic and derived units . . . . . . . . . . . . 11.4 Values of some physical constants . . . . . . . 11.5 Useful masses . . . . . . . . . . . . . . . . . . 11.6 Astronomical constants . . . . . . . . . . . . 11.7 Mathematical constants . . . . . . . . . . . . 11.8 The Greek alphabet . . . . . . . . . . . . . . iv

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64 64 65 65 67 67 69 69 70

Thoughout the handbook√the symbol j is used for the unit imaginary number i.e. j 2 = −1 or j = −1. Alternatively the symbol i is frequently used instead of j.

v

vi

Chapter 1

Algebra, Trigonometry and Geometry A polynomial of degree n is an expression of the form P (x) = an xn + an−1 xn−1 + ... + a2 x2 + a1 x1 + a0 Sigma

n P

ai = a1 + a2 + a3 + ... + an−1 + an

i=1

Factorial n! = n (n − 1) (n − 2) ...3.2.1 Finite series n X 1 k = n(n + 1) 2 k=1

n X

k2 =

k=1

n X

3

k =

k=1

1.1

1 n(n + 1)(2n + 1) 6

n X k=1

k

!2

=

1 2 n (n + 1)2 4

Indices and logarithms

If x > 0 then we have the following properties: x0 = 1,

x−1 = 1/x,

xm+n = xm xn ,

xm−n = xm /xn ,

xmn = (xm )n

If a > 0 and a 6= 1 then loga 1 = 0,

loga (x−1 ) = loga (1/x) = − loga x

loga x + loga y = loga (xy),

m loga x = loga (xm )

loga x − loga y = loga (x/y) 1

1

xn =

√ n

x

The logarithm and power functions are inverse functions, i.e. if x = loga y then y = ax and if x = ay then y = loga x Change of base loga x = logb x/ logb a Logarithms to base e, i.e. loge x, are often written ln x. Such logarithms are called natural logarithms. e is the exponential constant given by e = 1 + 1!1 + 2!1 + 3!1 + . . . = 2.71828 . . ..

1.2

Factors and roots of equations (a ± b)2 = a2 ± 2ab + b2 a2 − b2 = (a + b) (a − b)
a3 ± b3 = (a ± b) a2 ∓ ab + b2
an − bn = (a − b) an−1 + an−2 b + an−3 b2 + ... + abn−2 + bn−1

Quadratic equation ax2 + bx + c = 0 with roots α, β i h p α, β = −b ± b2 − 4ac /2a α + β = −b/a,

αβ = c/a

Cubic equation ax3 + bx2 + cx + d = 0 with roots α, β, γ α + β + γ = −b/a,

αβ + βγ + γα = c/a,

αβγ = −d/a

f (a) = 0 if and only if (x − a) is a factor of f (x). Remainder Theorem Suppose P (x) = an xn + an−1 xn−1 + . . . + a1 x + a0 is a polynomial of degree n and that a is a root of the equation P (x) = 0. Then (x − a) is a factor of P (x).

1.3

Partial fractions

If the degree of the polynomial f (x) is less than that of the denominator then: f (x) (x + a) (x + b) ... f (x) 2 (ax + bx + c) (dx + e) f (x) (ax + b) (cx + d)2

= = =

A B + + ... x+a x+b Ax + B C + 2 ax + bx + c dx + e A B C + + ax + b cx + d (cx + d)2

2

1.4

Permutations and combinations

The number of ways of selecting r objects from n objects with due regard to order is n! n Pr = (n − r)! The number of ways of selecting r objects from n objects without regard to order is   n! n (sometimes written n Cr ) = r (n − r)!r! Properties     n n = , r n−r



n+1 r



=



n r



+

Binomial Theorem For any positive integer, n,  n  X n (a + b) = an−r br r n

r=0

1.5

Trigonometric functions

Degrees and Radians π ◦ ◦ 360◦ = 2π rad 1◦ = 180 rad 1 rad = ( 180 π ) ≈ 57.296 cos nπ = (−1)n , sin nπ = 0 cos [(2n + 1) π/2] = 0,

sin [(2n + 1) π/2] = (−1)n

√ cos (π/4) = sin (π/4) = 1/ 2 cos (π/3) = sin (π/6) = 1/2 √ cos (π/6) = sin (π/3) = 3/2 cos (π/2 − θ) = sin θ,

sin (π/2 − θ) = cos θ

Trigonometric identities cos−1 x = π/2 − sin−1 x, tan−1 x = π/2 − tan−1 (1/x) sin2 x + cos2 x = 1, 1 + tan2 x = sec2 x 1 + cot2 x = cosec 2 x sin (x + y) = sin x cos y + cos x sin y sin (x − y) = sin x cos y − cos x sin y 3



n r−1



cos (x + y) = cos x cos y − sin x sin y cos (x − y) = cos x cos y + sin x sin y tan x + tan y 1 − tan x tan y tan x − tan y tan (x − y) = 1 + tan x tan y

tan (x + y) =

cos 2x = cos2 x − sin2 x = 2 cos2 x − 1 = 1 − 2 sin2 x 2 tan x sin 2x = 2 sin x cos x, tan 2x = 1 − tan2 x

sin 3x = 3 sin x − 4 sin3 x

cos 3x = 4 cos3 x − 3 cos x,

x−y x+y cos 2 2 x+y x−y sin x − sin y = 2 cos sin , 2 2 x+y x−y cos x + cos y = 2 cos cos 2 2 x+y x−y cos x − cos y = −2 sin sin 2 2

sin x + sin y = 2 sin

cos x cos y = 12 [cos (x + y) + cos (x − y)] sin x sin y = 12 [cos (x − y) − cos (x + y)] sin x cos y = 12 [sin (x + y) + sin (x − y)] sin2 x = 12 (1 − cos 2x)   p √ sin−1 x ± sin−1 y = sin−1 x 1 − y 2 ± y 1 − x2   p √ cos−1 x ± cos−1 y = cos−1 xy ∓ 1 − x2 1 − y 2

cos2 x =

1 2

(1 + cos 2x) ,

tan−1 x ± tan−1 y = tan−1 [(x ± y) / (1 ∓ xy)]

x If t = tan , then 2 2t sin x = , 1 + t2

cos x =


cos x = ejx + e−jx /2,

De Moivre’s Theorem

1 − t2 , 1 + t2

tan x =

2t 1 − t2


sin x = ejx − e−jx /2j

(cos x + j sin x)n = cos nx + j sin nx ejx = cos x + j sin x,

e−jx = cos x − j sin x

4

1.6

Hyperbolic functions

cosh x = (ex + e−x ) /2, sinh x = (ex − e−x ) /2 ex = cosh x + sinh x, e−x = cosh x − sinh x e2x − 1 sinh x ex − e−x 1 − e−2x = = x = cosh x e + e−x e2x + 1 1 + e−2x   √ sinh−1 x = ln x + x2 + 1   √ cosh−1 x = ± ln x + x2 − 1   1 1+x −1 tanh x = ln 2 1−x

tanh x =

Relationship with trigonometric functions cos jx = cosh x, cosh jx = cos x sin jx = j sinh x, sinh jx = j sin x Osborne’s rule An identity involving hyperbolic functions may be obtained from the equivalent trigonometric identity by replacing the trigonometric functions with the corresponding hyperbolic functions and changing the sign of a product (or implied product) of two sines. e.g. cosh2 x − sinh2 x = 1, 1 − tanh2 x = sech2 x N.B. The Maclaurin series in powers of x for the hyperbolic functions may be found from the Maclaurin series for the corresponding trigonometric function by changing the sign of a product of two x’s e.g. x2 x4 x2n cosh x = 1 + + + ··· + + ··· 2! 4! (2n)! sinh−1 x = x −

1.7

x3 3x5 5x7 (2n)! x2n+1 + − + . . . + (−1)n 2n + ... 6 40 112 2 (n!)2 (2n + 1)

Geometry

5

Pythagoras’ theorem a2 + b2 = c2 Theorems The angle sum of a triangle is 180◦ or π rad. The sum of the interior angles of an n-sided polygon is (2n − 4) × 90◦ or (n − 2)π rad. The exterior angle of a triangle is equal to the sum of the interior opposite angles. The angle subtended by a diameter at the circumference of a circle is 90◦ . The angle subtended by a chord at the centre of a circle is twice the angle subtended by the same chord in the opposite segment. The diagonals of a parallelogram bisect each other.

1.8

Conic sections

A conic section is the locus of a point that moves in a plane so that the ratio (the eccentricity) of its distance from a fixed point (the focus) in the plane, to its distance from a fixed line (the directrix), is a constant, ǫ.

1. Parabola (ǫ = 1) ; focus at (a, 0), directrix x = −a Cartesian equation: y 2 = 4ax Parametric equation: x = at2 , y = 2at 2. Ellipse (ǫ < 1); foci at (±aǫ, 0), directrices x = ±a/ǫ Major axis of length 2a, minor axis of length 2b
1 x2 y 2 Cartesian equation: 2 + 2 = 1 with b = a 1 − ǫ2 2 a b Parametric equation: x = a cos θ, y = b sin θ 3. Hyperbola (ǫ > 1); foci at (±aǫ, 0), directrices at x = ±a/ǫ
1 x2 y 2 Cartesian equation: 2 − 2 = 1 with b = a ǫ2 − 1 2 a b Parametric equation: x = a sec θ, y = b tan θ Rectangular hyperbola referred to its asymptotes as axes: xy = c2 4. Circle (ǫ = 0) Cartesian equation x2 + y 2 = a2 Parametric equation x = a cos θ, y = a sin θ

6

The polar equation for these three conic sections with the pole at a focus is 1 = 1 + ǫ cos θ r The general conic The equation ax2 + 2hxy + by 2 + 2gx + 2f y + c = 0 represents: a circle if a = b and h = 0 a pair of straight lines if af 2 + bg2 + ch2 = 2f gh + abc a parabola if h2 = ab an ellipse if h2 < ab a hyperbola if h2 > ab a rectangular hyperbola if a + b = 0

1.9

Mensuration

Circle, radius r: perimeter is 2πr, area is πr 2 For a segment of angular width θ (radians), arc length is rθ and area is 12 r 2 θ p Ellipse, axes 2a and 2b: perimeter is approximately 2π (a2 + b2 ) /2, area is πab Cylinder , radius r, height h: surface area is 2πr (h + r), volume is πr 2 h Cone, base radius r, height h, slant height l: curved surface area is πrl, volume is πr 2 h/3 Sphere, radius r: area is 4πr 2 , volume is 4πr 3 /3, area cut off by parallel planes distance h apart is 2πrh Triangle, ABC sides a, b, c: area of triangle isp ∆ = 21 bc sin A = s (s − a) (s − b) (s − c), (Heron’s formula) where 2s = a + b + c Radius of circumcircle is R = abc/4∆ Radius of inscribed circle is r = ∆/s sine rule

a b c = = = 2R sin A sin B sin C

cosine rule a2 = b2 + c2 − 2bc cos A p sin (A/2) = p{(s − b) (s − c) /bc} cos (A/2) = {(s − a) /bc}

1.10

Complex numbers

The algebra of complex numbers follows the same rules as that for real numbers with the addition of the unit imaginary number j which has the 7

property j 2 = −1 or j =

√ −1.

Cartesian form z = x + jy where x is called the real part of z, Re(z), and y is called the imaginary part of z, Im(z). The Argand diagram is a geometric representation of the complex number z = x + jy, the point (x, y) represents z. Polar form z = r (cos θ + j sin θ) where (r, θ) are the polar coordinates of (x, y) Exponential form z = rejθ Complex conjugate z¯ = x − jy = r (cos θ − j sin θ) = re−jθ Modulus |z| =

p

x2 + y 2 ,

z¯ z = r 2 = x2 + y 2

Argument (principal value) arg z = θ, (−π < θ ≤ π). De Moivre’s theorem (cos θ + j sin θ)n = cos nθ + j sin nθ Euler’s formulae ejθ = cos θ + j sin θ,
e−jθ = cos θ − j sin θ
1 cos θ = 21 ejθ + e−jθ , sin θ = 2j ejθ − e−jθ

Complex roots If z = rejθ then the n complex roots of z are given by z 1/n = r 1/n exp{j

θ + 2kπ } k = 0, 1, 2, . . . , n − 1 n

These roots are equally spaced around the circle, radius r 1/n , centred on the origin. Fundamental theorem of algebra A polynomial, P (z), of degree n given by P (z) = an z n + an−1 z n−1 + · · · + a1 z + a0 (an 6= 0) can be factorized into n complex factors: P (z) = an (z − z1 ) (z − z2 ) . . . (z − zn ) . The numbers z1 , z2 , . . . , zn are called the roots of the equation P (z) = 0. If the coefficients a0 , a1 , . . . , an are all real then the complex roots occur in conjugate pairs.

8

Complex variable If f (z) = u(x, y) + jv(x, y) is an analytic function of the complex variable z = x + jy, then u and v satisfy the Cauchy-Riemann equations ∂v ∂u = ∂x ∂y

1.11

∂v ∂u =− ∂y ∂x

Inequalities

Basic rules based on ‘greater than’ (the same rules apply to >, y ⇔x+a>y+a if p > 0 then x > y ⇔ px > py if n < 0 then x > y ⇔ nx < ny if a > b and x > y then a + x > b + y if x > y and y > z then x > z √ √ √ x+y ≤ x+ y √ √ if x, y > 0 then x < y ⇔ x2 < y 2 ⇔ x < y if x > 1 then xm > xn ⇔ m > n

if 0 < x < 1 then xm > xn ⇔ m < n if x > 1 then x1/m > x1/n ⇔ m > n

Bernoulli’s inequality If x > −1 then (1 + x)n ≥ 1 + nx Arithmetic mean

1 2 (x

+ y)

Geometric mean 1 2 (x

+ y) ≥

√

xy

√ xy

Triangle inequality |x + y| ≤ |x| + |y| |x| − |y| ≤ ||x| − |y|| ≤ |x − y| Cauchy Schwarz inequality |u.v| ≤ kukkvk Minkowski inequality ku + vk ≤ kuk + kvk

9

Chapter 2

Determinants and Matrices 2.1

Determinants

a 2 × 2 : c a11 3×3 : a21 a31

b = ad − bc d a12 a13 a a a22 a23 = a11 22 23 a a33 32 a32 a33 a11 a12 · · · a1n a21 a22 · · · a2n n×n:∆= . .. .. .. . . an1 an2 · · · ann

−a12 a21 a23 a31 a33

+a13 a21 a22 a31 a32



The minor , αij , of the element aij is the (n − 1)th order determinant formed from ∆ by omitting the row and the column containing aij . The cofactor , Aij , of the element aij is given by Aij = (−1)i+j αij . The value of the n × n determinant is ∆ = ai1 Ai1 + ai2 Ai2 + . . . + ain Ain (expansion by ith row) = a1j A1j + a2j A2j + . . . + anj Anj (expansion by jth column) valid for i, j = 1 to n. Properties 1. The value of the determinant is unchanged if its rows and columns are interchanged. 2. The value of a determinant is unchanged if a multiple of any row (or column) is added to any other row (or column). 3. If the elements of two rows (or columns) are proportional, i.e. linearly dependent, then the value of the determinant is zero. 10

4. If the elements of two rows (or columns) are interchanged then the resulting determinant has the same numerical value but its sign is changed. 5. If the elements of a row (or column) are all multiplied by a constant k, then the value of the determinant is also multiplied by k. Cramer’s rule (efficient for only two equations in two unknowns or three equations in three unknowns). a11 a12 a13 ∆ = a21 a22 a23 a31 a32 a33

a11 x1 + a12 x2 + a13 x3 = h1 a21 x1 + a22 x2 + a23 x3 = h2 a31 x1 + a32 x2 + a33 x3 = h3 h1 a12 a13 ∆1 = h2 a22 a23 h3 a32 a33 The solution is

,

a11 h1 a13 ∆2 = a21 h2 a23 a31 h3 a33

x1 = ∆1 /∆,

2.2

x2 = ∆2 /∆,

,



a11 a12 h1 ∆3 = a21 a22 h2 a31 a32 h3

x3 = ∆3 /∆

Matrices

The m × n matrix is written as  a11  a21  A= .  ..

a12 a22 .. .

··· ···

a1n a2n .. .

am1 am2 · · ·

amn

    

or A = [aij ] Transpose matrix AT = [aji ] is n × m A is called a symmetric matrix if AT = A If A and B are of the same order then αA + βB = [αaij + βbij ] If A is m × r and B is r × n then " r # X AB = aik bkj is m × n k=1

Unit matrix or identity matrix I = [δij ] where the Kronecker δ is given by  0, i 6= j δij = 1, i = j 11

Null matrix or zero matrix 0 = [0] For a square matrix A if det A 6= 0 then the inverse matrix , A−1 , exists and is given by A−1 = det1 A adj A, where adj A is the transposed matrix of cofactors. The inverse matrix has the property AA−1 = A−1 A = I If det A = 0 then A is said to be singular. A is called an orthogonal matrix if AT = A−1 For a 2 × 2 matrix  −1   1 a b d −b = c d ad − bc −c a For an n × n matrix det A = det AT and det (kA) = kn det A The rank of a matrix, A, is the largest non-zero sub-determinant of A. A set of vectors {v1 , v2 , . . . , vn } for a finite dimensional space is said to be linearly dependent if there exist scalars α1 , α2 , . . . , αn , not all zero such that α1 v1 + α2 v2 + . . . + αn vn = 0 otherwise the set is linearly independent. Suppose that {v1 , v2 , . . . , vn } is an arbitrary basis for a finite dimensional inner product space. Then the orthogonal basis {w1 , w2 , . . . , wn } may be obtained by the Gram Schmidt process wn = vn −

2.3

< vn , w1 > < vn , w2 > < vn , wn−1 > w1 − w2 −, . . . , − wn−1 2 2 ||w1 || ||w2 || ||wn−1 ||2

Systems of equations

For a system of m linear equations in n unknowns we write in matrix form Ax = b The explicit form is a11 x1 + a12 x2 + a21 x1 + a22 x2 +

··· ··· .. .

am1 x1 + am2 x2 + · · ·

+a1n xn +a2n xn

= =

b1 b2

+amn xn = bm

1. If rank [A,b] > rankA then there is no solution. 2. If rank [A,b] = rankA = n then there is a unique solution. In the case m = n, the solution is given by x = A−1 b. 12

3. If rank [A,b] = rankA < n then there is an infinite number of solutions. It follows from 2. and 3. that, for a square matrix A, the homogeneous set of equations Ax = 0 has a non-trivial solution if and only if detA = 0.

2.4

Eigenvalues and eigenvectors

For a square matrix A, the eigenvalues λi and corresponding eigenvectors xi are given by Axi = λi xi Thus the eigenvalues satisfy det [A − λI] = 0 Properties Matrix pA AT Eigenvalues pλi λi

A−1 Ak A + qI [A + pI]−1 1/λi λki λi + q 1/ (λi + p)

In each case the corresponding eigenvectors are xi . The trace of the matrix A, is given by trA = a11 + a22 + . . . + ann X Y λi = trA, λi = det A

Suppose that λ is an eigenvalue of A then λ = 0 iff det A = 0.

13

Chapter 3

Vector Algebra 3.1

Vector addition

Triangle law a+b = c ˆ Component form a + b = (a1 + b1 )ˆı + (a2 + b2 ) ˆj + (a3 + b3 ) k

3.2

Vector products

θ is the angle between the two vectors a and b, n ˆ is the unit vector perpendicular to both a and b in the sense that a, b and n ˆ , in that order, form a right-handed system. a.b = ab cos θ = b.a, 14

a×b = ab sin θˆ n

In component form ˆ a = a1ˆı + a2ˆj + a3 k,

ˆ b = b1ˆı + b2ˆj + b3 k

a.b = a1 b1 + a2 b2 + a3 b3 ˆ ˆı ˆj k a×b = a1 a2 a3 = −b×a b1 b2 b3 p Magnitude: |a| = a = a21 + a22 + a23 1 Unit vector in the direction of the vector a is ˆ a= a= |a| a1 a2 a3 a. (b×c) = (a×b) .c = b1 b2 b3 = [a b c] c1 c2 c3 a × (b×c) = (a.c) b − (a.b) c

(a×b) × c = (a.c) b − (b.c) a a.c a.d (a×b) . (c×d) = b.c b.d

(a×b) × (c×d) = [a b c] c − [a b c] d

3.3

Polar coordinates in two and three dimensions

Plane polar coordinates (r, θ) x = r cos θ,

15

y = r sin θ

Cylindrical polar coordinates (R, φ, z) x = R cos φ,

y = R sin φ



 ˆ ˆ ˆ Unit vectors R, φ, k form a right-handed system. Relationships with Cartesian unit vectors: ˆ = cos φˆı + sin φˆj, φ ˆ = − sin φˆı + cos φˆj R

Spherical polar coordinates (r, θ, φ) x = r sin θ cos φ,

y = r sin θ sin φ,

z = r cos θ

  ˆφ ˆ form a right-handed system. Unit vectors ˆ r, θ, Relationship with cylindrical unit vectors: ˆ θˆ = − cos θ R ˆ ˆ + cos θ k, ˆ + sin θ k ˆ r = sin θ R 16

Chapter 4

Calculus 4.1

Rules for manipulation of derivatives and integrals

Differentiation d du dv (au + bv) = a +b Linearity rule dx dx dx Product rule Quotient rule Leibniz rule n

d du dv (uv) = v+u dx dx dx   d u du dv = v −u /v 2 dx v dx dx n

D (uv) = uD v +



n 1



(Du) D

n−1

v+



n 2



(Du) D n−2 v + . . . + (D n u) v

 n  X d n = (D r u) D n−r v, where D ≡ r dx r=0

Chain rule for ordinary differentiation (function of a function rule) Leibniz notation y = y (x) and x = x (u) ; dy dy dx = du dx du Function notation

[F (g (x))]′ = F ′ (g (x)) g′ (x)

Chain rule for partial differentiation F (u, v) = f (x, y) with x = x (u, v) , y = y (u, v) ∂f ∂x ∂f ∂y ∂F = + ; ∂u ∂x ∂u ∂y ∂u

∂F ∂f ∂x ∂f ∂y = + ∂v ∂x ∂v ∂y ∂v 17

Total differential df =

∂f ∂f dx + dy ∂x ∂y

Integration Integration by parts Z

dv u dx = uv − dx

Z

v

du dx dx

Integration by substitution Z Z dx f (x) dx = F (u) du where F (u) = f (x (u)) du Z Z

f ′ (x) dx = ln |f (x)| f (x)

f ′ (x) [f (x)]n dx =

[f (x)]n+1 , n+1

Differentiation of integrals Z v(x) If I (x) = F (x, t)dt, then

n 6= −1

u(x)

dI dv du = F (x, v(x)) − F (x, u(x)) + dx dx dx

18

Z

v(x) u(x)

∂F (x, t)dt ∂x

4.2

Standard derivatives and integrals Table of derivatives and integrals 1

df /dx

f (x)

F (x) =

Z

f (x)dx

(add arbitrary constant where necessary) nxn−1

xn

xn+1 (n 6= −1) n+1

(4.1)

1 x2 ex

1 x ex

ln |x|

(4.2)

ax ln a

ax

−

1 x

ln x

ex ax ln a

(4.3) (4.4)

x(ln x − 1)

(4.5)

1 x ln b cos x

logb x

x(logb x − logb e)

(4.6)

sin x

(4.7)

− sin x

cos x

− cos x sin x

(4.8)

ln | sec x|  ln | sec x + tan x| ln | tan(x/2 + π/4)|  ln |cosecx − cot x| ln | tan(x/2)|

(4.9)

2

sec x

tan x

sec x tan x

sec x

−cosecx cot x

cosecx

−cosec2 x cot x 1 √ sin−1 x 2 1−x −1 √ cos−1 x 2 1−x 1 tan−1 x 1 + x2 cosh x sinh x sinh x

cosh x

sech2 x

tanh x

ln | sin x|

x sin−1 x + x cos−1 x − x tan−1 x −

p

1 − x2

p

1 − x2

1 ln(1 + x2 ) 2

(4.10) (4.11) (4.12) (4.13) (4.14) (4.15)

cosh x

(4.16)

sinh x  ln cosh x ln(ex + e−x )

(4.17) (4.18) continued. . .

19

Table of derivatives and integrals 2

df /dx

−sechx tanh x −cosechx coth x −cosech2 x 1 1 + x2 1 √ 2 x −1 1 1 − x2 √

x (a2

3

− x2 ) 2

f (x)

sechx

cosechx coth x sinh−1 x cosh−1 x tanh−1 x 1 √ 2 a − x2

−2x (x2 + a2 )2 a2 − 2x2

1 x2 + a2 1 √ x x2 − a2

−x

3

1 √ 2 a + x2

3 2

1 √ 2 x − a2

3

x2 (x2 − a2 ) 2 (a2

+ x2 ) 2 −x

(x2 − a2 ) (a2

2x − x2 )2

(a2

1 − x2 )

F (x) =

Z

f (x)dx

(add arbitrary constant where necessary)   2 tan−1 (tanh x/2) 2 tan−1 ex (4.19)  tan−1 (sinh x)   ln | tanh(x/2)| − ln |cosechx + coth x| (4.20)  ln |(ex − 1)/(ex + 1)|  ln | sinh x| (4.21) ln |ex − e−x | p (4.22) x sinh−1 x − 1 + x2 x cosh−1 x −

p

x2 − 1

1 x tanh−1 x − ln(1 − x2 ) 2  x −1  sin a  − cos−1 x a 1 −1 x tan a a x 1 sec−1 a a ( x sinh−1 a √ ln |x + x2 + a2 | ( x cosh−1 a √ ln |x + x2 − a2 |  1 −1 x    a tanh a a + x  1   ln 2a a − x

20

(4.23)

(4.24) (4.25) (4.26) (4.27) (4.28) (4.29)

(4.30)

continued. . .

Table of derivatives and integrals 3

df /dx

f (x)

F (x) =

Z

f (x)dx

(add arbitrary constant where necessary) p √ xp 2 a2 x √ x ± a2 ± x2 ± a2 ln |x + x2 ± a2 | (4.31) 2 2 x2 ± a2 p √ −x a2 x x √ a2 − x2 + a2 − x2 sin−1 (4.32) 2 2 2 2 a a −x eax (a cos bx + b sin bx) (4.33) eax (a cos bx − b sin bx) eax cos bx a2 + b2 eax eax (a sin bx + b cos bx) eax sin bx (a sin bx − b cos bx) (4.34) a2 + b2

4.3

Definite integrals

Wallis’s formulae (reduction formulae also hold if upper limit is π or 2π.) Sn = Cn =

π 2

Z

Z0 π 2

0

Im,n =

Z

π 2

n−1 Sn−2 n n−1 Cn−2 . cosn θdθ = n

sinn θdθ =

sinm θ cosn θdθ =

0

S n = Cn =

m−1 n−1 Im−2,n = Im,n−2 . m+n m+n

(n − 1)(n − 3)(n − 5) . . . p; n(n − 2)(n − 4) . . .

(m − 1)(m − 3) . . . (n − 1)(n − 3) . . . q; (m + n)(m + n − 2)(m + n − 4) . . .  π  π n even 2 2 both m and n even where p = q= 1 n odd 1 otherwise For all integer values of m and n: Z π sin mθ cos nθdθ = 0 Im,n =

−π

Z

π

sin mθ sin nθdθ = −π

Z

π

cos mθ cos nθdθ = πδmn ,

−π

(δmn is the Kronecker δ see page 11)

21

m, n 6= 0

Z Z

∞

−x2

e

dx =

√

Z

π;

−∞ ∞ −∞

cos ax dx = πe−|a| ; 1 + x2

Z

∞

−(ax2 +bx)

e

dx =

−∞ ∞ −∞

r

π exp(b2 /4a) a

a>0

sin ax dx = 0 1 + x2

Error function 2 erf(x) = √ π Complementary error function

Z

x 0

2 erfc(x) = 1 − erf(x) = √ π erf(0) = erfc(∞) = 0,

2

e−t dt

Z

∞

2

e−t dt

x

erf(∞) = erfc(0) = 1

For the Chebyshev polynomials Tn (x), Tm (x) (see page 57)  Z 1 n 6= m  0 Tn (x)Tm (x) √ dx = π n=m=0  1 − x2 −1 π/2 n = m 6= 0 Gamma function Z Γ(t) =

∞

e−x xt−1 dx

t 6= 0, −1, −2, . . .

0

Γ(t) = (t − 1)Γ(t − 1) = (t − 1)(t − 2)(t − 3) . . . Γ(t − [t]) √ Γ( 21 ) = π Γ(n + 1) = n! when n is an integer

Beta function B(s, t) =

Z

0

1

xs−1 (1 − x)t−1 dx = Γ(s)Γ(t)/Γ(s + t)

Legendre polynomial Z 2π p
n 1 1 dn Pn (x) = [x + x2 − 1 cos θ]n dθ = n x2 − 1 n 2π 0 2 n! dx

P0 (x) = 1,

P1 (x) = x,

P2 (x) = (3x2 − 1)/2,

P3 (x) = (5x3 − 3x)/2, . . .

Generating function 1

(1 − 2xh + h2 )− 2 = 22

∞ X

n=0

Pn (x)hn

Pn (x) is a solution of Legendre’s equation (1 − x2 )y ′′ − 2xy ′ + n(n + 1)y = 0. A second, linearly independent, solution is the Legendre function of the second kind Qn (x) which is not a polynomial. Bessel function Jn (x) =

xn √ 2n−1 πΓ(n + 12 )

Z

π 2

cos(x sin θ) cos2n θdθ.

0

Jn (x) is a solution of Bessel’s equation x2 y ′′ + xy ′ + (x2 − n2 )y = 0 A second, linearly independent, solution, Yn (x), is the Bessel function of the second kind. The modified Bessel function, In (x), satisfies the equation x2 y ′′ + xy ′ − (x2 + n2 )y = 0 and a second linearly independent solution is Kn (x), the modified Bessel function of the second kind.

4.4

Radius of curvature of a curve

1 ds . Curvature: κ = . dψ ρ [1 + (y ′ )2 ]3/2 Cartesian coordinates: ρ = . |y ′′ | ((x) ˙ 2 + (y) ˙ 2 )3/2 Parametric coordinates: ρ = |x¨ ˙y − x ¨y| ˙ Intrinsic coordinates: ρ =

4.5

Stationary points

Functions of one variable The function f (x) has stationary points given by f ′ (x) = 0. The stationary point, (x0 , f (x0 )), is a local maximum if f ′′ (x0 ) < 0 or a local minimum if f ′′ (x0 ) > 0. If f ′′ (x0 ) = 0 then the second derivative test is inconclusive and we consider the sign of f ′ (x ± ǫ) where ǫ is small and positive. If f ′′ (x0 ) = 0 and f ′ (x0 ± ǫ) are both positive or both negative then x0 is a point of inflection. Functions of more than one variable Stationary points occur when the first partial derivatives vanish. For a function of two variables, f (x, y), stationary points are given by fx = fy = 0. The Hessian is given by fxx fxy H(x, y) = fyx fyy The stationary point is: a maximum if fxx < 0, fyy < 0 and H > 0, 23

a minimum if fxx > 0, fyy > 0 and H > 0, a saddle point if H < 0. Lagrange multipliers To find the stationary point of f (x, y) subject to the constraint g(x, y) = 0, find the stationary points of the function φ(x, y; λ) = f (x, y) + λg(x, y).

4.6

Limits and series tan x = 1, x→0 x

sin x = 1, x→0 x

lim

lim

α

lim x ln x = 0,

x→0

lim x

x→∞

De L’Hˆ opital’s rule

−α

lim

n→∞



1+

lim xα e−x = 0,

ln x = 0,

x n = ex . n

x→∞

(in all cases α > 0).

f (x) f ′ (x) = lim ′ , provided g′ (a) 6= 0. x→a g(x) x→a g (x)

If f (a) = g(a) = 0, then lim Taylor’s theorem

f (x) = f (a) + (x − a)f ′ (a) + where ǫn =

(x − a)n (n) (x − a)2 ′′ f (a) + . . . + f (a) + ǫn 2! n!

(x − a)n+1 (n+1) f (c) for some c ∈ (a, x). (n + 1)!

Taylor’s series for a function of one variable f (x) = f (a) + (x − a)f ′ (a) +

(x − a)n (n) (x − a)2 ′′ f (a) + . . . + f (a), 2! n!

or, writing x = a + h, f (a + h) = f (a) + hf ′ (a) +

h2 ′′ hn (n) f (a) + . . . + f (a) + . . . 2! n!

Maclaurin series (put a = 0, h = x in Taylor’s series) f (x) = f (0) + xf ′ (0) +

x2 ′′ xn (n) f (0) + . . . + f (0) + . . . 2! n!

Power series If the power series a0 + a1 x+ a2 x2 + . . . + an xn + . . . =

∞ X

an xn is convergent

n=0

for |x| < R then R is called the radius of convergence of the series. R = lim

n→∞

24

|an+1 | |an |

Some useful series: x2 x3 x4 xn + − + . . . + (−1)n+1 + . . . , −1 < x ≤ 1 2 2 3 3 4 4 n n x x x x − − − ... − − . . . , −1 < x ≤ 1 ln(1 − x) = −x − 2 3 4 n

ln(1 + x) = x −

n(n − 1) 2 n(n − 1)(n − 2) 3 x ± x + ... − 1 < x < 1 2! 3! [This is the binomial series. If n is a positive integer the series terminates after n + 1 terms, otherwise it converges if and only if |x| < 1.] (1 ± x)n = 1 ± nx +

(1 ∓ x)−1 = 1 ± x + x2 ± x3 + . . . + x2n ± x2n+1 + . . . , |x| < 1 x2 x3 x4 xn ex = 1 + x + + + + ... + + . . . , all x 2! 3! 4! n! x3 x5 x7 (−1)n x2n+1 sin x = x − + − + ... + + . . . , all x 3! 5! 7! (2n + 1)! x2 x4 x6 (−1)n x2n cos x = 1 − + − + ... + + . . . , all x 2! 4! 6! (2n)! x3 2x5 x7 π tan x = x + + + + . . . , |x| < 315 2 π 3 −1 15 sin−1 x = − cos x 2 x3 3x5 5x7 (2n)! x2n+1 = x+ + + + . . . + 2n + . . . , |x| < 1 6 40 112 2 (n!)2 (2n + 1) 3 5 2n+1 x x x tan−1 x = x − + + . . . + (−1)n+1 + . . . , |x| < 1 3 5 (2n + 1) To obtain the series for the corresponding hyperbolic functions, see the note following Osborne’s rule on page 5. Taylor’s series for a function of two variables f (a + h, b + k) = f (a, b) + {hfx (a + b) + kfy (a, b)} 1 + {h2 fxx (a, b) + 2hkfxy (a, b) + k2 fyy (a, b)} + . . . 2!   ∞ X 1 ∂ ∂ n = h +k f (a, b) n! ∂x ∂y n=0

4.7

Multiple integration Z Z

R

f (x, y)dxdy =

Z Z

  x, y dudv f (x(u, v), y(u, v)) J u, v ′

R

where the region R in the xy-plane is mapped to the region R′ in the uvplane and the Jacobian of the transformation is given by   xu y u x, y J = xv y v u, v 25

Plane polar coordinates J = r, so that dA = dxdy = rdrdθ

Volume and surface integrals Cylindrical polar coordinates dS = Rdφdz and dV = RdRdφdz

26

Spherical polar coordinates dS = r 2 sin θdθdφ and dV = r 2 sin θdrdθdφ

Line integrals Green’s theorem in the plane  Z Z Z  ∂Q ∂P − dxdy (P dx + Qdy) = ∂x ∂y R C where C is the closed curve bounding the region R. Z ∂Q ∂P (P dx + Qdy) is independent of the path if and only if = . ∂x ∂y C The line integral of F is independent of the path C joining A to B, if and only if there exists a potential function φ such that F = gradφ, or equivalently, if and only if curlF ≡ 0. In this case Z

4.8

F.ds = CAB

Z

B

A

dφ = φB − φA

Applications of integration

Arc length of a curve Z bp 1 + (y ′ )2 dx Cartesian form: a Z θ2 p Polar form: r 2 + (dr/dθ)2 dθ θ1 Z t2 p Parametric form: x˙ 2 + y˙ 2 dt t1

27

Solid of revolution The surface area of the solid of revolution obtained by one complete revolution of the curve y = f (x) about the x-axis is given by Z

2π

b

y a

p

1 + (y ′ )2 dx

The volume of the solid of revolution obtained by one complete revolution of the curve y = f (x) about the x-axis is given by π

b

Z

y 2 dx

a

Centroid The centroid of the volume of the solid of revolution above is at (¯ x, 0) where x ¯=

Z

b

2

xy dx

Z

b

y 2 dx

a

a

The centroid of the area between the curve y = f (x), the x−axis and the lines x = a, x = b is given by  Z b Z b Z b Z b 2 ydx , y¯ = y dx 2 ydx x ¯= xydx a

a

a

a

Mean value The mean value of f (x) between x = a, x = b is 1 b−a

Z

b

f (x)dx

a

Root mean square value The RMS of the function f (x) between x = a, x = b is given by 

1 b−a

Z

b

2

[f (x)] dx

a

 21

Moments of area and inertia The first and second moments of a plane area A about an axis are given respectively by Z Z Z Z rdA and r 2 dA A

A

where r is the distance from the axis of the element dA.

28

The moment of inertia, I, of a body, of density ρ and volume V , about an axis is given by Z r 2 ρdV V

where r is the distance from the axis of the element dV .

Parallel axes theorem If IG is the moment of inertia about an axis through the centroid and I is the moment of inertia about a parallel axis distance d away, then I = IG + md2 Table of moments of inertia 1 Uniform Body mass m

Axis

M.of I.

Perpendicular to bar through one end Perpendicular to bar through centroid

ma2 3 4ma2 3

2. Rectangular lamina sides 2a and 2b

Parallel to side 2b through centroid Perpendicular to plane through centroid

ma2 3 m(a2 + b2 ) 3

3. Rectangular solid edges 2a and 2b, depth 2d

Perpendicular to face 4ab through centroid

m(a2 + b2 ) 3 4m(a2 + b2 ) 3

1. Bar length 2a

Coinciding with edge 2d 4. Solid sphere radius r

Diameter Tangent

5. Spherical shell radii R and r

Diameter Tangent

2mr 2 5 7mr 2 5 2m(R5 − r 5 ) 3(R3 − r 3 ) m(7R5 − 5R2 r 3 − 2r 5 ) 5(R3 − r 3 ) continued. . .

29

Table of moments of inertia 2 Uniform Body mass m 6. Disc radius r

Axis

M.of I.

Diameter Perpendicular to disc through centroid

7. Annular ring radii R and r

Diameter Perpendicular to plane of ring through centroid

8. Solid cylinder radius r length 2a

Axis Perpendicular to axis through centroid

9. Cylindrical shell radii R and r length 2a

Axis Perpendicular to axis through centroid

mr 2 4 mr 2 2 m(R2 + r 2 ) 4 m(R2 + r 2 ) 2 mr 2 2

m

a2 r 2 + 3 4



m(R2 + r 2 ) 2  2  a R2 + r 2 m + 3 4

For entries 2, 6 and 7 the second moment of area J is given by J = IA/m where A is the area.

30

4.9

Kinematics and dynamics

Kinematics Position vector r = xi + yj + zk (Cartesian coordinates) = rˆ r (Plane polar coordinates) Velocity vector ˆ (Cartesian coordinates) r˙ = xˆ ˙ ı + y˙ˆj + z˙ k ˙ ˆ = rˆ ˙ r + rθθ (Plane polar coordinates) ˆ = s˙ t (Intrinsic coordinates) Acceleration vector ˆ ¨r = x ¨ˆı + y¨ˆj + z¨k (Cartesian coordinates) 1 d ˙ θˆ (Plane polar coordinates) = (¨ r − r θ˙ 2 )ˆ r+ (r 2 θ) r dt s2 = s¨ˆ t+ n ˆ (Intrinsic coordinates) ρ For uniform motion with angular velocity ω in a circle of radius a, speed is v = aω and acceleration is v 2 /a = aω 2 directed towards the centre. Dynamics Newton’s second law for a particle of mass m moving under the influence of an external force F F = m¨r For a rigid body moving about an axis with angular velocity ω and external torque Γ we have Γ = I ω˙ where I is the moment of inertia about the axis.

31

Chapter 5

Ordinary Differential Equations 5.1

First order equations dy = F (x, y) or y ′ = F (x, y) dx

Variables separable dy = f (x)g(y) dx

(g(y) 6= 0)

General Solution is Z

1 dy = g(y)

Z

f (x)dx + constant

Linear equation (integrating factor method) dy + P (x)y = Q(x) dx R Integrating factor: µ(x) = exp( P (x)dx) d Then (µ(x)y) = µ(x)Q(x) and the dx General Solution is µ(x)y = Initial-value problem Find the general solution of

Z

µ(x)Q(x)dx + constant

dy = F (x, y) subject to y(x0 ) = y0 dx i.e. solve the differential equation containing one arbitrary constant and determine the value of the constant to satisfy the initial condition that y = y0 when x = x0 . 32

5.2

Second order equations

The general linear equation with constant coefficients a

dy d2 y +b + cy = f (x) or ay ′′ + by ′ + cy = f (x) dx2 dx

Complementary function yc Solution of the associated homogeneous equation ay ′′ + by ′ + cy = 0

(a, b, c real)

Auxiliary equation aλ2 + bλ + c = 0 Three cases: Two real distinct roots, λ1 , λ2 Linearly independent solutions eλ1 x and eλ2 x ; yc = C1 eλ1 x + C2 eλ2 x One real repeated root, λ1 Linearly independent solutions eλ1 x and xeλ1 x ; yc = eλ1 x (C1 + C2 x) Complex conjugate roots, α ± jβ Linearly independent solutions eαx cos βx and eαx sin βx; yc = eαx (C1 cos βx + C2 sin βx) Particular integral yp Any solution of the non-homogeneous equation ay ′′ + by ′ + cy = f (x) Method of undetermined coefficients Choose a suitable trial function for yp according to the right-hand-side, f (x). f (x) trial function constant constant px + q lx + m 2 2 px + qx + r lx + mx + n polynomial degree n an xn + an−1 xn−1 + . . . + a1 x1 + a0 eωx keωx p cos ωx m cos ωx + n sin ωx q sin ωx m cos ωx + n sin ωx jωx Re[(p − jq)e ] Re[zejωx ] Beware: If the rhs includes one of the complementary function linearly independent solutions then choose x times the appropriate trial function.

33

General Solution The sum of the complementary function and a particular integral y = yc + yp . Initial-value problem ay ′′ + by ′ + cy = f (x) subject to y(x) = y0 , y ′ (x0 ) = y0′ Find the general solution containing two arbitrary constants and then use the initial conditions to determine the values of the constants.

5.3

Higher order equations

The general linear equation with constant coefficents an y (n) + an−1 y (n−1) + . . . + a2 y ′′ + a1 y ′ + a0 y = f (x) The complementary function, yc , is found in a similar manner to that on page 33 using the auxiliary equation an λn + an−1 λn−1 + . . . + a1 λ + a0 = 0 A particular integral, yp , is also found in a similar manner to that on page 33 using suitable trial functions. The general solution is y = yc + yp .

34

Chapter 6

Fourier Series Full-range series f (x) is defined for −l < x < l. The Fourier series for f (x) is given by: ∞ ∞ X 1 nπx X nπx F (x) = a0 + an cos + bn sin 2 l l n=1 n=1

where

1 a0 = l an =

1 l

Z

l

f (x) cos −l

Z

l

f (x)dx,

−l

nπx dx; l

bn =

1 l

Z

l

f (x) sin

−l

nπx dx l

F (x) extends f (x) periodically with period 2l. F (x) = f (x) at all points, −l < x < l, at which f is continuous. If f is discontinuous at x0 , then F (x0 ) = {f (x0 +) + f (x0 −)}/2. Half-range series f (x) is defined for 0 < x < l. The Fourier half-range cosine series is: ∞

X nπx 1 Fc (x) = a0 + an cos ; 2 l

2 an = l

n=1

Z

l

f (x) cos 0

nπx dx. l

Fc (x) extends f (x) periodically as an even function with period 2l. The Fourier half-range sine series is Z ∞ X nπx 2 l nπx Fs (x) = bn sin ; bn = f (x) sin dx. l l 0 l n=1 Fs (x) extends f (x) periodically as an odd function with period 2l. The Fourier series of some frequently occurring functions are given on page 47. 35

Complex Fourier series f (x) is defined for −l < x < l. The complex Fourier series for f (x) is given by F (x) =

∞ X

cn ejnπx/l

n=−∞

where  n>0  (an − jbn )/2 1 −jnπx/l cn = f (x)e dx = a /2 n=0  0 2l −l (a−n + jb−n )/2 n < 0 Z

l

The constant term, i.e. interval (−l, l).

1 2 a0

or c0 , is the average value of f (x) over the

Parseval’s theorem Z ∞ ∞ X 1 2 X1 2 1 l 2 2 (a + bn ) = |cn |2 [f (x)] dx = a0 + 2l −l 4 2 n n=−∞ n=1

36

Chapter 7

Vector Calculus 7.1

grad, div and curl ∂ ∂ ˆ∂ Nabla operator ∇ = ˆı + ˆj +k ∂x ∂y ∂z

=(

gradφ ≡ ∇φ =

∂φˆ ∂φ ˆ ∂φ ˆı + j+ k ∂x ∂y ∂z

divF ≡ ∇.F =

∂F1 ∂F2 ∂F3 + + ∂x ∂y ∂z

ˆj ˆ ˆı k curlF ≡ ∇ × F = ∂/∂x ∂/∂y ∂/∂z F1 F2 F3

∂F3 ∂F2 ∂F1 ∂F3 ˆ ∂F2 ∂F1 ˆ − )ˆı + ( − )j + ( − )k ∂y ∂z ∂z ∂x ∂x ∂y

∂φ n ˆ ∂n where n ˆ is the unit vector normal to the surface φ = constant. grad φ =

Directional derivative. The rate of change of a scalar field, φ, in the direction of the unit vector n ˆ is given by ∂φ = grad φ.ˆ n ∂n If δV is volume bounded by the surface δS then Z 1 gradφ = lim φdS δV →0 δV δS Z 1 divA = lim A.dS δV →0 δV δS 37

1 curlA = lim δV →0 δV Laplacian operator ∇2 ≡ ∇.∇ =

Z

δS

dS × A

∂2 ∂2 ∂2 + + ∂x2 ∂y 2 ∂z 2

Vector identities ∇(φψ) = φ∇ψ + ψ∇φ

∇(φ + ψ) = ∇φ + ∇ψ

∇.(φA) = A.∇φ + φ∇.A

∇.(A+B) = ∇.A + ∇.B

∇ × (φA) = ∇φ × A + φ∇ × A ∇ × (A+B) = ∇ × A + ∇ × B ∇.(A × B) = (∇ × A).B − A.(∇ × B) ∇ × (∇φ) = 0 ∇ × (∇ × A) = ∇(∇.A) − ∇2 A

∇.(∇ × A) = 0

∇2 (φ + ψ) = ∇2 φ + ∇2 ψ

∇2 (φψ) = φ∇2 ψ + 2∇φ.∇ψ + ψ∇2 φ

∇(A.B) = (A.∇)B + (B.∇)A + A × (∇ × B) + B × (∇ × A) ∇ × (A × B) = A(∇.B) − B(∇.A) + (B.∇)A − (A.∇)B Cylindrical polar coordinates 1 ∂ψ ˆ ∂ψ ˆ ∂ψ ˆ R+ φ+ k ∂R R ∂φ ∂z 1 ∂ ∂ψ 1 ∂2ψ ∂2ψ (R )+ 2 2 + ∇2 ψ = R ∂R ∂R R ∂φ ∂z 2

∇ψ =

∇.A =

1 ∂ 1 ∂Aφ ∂Az (RAR ) + + R ∂R R ∂φ ∂z

∇×A =



∂Aφ 1 ∂Az − R ∂φ ∂z



ˆ+ R



38

∂AR ∂Az − ∂z ∂R



ˆ + 1 φ R



∂ ∂AR (RAφ ) − ∂R ∂φ



ˆ k

Spherical polar coordinates 1 ∂ψ ˆ ∂ψ 1 ∂ψ ˆ ˆ r+ θ+ φ ∂r r ∂θ r sin θ ∂φ 1 ∂ ∂ψ 1 ∂ ∂ψ 1 ∂2ψ ∇2 ψ = 2 (r 2 )+ 2 (sin θ )+ 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2 ∇ψ =

∇.A =

1 ∂ 2 1 ∂ 1 ∂Aφ (r Ar ) + (sin θAθ ) + 2 r ∂r r sin θ ∂θ r sin θ ∂φ

1 ∇×A = r sin θ



1 + r

∂ ∂Aθ (sin θAφ ) − ∂θ ∂φ 

∂ ∂Ar (rAθ ) − ∂r ∂θ





1 ˆ r+ r sin θ



 ∂Ar ∂ − sin θ (rAφ ) θˆ ∂θ ∂r

ˆ φ

General orthogonal curvilinear coordinates Transformation from x = (x1 , x2 , x3 , ) to u = (u1 , u2 , u3 ) where xi = xi (u1 , u2 , u3 ) i = 1, 2, 3 ˆ1 , du2 h2 e ˆ, du3 h3 e ˆ3 ) dx = (du1 h1 e where hi are scale factors (metric coefficients) ˆi are unit vectors e ∂x ∂x i = 1, 2, 3 ˆi = hi = and hi e ∂ui ∂ui  1/2 Element of arc dl = h21 (du1 )2 + h22 (du2 )2 + h23 (du3 )2 Area element on u1 -coordinate surface dA = h2 h3 du2 du3 Volume element dV = h1 h2 h3 du1 du2 du3 Vector calculus gradψ = ∇ψ =

1 ∂ψ 1 ∂ψ 1 ∂ψ ˆ1 + ˆ2 + ˆ3 e e e h1 ∂u1 h2 ∂u2 h3 ∂u3

ˆ 1 + A2 e ˆ 2 + A3 e ˆ3 A = A1 e   1 ∂ ∂ ∂ divA = ∇.A = (A1 h2 h3 ) (A2 h3 h1 ) (A3 h1 h2 ) h1 h2 h3 ∂u1 ∂u2 ∂u3 h1 e ˆ1 h2 e ˆ2 h3 e ˆ3 ∂ 1 ∂ ∂ curlA = ∇ × A = h1 h2 h3 ∂u1 ∂u2 ∂u3 h1 A1 h2 A2 h3 A3        1 ∂ h2 h3 ∂ψ ∂ h3 h1 ∂ψ ∂ h1 h2 ∂ψ 2 ∇ ψ= + + h1 h2 h3 ∂u1 h1 ∂u1 ∂u2 h2 ∂u2 ∂u3 h3 ∂u3

39

7.2

Integral theorems of the vector calculus

V is the volume bounded by the closed surface S Gauss’s divergence theorem Z Z divFdV = F.dS V

S

Green’s theorem First form: Z Z 2 φgradψ.dS (φ∇ ψ + gradφ.gradψ)dV = V

S

Second form: Z

V

2

2

(φ∇ ψ − ψ∇ φ)dV =

Z

(φ S

∂ψ ∂φ − ψ )dS ∂n ∂n

C is the curve bounding the open surface S

Stokes’s theorem Z

S

curlF.dS =

Z

F.ds = C

40

Z

C

F.dr

Chapter 8

Tables of Transforms 8.1

The Laplace transform L[f (t)] =

Z

0

∞

−st

e

f (t)dt or L[x(t)] =

Z

∞

e−st x(t)dt

0

= F (s)

= x ¯(s)

f (t) = L−1 [F (s)]

x(t) = L−1 [¯ x(s)]

Table of Laplace transforms 1 f (t) L

−1

[F (s)] 1 t tn

L[f (t)] F (s) 1 s 1 s2 n! sn+1

(8.1) (8.2) (8.3)

(n a positive integer) tλ (λ > 1) e−at sin bt cos bt sin(at + φ)

Γ(λ + 1) sλ+1 1 s+a b s 2 + b2 s s 2 + b2 s sin φ + a cos φ s2 + a2

(8.4) (8.5) (8.6) (8.7) (8.8) continued. . .

41

Table of Laplace transforms 2 f (t) L

−1

[F (s)]

sinh bt cosh bt t sin bt t cos bt sin bt − bt cos bt sin bt + bt cos bt e−at tn

L[f (t)] F (s)

b − b2 s 2 s − b2 2bs 2 (s + b2 )2 s 2 − b2 (s2 + b2 )2 2b3 (s2 + b2 )2 2bs2 (s2 + b2 )2 n! (s + a)n+1 s2

(8.9) (8.10) (8.11) (8.12) (8.13) (8.14) (8.15)

(n a positive integer) e−at sin bt e−at cos bt e−at sinh bt e−at cosh bt H(t) H(t − a) H(t) − H(t − a) δ(t) δ(t − a) [t] sin at t

b (s + a)2 + b2 s+a (s + a)2 + b2 b (s + a)2 − b2 s+a (s + a)2 − b2 1 s −as e s 1 − e−as s 1 e−as e−as s(1 − e−s ) a tan−1 s

(8.16) (8.17) (8.18) (8.19) (8.20) (8.21) (8.22) (8.23) (8.24) (8.25) (8.26) continued. . .

42

Table of Laplace transforms 3 f (t) L

L[f (t)]

−1

[F (s)] √ cos 2 at √ πt   a √ erfc 2 t a 2 √ e−a /4t 2 πt3 2 e−a /4t √ πt r   t −a2 /4t a √ 2 e − aerfc π 2 t 2a 2 2 √ e−a t π αf (t) + βg(t)

F (s)

e

e−a s √ s

(8.30)

e−a s √ s3 s 2 2 erfc es /4a 2a αF (s) + βG(s)

(8.33)

e−as F (s)

(8.34)

F (s + a)

(8.35)

sF (s) − f (0) sf (0) −

f ′ (0)

(8.31) (8.32)

(8.36) (8.37)

sn F (s) − sn−1 f (0) − sn−2 f ′ (0) −

t

f (u)du

0

−tf (t)

. . . − f n−1 (0) F (s) s ′ F (s)

(8.39)

F (n) (s)

(8.41)

n

(−t) f (t) f (t) t 0

(8.29)

s2 F (s) −

f (t)

Z

s

√

′′

Z

s

√

f (t) (t)

(8.28)

√

′

f

e−a s e−a

f (t)

(n)

(8.27)

√

H(t − a)f (t − a) −at

e−a/s √ s

Z

t

f (u)g(t − u)du

∞

F (u)du

(8.38)

(8.40)

(8.42)

s

F (s)G(s)

(8.43)

(convolution integral) 1 1 − e−T s

f (t + T ) = f (t) n X P (αk ) exp(αk t) Q′ (αk )

Z

T

e−su f (u)du (8.44)

0

P (s)/Q(s)

(8.45)

k=1

P polynomial of degree less than n Q(s) = (s − α1 )(s − α2 ) . . . (s − αn ) where α1 , α2 , . . . , αn are all distinct

43

Limiting theorems

lims→∞ F (s) = 0,

limt→∞ f (t) = lims→0 sF (s)

Inversion formula 1 2πj

Z

γ+j∞

st

e F (s)ds = γ−j∞



f (t) t > 0 0 t