ANSWERS

Chapter 1

1.6.1. xj3- x3 /9,.0332224 (from series) and.0332222 (calculator).

1.6.2. (i) 3x2 cos(2 + :r3), (ii) -2 cos[cos(2:r )] sin(2:r ), (iii) 3 sec2 :r tan2 :r,

(iv) tanh:r, (v) 1/(1 + x 2), (vi) 1/(1- x2), (vii) 0, (viii) 1/(1 + cosx). 1.6.3. 100e·12 ~ $112.72.

1.6.4. v = tanh 8

1.6.7. Square of side L/4.

1.6.9. 1,3

1.6.12. -! 1.6.13. x = 1 is a minimum; :r = -1 is a maximum.

Chapter 2

2.1.2. :r lnx- x

2.2.1. sin-1 ~- sin-1 7 2.2.3. 2

2.2.5. ln(3/2)

2.2.6. 1/3

2.2.7. 1/(ln2)

2.2.9. /a= 1/(2a2 ), /4 = ~~ 2 2 1 2Gk G2-k2

• .) • (G~H~):I; (G2+k~)~

JSI

3S2

Chapter 3

3.1.2. /z = 3x2 + 2xy5, / 11 = 5x2y4 + 4y3, fz 11 = 10xy4 = /11z

3.1.5. d = 4/../5 3.2.5. I = M R2 /2, (3M R2)/5

3.2.8. v = 12811"

Chapter 4

4.2.3. Diverges

4.2.4. (i) ratio test is inconclusive, integral test says divergent.

(ii) r = e3 /27, convergent by ratio test. (iii) Divergent by integral test. (iv) Ratio test inconclusive, integral test says divergent. (v) r = e, divergent. (vi) Ratio test inconclusive, integral test says divergent. 4.2.5. (i) C, (ii) C, (iii) C (iv) C.

4.2.6. C, C, D, D, D.

Answers

4.3.1. R = (i)../2, (ii) 1, (iii) 1, (iv) 1, (v) Converges for lxl > 1, (vi) lxl > !· (1 v2 3v4 5v& ) 4.3.4.E=m + 2 +s+16+ ... ,P=vE.

4.3.6. T = 21rffg(1 + k: + ... ), 6T/T = 1/16.

4.3.7. (i) ~[1 +x- x2/2- x3 /6 +x4/24], (ii) ~[1 +x +x2 /2+ x3/6+ x4/24],

(iii) ln2 + x/2- x2j8 +x3j24- x4 j64.

Chapter 5

5 2 3 - 137 761 . . . . z - 35i7 - 35T7' 5.2.4. (i) Rez = 6/25, lmz = -8/25, lzl = 2/5, z• = 6/25 + 8/25i, 1/z = 3/2 + 2i, (ii) Rez = -7, Imz = 24, lzl = 25, z• = -7- 24i, 1/z = -(7/625}- (24j625)i, (iii) Rez = -7/25, lmz = 24/25, lzl = 1, z• = -(7 /25) - (24/25)i, 1/ z = -(7 /25) - (24/25) i, (iv) Rez = !.=f'i, lmz = ~. jzj = 4, 1/z = ¥ -i~,

Answers 353

(v)Rez=cosO, lmz=sinO, 11z=z*=cosfl-isinfl. 5.3.2. (i) Z1 = eiw/4, Z2 = 2e-iw/6, Z1Z2 = 2eiw/12, Z1IZ2 = .5e5wi/12.

(ii) z1 = e2iarctan4/3, z2 = .se-iw/2.

5.3.3. lz1 + z2l = 7.95, Phase is .98 radians.

5.4.4. z = 1- i = ._!2e-i"/4 , Io = 1001./2, current leads by 7rl4, resonance at

w = 223.6 rads. 5 4 5 Z - 31t33i . . . - 41 .

5.4.6. z = l±iR(..,OR 1/(..,L)) ·

5.4.7. Q(t) = e-6t[4cos8t + 3sin8t]- 4cos10t

Chapter 6

6.1.2. u., = -v11 u 11 = v.,

6.1.3. f.,= /y 6.1.7. Poles at (±1 ± i)l./2, double pole at z = ±i 6.1.8. Ur = vslr, Vr = -uslr 6.1.12. (i) f = z3, (ii) f = eiz, (iii) u is not harmonic.

6.l.l3. Hint: Relate the Laplacian to a:::r• 6.l.l4. Hint: Consider /2.

6.2.3. sinxcoshy,cosxsinhy, Jsin2x +sinh2y, [x = n1r,y = 0]

6.2.4. z = n1r, (n + 112)7r, in1r, i(n + 112)7r

6.2.9. e2"in/N, n = 0, 1, ... N - 1. Roots add to zero.

6.2.10. 3 + 4i, 12 + 5i

6.2.11. (i) ±(1 + i)l ./2, (ii) ±(2 + i) 6.2.12. In 2 + (2m + 1 )i1r

6.2.13. (i) eosin 3 + i sin In 3 (ii) i1r 12 (iii) ±5ei"/4 (iv) ±eiw/3

6.2.14. (i) Repeated twice: ±i. (ii) Repeated twice: [eiw/3, -1,e-iw/3]

6.2.15. J2ei""/4 • The cube roots are (2)116 (ei""/12,ei(w/12)±(2wi/3))

6.2.16. e-wl2n+l/2)

6.4.1. 1rle, -1re 6.4.5. (i) 1r I (2312), (ii) 0,0 (iii) 1r I 18, (iv)-1r /3, (v) 0, (vi) 1r I 4

6.4.8. 1r I (27 e3)

6.5.3. Yes

354 Answers

6.5.4. No

Chapter 7

7.4.3. (i) 516, 0. 7.5.1. h = v-=R~2 ---x"ll'2---y72, Vh = -i ..: - j v

\fR2-z2-fl2 Jn2-z2-fl2

7.5.4. Cylindrical: hp = 1, hq, = p h 8 = 1. Spherical: hr = 1, h9 = r, hq, = rsinB. 7.5.5. (ii) v'la, (iii) 12/ v'la, (iv) 2. 7.5.7. (i) Towards the origin (ii) AT= -3v'2/10 7.5.8. (i) 2J3, (ii) v'3e3, (iii) J3, (iv) All gradients were radial, this direction is perpendicular to radial. 7.6.2. (i) no, (ii) yes, (iii) yes (iv) yes 7.6.3. (i) 1, (ii) 1, (iii) Possibly. (iv) rjl = x2y.

7.6.4. (i) 1, (ii) rjl = x3y

7.6.5. (i) 1/2, (ii) no. 7.6.11. (I) 0, (ii), 0, (iii) conservative, F = Vr/1, rjl = (x2 + y2)/2

7.6.13. 211" 7.6.14. The curl has no component in the plane containing the contours for two line integrals. 7.7.1. -sinx+2z

7.7.3. 1

7.7.4. ~ 7.7.6. (i) 47rR3 • (ii) 21rR3 , (iii) 3.

Chapter 8

[ 6 8 ] 2 [ 7 10 ] 8.1.7. M + N = 10 12 M = 15 22

[ 19 22 ] [ -4 -12 ] M N = 43 50 IM' N) = 12 4 . 8.3.4. {1, 2, -1); ( -3, -4, 8)

Answers

[ _1 ~~ 1 l

8.3.5. ~~ ~2 ~2 .

Chapter 9

9 .I. I. yes, real scalars, no, U 1 + U 2 is not unitary, yes with integer field 9.1.2. yes, yes,no

9.1.5. No: 13} = 11}- 212) 9.2.1. (i) VJ = (2 + i(1- J3))j../2, V/1 = ( J3 + 1)ij../2. (ii) VJ = (i -1)(i + J3)(v'6 -1)/4, Vl/ = 1 + .fii2. 9.2.2. !(3i + 4j) ((104}2 + (78)2)-112[104i- 78j)

9.2.4. When IV)= c!W} 9.2.6. Q = !jl. {3 = - 3jfi ''Y = 8

9.5.2. [eigenvalue)( eigenvector components) (1) (0, 0, 1} (e±iB] ~(1, =fi, 0}

3SS

9.5.3. The answers are given in the same fonnat as the matrices in the assigned problem. The three eigenvalues and eigenvectors for each matrix are given one below another.

(1) (1, 0, 0} [2] *( -5, -2, 1} [4) ?10(1, 0, 3}

[-1) ~ (-1,0,1} (0] (0, 1,0} [1] ~(1,0,1)

[-11-}a (1, -2, 1} [1]~ (-1,0,1) [2] 73(1, 1, 1}

[0] ~(-1,0,1} [2] ~(-1,0,1) (2] ~(-J3,0,3) [ -v'2j !(1, -../2, 1) [2(1- ../2)] ' !(1, -../2, 1) (3] (0, 1, 0) [../2] }(1, ../2, 1} [2(1 + ../2)] !(1, ../2, 1} (6] !<J3.o, 1}

9.5.7. (OJ ~( -1, 0, 1} (2] ~(1, 0, 1} [2] (0, 1, 0}

9.5.8. (1] (1, 0, 0} (0, 0] (a2 + b2)-112 (0, a, b)

[-1] ~(1, 1, 1} (1] ~( -1, 0, 2) (2) (0, 0, 1) 9.5.9. [2] ~(-1,0,1) (1] (0,1,0) (2] ~(-1,1,0}

(2] ~(-1,1,0} (6) ~(2,0,1) (4] ~(1,1,0)

9.5.10. (-1) ~(-1, 2, 1) (2) ~(-1, -1, 1) (3) ~(1,0, 1} for N

356

9.5.11. (a) [1, 0, -1], (b) [1, 0, -1], (c) (!, ~· !), (d) P(1) = t = P(-1), P(O) = !; (1,0,0); no, (e) 2, (f) two, (g) IV)= 71'4(1,2,3) P(1) = 114 P(O) = 1~ P(-1) = 14 Average=-~= (VIS.z:!V).

9.6.1. lx(t)) = 1 [ cos ~t +cos J3k/mt ] 2 cos ~t- cos J3k/mt

9.6.2. Same as in worked example, but with J3k/m ~ J5k/m.

Answers

9.6.3. The eigenfrequencies (in square brackets) are followed by unnormalized eigenvectors:

(V2) (-1,0,1); (J2-J2] (1,J2,1); [J2+J2] (1,-J2,1).

9.6.4. (x = O,y = 1) saddle point,!"= [ ~1 =~ ] Eigenvalue and eigenvector [ - 1ji5) = (1, ¥> 9.6.7. (-12/7),32/7} is a maximum.

9.7.3. (a) J(x) = t(I- Eodd w!l~2e2min/L). (b) w: Look at x = L/2 4 1 i ~ ein~ 9.7 .. (a) 2 + ;r L.Jodd ,..-.

IJ.I ~oo (-1)" (1-n'lri~slnh 1 iwnz 1u1 LJ-oo 1+n w!l e

lOOi e 200n wit 9.7.5. I= En#- 0 60ntr+i(200n!lw'-2500)

9 7 8 (in J e-1 + 2 ~oo e-(-1)" • • • It = e L.Jl e(I+n2w2) COS n11'X

J _ . h 1 [1 + 2 ~oo (-1)"cosnwz] - sm L.Jl t+n2w2 •

9 7 12 ·'·( t) 8h ~oo 1 • mwz • mw mwvt • • • 'I' x, = iY L.Jl ~ sm -r;- sm 2 cos -r.;-·

9.7.13. u(x,t) =~Ego 12t2~~;)]' sin(2+4n)xcos{2+4n)t.

9.7.14. ,p(x, t) = 2: E~ !;(I- cos ";)sin "L:z cos mzt 9.9.2.(a) Iij = 4m6ij

(b) ' [ !1 ~1 =~ l -1 -1 2

[OJ (1, 1,1} Masses lined up along axis through origin. ( r, 3rJ ( -1, 0, 1) and ( -1, 1, 0) Anything in plane perpendicular to (1,1,1).

Chapter 10

10.2.4. 2coshwt

Answers 357

10.2.5. Aet + Beit + ce-it

10.2.8. (i)(1+t)e-t, (ii) A exp[~t]+B exp[-~t]+C exp[7:ftJ+D exp[-jtit],

(iii) Ae5t + (Bt + C)e-t, (iv) (A+ Bt)e-t + Ce4t + De-4t + Ee4it + De-4it.

10.2.10. (i) Critically damped (ii) e-t[3- St]- 3cos2t + 4sin2t (iii) e-t[4-

4t]- 3cos2t + 4sin 2t (iv) e-t[3- 4t]- 3cos 2t + 4sin2t 10.2.11. (i) (2/9)e2"' + (7/9)e-z + "'~"', (ii) e"'- sinx, (iii) cos4x + 2xsin4x,

(iv) cosh x + z slnhz

10.3.3. y = !-} + b 10.3.4. y = x2 + 1 - e-z

10.3.5. y = ~(coshx + 2- cosh 1). 10.3.6. y = (1/2)(x- 1)(x2 - 2x + C- 2ln lx- 11)

10.3.7. y = {1/2)(x - 1)(x2 + 2C) 10.3.8. y = x + c~ 10.3.9. a: y = (1 + Ce"'212)-1 b: y = (x2 + Cx )-i 10.4.8. a: y = (1 + x)(A + B lnx)

b· - <t+~) · Y- z z-1 c: y = Ax3 + B x - 3

d· y- A(1- 4x + l!x2 - .a!x3 + Ax4 · · ·) + Bx112(1- ~x + ..!.x2 - Ax3 + 32 - 3 45 315 , 15 315 2835x4 .. ·) e: y = x(A +Blnx)

(2z)2 (2z)13 ~2z)~ 3/2 2z (2z)~ (2z)3 f: y=A(1+2x-2!+ 3·3- ·5·4 + .. ·J+Bx [1-T+5·7·2 -IT.Q.3T+

(2z)4 + J 5·7·9·11·41 • • • 3 4

g: y = A(x2 +2x) + B((x2 +2x)lnx + 1 +5x- ~ + ; 2 + ... ] h: y = ..&.. + B[.k!u + 1 +x] 1-z 1-z 10.4.9. (i): y = x2(x +C)

4

(ii): y = 'ir + ~ 10.4.10. Lo = 1

£1 = 1- X L _ 2-4ztz2 2- 2

L _ 6-18zt9z2-z3 3- 6

10.5.6. u(x,t) = ~ L~ ~sin (2m1l)11'ze-<2,..~~~2 .. \ 1o.s.s. (iJ u(x,y) = 2: E~ <-lr+l sin "1:1: e-!!fl

(ii) u(x,y) = ~ E~ 4"l~ 1 sin 2"£"' exp [- 2"i!!] 10 S 9 ••("' y) _ ~" 1 st'n(!!U) sinh",.. a · • • .. "'' - 11' L....n=odd n a sin nw a

INDEX

Absolute convergence, 80, 118 Absolute value, 92 Action. 309 Active transformation, 208 Adjoint matrix, 220 Adjoint operation, 254 Ampere's Law, 198 Analytic continuation, 142 Analytic function, I 07, I 09

analytic continuation of, 142 branch points of. 113 Cauchy--Riemann Equations for (CRE), 109 derivative of, 114 diagnostic for, 1 09 domain of analyticity of, II 0 essential singularities of, 113 exp(z), 119 harmonic functions--relation to. 114 hyperbolic functions. 119 integrals of, 128 Laplace's equation--relation to, 115 ln(z). 121 meromorphic functions. 113 permanence of functional relations, 143 poles of. Ill power series definition of. 116 removable singularity of, 113 residue theorem for, 132, 134 singularities of, Ill Taylor series for, 139 trigonometric functions, 115

Anti-analytic functions, 109 Antiderivative, 37 Antihermitian, 221 Antilinearity of inner product, 240 Antisymmetric tensor, 297 A rea vector, 15 5 Axioms of vector space, 230

Bernoulli's equation, 318, 327 Bessel's equation, 327 Bessel functions, 335 Binomial Theorem, 18 Box product, 155; see also Scalar triple product

Bra, 239, 253 Branch point, It 3

of In function, 124

Cauchy's Theorem, 132, 134 Cauchy--Riemann Equations (CRE), 109 Cayley--Hamilton Theorem, 263 Chain rule, 3 Characteristic polynomial, 257 Circulation. 172 Cofactor matrix, 215 Column vector, 206 Commutator, 21 0 Comparison test for series, 77 Complementary function. 103. 313 Complex numbers. 89

absolute value of. 92 application to LCR circuit. 98 argument of. 96 cartesian form of. 90 complex conjugate of. 91 Euler's identity for. 95 imaginary part of. 90 phase of. 96 polar form of. 94 purely imaginary. 90 real part of. 90 unimodular, 97

Complex variable, 1 07; see also Analytic function Complimentary function, 313 Components of vector, 236 Conditional convergence, 80 Conservative field, 162 Constraints, 55 Continuity equation

for charge, 193 for heat, 336

Continuity of function in one variable, I in two variables, 1 07

Contraction of tensors, 296 Contravariant tensor, 299 Convergence of series, 75

absolute convergence, 80, 118

359

360

Convergence of series (cont.) conditional convergence, 80 complex series, 116 radius of, 118

Coupled mass problem, 267 Covariant tensor, 299 Cross product, I S 3

in terms of components, I 56 as a determinant, 217

Curl, 172, 174, 177 as a determinant, 217 in noncartesian coordinates, 181

Current density, 162 Cylindrical coordinates, 66

D operator, 187, 279 eigenbasis of -iD, 279 Fourier series, 281

D2 operator, 284 Dagger, 220 de Moivre's Theorem, 98 Definite integral, 34

lower limit of, 24 upper limit of, 24

Degeneracy, 263 Dependent variable, I Derivative, 2 Determinant, 212, 214 Determinants, 214

and cross product, 217 and curl, 217

Differential calculus, I chain rule, 3 dependent variable, I differentials, 29 exponential and log functions, S hyperbolic functions, 13 implicit differentiation, 28 independent variable, I inverse hyperbolic functions, 18 inverse trigonometric functions, 22 L'Hopital's rule, 24 linearity of derivative, 3 ln(l + y), IS plotting functions, 23 radian measure, 19 series for hyperbolic functions, 19 Taylor series, 8 trigonometric functions, 19

Differential equations, 305 for damped oscillator, 311

Differential equations (cont.) Frobenius method, 318 Green's function method for, 344 heat equations, 336 linear, 306 nonlinear, 306 order of, 306

Index

ordinal)', 305; see also Ordinal)' differential equations

partial, 305; see also Partial differential equations

superposition principle for, 306 wave equations, 329

Differential operators, 187 Differentials, 29 Dimensionality, 235, 246 Dirac delta function, 197, 288 Dirac matrices, 227 Displacement current, 199 Divergence, 182

in other coordinates, 186 Domain

of analyticity, II 0 of integration, 61 simply connected, 129

Dot product, IS I in terms of components, IS I and inner product, 23 7

Drum circular, 332 square, 330

Dual spaces, 253 Dummy variable, 43

e-the base of natural logs power series definition, 7 second definition, 12

Eigenfunction, 280 Eigenvalue problem, 255

characteristic polynomial, 257 of commuting operators, 263 of coupled masses, 263 of D 2 operator, 284 degenerate case, 263 of -iD operator, 279 of quadratic forms, 274

Eigenvalues, 255 Eigenvectors, 255 Einstein convention, 296 Electrodynamics, 193

Index

Electrodynamics (cont.) Ampere's law, 198 continuity equation, 193 displacement current, 199 Lorentz force law, 194 Maxwell's equations, 200 scalar potential, 20 I vector potential, 20 I wave equation, 200, 202

Epsilon neighborhood, 110 Essential singularity, 113

relation to Taylor series, 142 Euler's identity, 94 et eye/, ISS Exponential function

exp(elephant), 94 exp(x), S, 12 exp(z), 119

Field scalar and vector, I 58 vector space, 231

Field of vector space, 231 First order ordinary differential equation, 31 S Flux, 163 Fourier integrals, 287 Fourier series, 281

coefficients of, 281 completeness of, 292 convergence of, 286 exponential form, 281 trigonometric, 285 uniform convergence of, 286

Fourier transform, 287 Frobenius method, 318

indicia! equation, 325 recursion relation in, 318 with singular coefficients, 324

Function spaces, 277 eigenvalue problem of -iD, 279 eigenvalue problem of D2, 284 inner product in, 278 orthonormal basis for, 279

Gamma function, 43 Gauge transformation, 202 Gauss's Theorem, 184 Gaussian integral, 71 Gibbs' oscillation, 338 Gradient, 167

and Lagrange multipliers, 170

Gradient (cont.) in noncartesian coordinates, 177

Gram-Schmidt Theorem, 240, 244 applied to Legendre polynomials, 324

Green's function method, 344 Green's Theorem, 174

Hamiltonian, 273 Harmonic function, II S Heat equation, 336

derivation of, 337 in d= 1, 337 ind=2,340 in polar coordinates, 343

Hermite polynomials, 323 Hermitian, 221 Hermitian operators

and coupled masses, 267 diagonalization of, 294 and orthogonal functions, 279 orthogonality of eigenvectors, 260 and quadratic forms, 274 reality of eigenvalues, 260

Higher derivative vector operators, 189 Hilbert space, 278 Homology, 191 Hyper complex numbers, 125 Hyperbolic functions, 13

in complex plane, 119 Power series for, 19 relation to trigonometric functions, 121

Imaginary part, 90 Impedance, I 0 I Implicit differentiation, 28 Indefinite integral, 39 Independent variable, I Indicia! equation, 325 Inner product spaces, 237, 239

postulates of, 239 Integral calculus, 28

in many variables, 61 Integral test for series, 78 Integrals

analytical way, 36 antiderivative, 37 basics, 33 change of variables in, 44 composition law for, 41 in cylindrical coordinates, 66

361

362

Integrals (cont.) definite, 34 dummy variables in, 43 and evaluation of areas, 37, 40 of gaussian function, 71 integrand, 34 integration by parts, 42 Jacobian, 4S linearity of, 41 lower limit of, 32 multiple, 61 numerical way, 36 parametric differentiation, 46 in polar coordinates, 64 primitive, 37 scale factor, 6S of solid angle, 70 in spherical coordinates, 66 tricks of the trade, 44 upper limit of, 34 variable of, 34

Integrand, 34 Integration by parts, 42 Integration variable, 34 InterVal of convergence, I 0, 81 Inverse hyperbolic functions, 18

miscellaneous problems in, 2S Inverse matrix, 211, 212, 216 Inverse trigonometric functions, 22

Jacobian, 4S, 6S as a determinant, 71 218

Kronecker delta, 227, 240

L'H6pital's rule, 24 Lagrange multipliers, SS

and gradients, 170 Laguerre's equation, 328 Laplace's equation, liS, 340, 344 Laplacian, 192 LCR circuit, 98

impedance, 101 mechanical analogy, 104 transients in, I 03

Legendre's equation, 324 Legendre polynomial, 324 Level surface, 168 Line integral, I S9 Linear combination, 3 Linear dependence, 233

Linearindependence,233 Linear operators, 24 7

adjoint of, 2S2 D and -iD operators, 279 definition of, 248 eigenvalue problem of, 2SS matrix elements of, 249 product of, 2S I, 2S2

Linear transformation, 209 Linear vector spaces, 229

adjoint operation, 2S4 axioms of, 230 basis for, 23S over complex field, 231 dimension of, 23S, 246 dual spaces, 2S3 eigenvalue problem, 2SS examples, 239

Index

expansion of vectors in, 242, 243 expansion of vecton in orthonormal basis

~2 •

In

field of, 231 function spaces, 277 generation of basis for, 279 Gram-Schmidt procedure for, 244 inner product for, 23 7, 239 linear operators acting on, 241 real, 231 Schwarz inequality, 246 Triangle inequality, 246

branch point of, 124 branches of, 124 ln(z), IS ln(z), 121

log; see In Lorentz force law, 194 Lorentz gauge condition, 202

Matrix adjoint, 220, 22S analogy to numben, 22S antihermitian, 221 antisymmetric, 220 commutator of, 210 dagger, 220 determinant of, 211 Dirac, 227 elements, 20S functions of, 222

Index

Matrix (cont.) hennitian, 221 inverse, 211, 216 null, 209 orthogonal, 222 Pauli, 226 product of, 206 for rotation, 208 simultaneous equations, 212 sum of, 206 symmetric, 220 table of properties, 222 trace of, 227 transpose, 21 S unit, 209 unitary, 221

Matrix inverse, 211, 212, 216 Matrix product, 206 Matrix sum, 206 Matrix transpose, 21 S Maxima, minima and saddle points, 274 Maxwell's equations, 200 Meromorphic functions, 113 Mixed derivative, 52 Modulus, 90 Moment of inertia tensor, 298

Natural logarithm, 7, 8 Norm of a vector, 149 Normal modes, 267, 272

of coupled masses, 267 in quantum mechanics, 273 of string, 289

Null vector, 149, 230

Ohm's Law, 295 Ordinary differential equation (ODE), 307

Bessel's equation, 327 classical oscillator, 311 complementary function, 313 Frobenius method, 318, 324 Hennite's equation, 322 initial conditions, 308 integration constants, 307 Laguene'sequation,328 Legendre's equation, 324 particular solution, 313 quantum oscillator, 320 with constant coefficients, 307 with variable coefficients (first order), 31 S with variable coefficients (second order), 318

Orthogonal coordinates, 6S Orthogonal matrix, 221 Orthonormal basis, 149, 239

expansion of a vector in, 242 Oscillator

action for, 309 analogies to LCR circuit, I 04 equations for, I 04, 311 quantum, 320 types of behavior, 311

Outer product, 297

Parseval's Theorem, 293 Partial derivative, S I Partial differential equation (POE), 329

heat equation in d = 1, 2, 336 polar coordinates, 334, 343 separation of variables, 329 solution by Green's functions, 344 wave equation in d = 1, 2, 329

Particular solution, I 03, 313 Passive transformation, 208 Pauli matrices, 226 Permeability of free space, 172 Permanence of relations, 143 Pennittivity of free space, 166 Pi (71"), 20 Planck's constant, 288 Plotting functions, 23 Poisson's solution, 344 Polar coordinates, 64 Polar form, 94 Polarizability tensor, 299 Poles

n-th order, 113 residue at, 112 simple, 112

Position vector, I 56 Potential, 175 Power series

absolute convergence, 80, 116 hyperbolic functions, 19, 119 in :z:, 80 in z, 116 interval of convergence of, I 0 81 ln(l + :z:), IS, 85 radius of convergence, 118 tests for convergence, 80 tricks for expansion in, 83 trigonometric functions, 22, 119

363

364

Primitive, 37 Principal axes, 298 Projection operator, 210 Propagator, 2 71

Quadratic forms, 274 and inner products, 2 77

Quantum oscillator, 320

Radians, 19 Radius of convergence, 118 Rank of tensor, 296 Rapidity, 27 Ratio test for series, I 0, 78 Real part, 90 Reciprocal vectors, 219 Recursion relation, 319

for Bessel functions, 327 for Hermite polynomials, 322 for quantum oscillator, 322

Removable singularity, 113 Residue, 112 Residue Theorem (Cauchy), 132, 134 Right-hand rule, I S4 Right-handed coordinates, lSI Roots in complex plane, 122 Rotation matrix, 208 Row vector, 206

Saddle point, S4 of quadratic forms, 274, 276

Scalar field, I S8 gradient of, 167

Scalar potential, 20 I Scalar triple product, ISS; see also Box product Scale factor, 6S Schwarz inequality, 246 Screened Poisson equation, 345 Screw rule, IS6 Second Order ODE, 318 Separation of variables, 329 Series, 75

absolute convergence, 80, 117 comparison test, 77 in complex variable, 117 conditional convergence, 80 geometric, 76 integral test, 78 ratio test, 78 tests for convergence, 77

Simultaneous equations, 212 Singularities, Ill Solid angle, 68, 70 Space-time interval, 26 Spherical coordinates, 66 Spin, 265 State vector, 269 Stationary point, S3

with constraints, 55 and quadratic forms, 274

Steady state response, 1 04 Stokes' Theorem, 179 String, 289

normal modes of, 289 separation of variables for, 329

Superposition principle, 100, 195, 306 Surface integral, I 59, 162, 164 Surface term in integrals, 42

Taylor series in a complex variable, 139 and essential singularities, 83, 141 single variable, 8, 10 two variables, 52

Tensors, 294 antisymmetric tensor, 297 conductivity, 296 contraction, 296 moment of inertia, 298 outer product, 297 polarizability, 299 rank of, 296

Time derivatives of vectors, 1 56 Torque, IS4 Trace of matrix, 227 Transients, I 03 Transpose of

matrix, 215 of a product, 220

Triangle inequality, 246 Trigonometric functions

and hyperbolic functions, 121 power series for, 22 in z, 19 in z, liS

Uncertainty Principle, 288 Unit matrix, 209 Unit vector, 149 Unitary matrix, 221

Index

Index

Unitary operators, 260

Vector calculus, 149 applications to electrodynamics, 193; see also

Electrodynamics conservative field, 162 curl ofvector field, 172 digression on integral theorems, 190 digression on vector identities, 190 divergence of vector field, 182 gradient of scalar field, 167 Green's theorem, 174 higher derivatives, 189 integral theorems in, 188 Laplacian, 192 line and surface integrals, 1 59 potential of conservative field, 175 Stokes' theorem, 179 vector analysis review, 149

Vector field, I 58 circulation of, 172 curl of, 172 divergence of, 182

Vector field (cont.) Gauss's Law for, 184 Green's Theorem for, 174 line integral of, 159 Stokes' Theorem for, 179 surface integral of, 162

Vector potential, 201 Vectors

cross product for, 153 dot product for, 151 introduction, 149 in polar coordinates, 157 time derivative of, I 54

Velocity vector, 156

Wave equation, 200 circular drum, 332 d= 1,2, 329 separation of variables, 329 square drum, 328

Weight functions, 324

Zero matrix, 209

365