Chow T.L. Mathematical Methods for Physicists: A Concise Introduction

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Proof: Expanding the determinant by the row whose terms are binomials, we

will see property (6) immediately.

(7) If we add to the elements of a row (or column) any constant multiple of the

corresponding elements in any other row (or column), the value of the determi-

nant is unaltered.

Proof: Applying property (6) to the determinant that results from the given

addition, we obtain a sum of two determinants: one is the original determinant

and the other contains two proportional rows. Then by property (4), the second

determinant is zero, and the proof is complete.

It is advisable to simplify a determ inant before evaluating it. This may be done

with the help of properties (7) and (2), as shown in the following example.

Example A2.6

Evaluate

D 

12421 93

2 ÿ37 ÿ1 194

ÿ2350ÿ171

ÿ3 177 63 234

To simplify this, we want the ®rst elements of the second, third and last rows all to

be zero. To achieve this, add the second row to the third, and add three times the

®rst to the last, subtract twice the ®rst row from the second; then develop the

resulting determinant by the ®rst column:

D 

1242193

0 ÿ85 ÿ43 8

0 ÿ2 ÿ123

0 249 126 513



ÿ 85 ÿ43 8

ÿ 2 ÿ123

249 126 513

We can simplify the resulting determinant further. Add three times the ®rst row to

the last row:

D 

ÿ85 ÿ43 8

ÿ 2 ÿ123

ÿ 6 ÿ3 537

Subtract twice the second column from the ®rst, and then develop the resulting

determinant by the ®rst column:

D 

1 ÿ43 8

0 ÿ 123

0 ÿ3 537



ÿ123

ÿ3 537

ÿ537 ÿ 23 ÿ3ÿ468:

546

APPENDIX 2 DETERMINANTS

By applying the product rule of diÿerentiation we obtain the following

theorem.

Derivative of a determinant

If the elements of a determinant are diÿerentiable functions of a variable, then the

derivative of the determinant may be written as a sum of individual determinants,

for example,

abc

ef g

hmn



efg

hmn



abc

hmn



abc

ef g

;

where a; b; ...; m; n are diÿerentiable functions of x, and the primes denote deri-

vatives with respect to x.

Problem A2.5

Show, without computation, that the following determinants are equal to zero:

0 a ÿb

ÿa 0 c

b ÿc 0

;

02ÿ3

ÿ204

3 ÿ40

Problem A2.6

Find the equation of a plane which passes through the three points (0, 0, 0),

(1, 2, 5), and (2, ÿ1, 0).

547

DERIVATIVE OF A DETERMINANT

Appendix 3

Table of * Fx



2

ÿt

dt:

548

x 0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359

0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753

0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141

0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517

0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879

0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224

0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549

0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852

0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133

0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389

1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621

1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830

1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015

1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177

1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319

1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441

1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545

1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633

1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706

1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767

2.0 0.4472 0.4778 0.4783 0.4788 0.04793 0.4798 0.4803 0.4808 0.4812 0.4817

2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857

2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890

2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916

2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936

2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952

2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964

2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974

2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981

2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4986 0.4986 0.4986

3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990

x 0.0 0.2 0.4 0.6 0.8

1.0 0.3413447 0.3849303 0.4192433 0.4452007 0.4640697

2.0 0.4772499 0.4860966 0.4918025 0.4953388 0.4974449

3.0 0.4986501 0.4993129 0.4998409 0.4999277 0.4999277

4.0 0.4999683 0.4999867 0.4999946 0.4999979 0.4999992

* This table is reproduced, by permission, from the Biometrica Tables for Statisticians, vol. 1, 1954,

edited by E. S. Pearson and H. O. Hartley and published by the Cambridge University Press for the

Biometrica Trustees.

Further reading

Anton, Howard, Elementary Linear Algebra, 3rd ed., John Wiley, New York, 1982.

Arfken, G. B., Weber, H. J., Mathematical Methods for Physicists, 4th ed., Academic Press,

New York, 1995.

Boas, Mary L., Mathematical Methods in the Physical Sciences, 2nd ed., John Wiley, New

York, 1983.

Butkov, Eugene, Mathematical Physics, Addison-Wesley, Reading (MA), 1968.

Byon, F. W., Fuller, R. W., Mathematics of Classical and Quantum Physics, Addison-

Wesley, Reading (MA), 1968.

Churchill, R. V., Brown, J. W., Verhey, R. F., Complex Variables & Applications, 3rd ed.,

McGraw-Hill, New York, 1976.

Harper, Charles, Introduction to Mathematical Physics, Prentice Hall, Englewood Cliÿs,

NJ, 1976.

Kreyszig, E., Advanced Engineering Mathematics, 3rd ed., John Wiley, New York, 1972.

Joshi, A. W., Matrices and Tensor in Physics, John Wiley, New York, 1975.

Joshi, A. W., Elements of Group Theory for Physicists, John Wiley, New York, 1982.

Lass, Harry, Vector and Tensor Analysis, McGraw-Hill, New York, 1950.

Margenus, Henry, Murphy, George M., The Mathematics of Physics and Chemistry,D.

Van Nostrand, New York, 1956.

Mathews, Fon, Walker, R. L., Mathematical Methods of Physics, W. A. Benjamin, New

York, 1965.

Spiegel, M. R., Advanced Mathematics for Engineers and Scientists, Schaum's Outline

Series, McGraw-Hill, New York, 1971.

Spiegel, M. R., Theory and Problems of Vector Analysis, Schaum's Outline Series,

McGraw-Hill, New York, 1959.

Wallace, P. R., Mathematical Analysis of Physical Problems, Dover, New York, 1984.

Wong, Chun Wa, Introduction to Mathematical Physics, Methods and Concepts, Oxford,

New York, 1991.

Wylie, C., Advanced Engineering Mathematics, 2nd ed., McGraw-Hill, New York, 1960.

549

Index

Abel's integral equation, 426

Abelian group, 431

adjoint operator, 212

analytic functions, 243

Cauchy integral formula and, 244

Cauchy integral theorem and, 257

angular momentum operator, 18

Argand diagram, 234

associated Laguerre equation, polynomials see

Laguerre equation; Laguerre functions

associated Legendre equation, functions see

Legendre equation; Legendre functions

associated tensors, 53

auxiliary (characteristic) equation, 75

axial vector, 8

Bernoulli equation, 72

Bessel equation, 321

series solution, 322

Bessel functions, 323

approximations, 335

®rst kind J

x, 324

generating function, 330

hanging ¯exible chain, 328

Hankel functions, 328

integral representation, 331

orthogonality, 336

recurrence formulas, 332

second kind Y

x see Neumann functions

spherical, 338

beta function, 95

branch line (branch cut), 241

branch points, 241

calculus of variations, 347±371

brachistochrone problem, 350

canonical equations of motion, 361

constraints, 353

Euler±Lagrange equation, 348

Hamilton's principle, 361

calculus of variations (contd)

Hamilton±Jacobi equation, 364

Lagrangian equations of motion, 355

Lagrangian multipliers, 353

modi®ed Hamilton's principle, 364

Rayleigh±Ritz method, 359

cartesian coordinates, 3

Cauchy principal value, 289

Cauchy±Riemann conditions, 244

Cauchy's integral formula, 260

Cauchy's integral theorem, 257

Cayley±Hamilton theorem, 134

change of

basis, 224

coordinate system, 11

interval, 152

characteristic equation, 125

Christoÿel symbol, 54

commutator, 107

complex numbers, 233

basic operations, 234

polar form, 234

roots, 237

connected, simply or multiply, 257

contour integrals, 255

contraction of tensors, 50

contravariant tensor, 49

convolution theorem Fourier transforms, 188

coordinate system, see speci®c coordinate system

coset, 439

covariant diÿerentiation, 55

covariant tensor, 49

cross product of vectors see vector product of

vectors

crossing conditions, 441

curl

cartesian, 24

curvilinear, 32

cylindrical, 34

spherical polar, 35

551

curvilinear coordinates, 27

damped oscillations, 80

De Moivre's formula, 237

del, 22

formulas involving del, 27

delta function, Dirac, 183

Fourier integral, 183

Green's function and, 192

point source, 193

determinants, 538±547

diÿerential equations, 62

®rst order, 63

exact 67

integrating factors, 69

separable variables 63

homogeneous, 63

numerical solutions, 469

second order, constant coecients, 72

complementary functions, 74

Frobenius and Fuchs theorem, 86

particular integrals, 77

singular points, 86

solution in power series, 85

direct product

matrices, 139

tensors, 50

direction angles and direction cosines, 3

divergence

cartesian, 22

curvilinear, 30

cylindrical, 33

spherical polar, 35

dot product of vectors, 5

dual vectors and dual spaces, 211

eigenvalues and eigenfunctions of Sturm±

Liouville equations, 340

hermitian matrices, 124

orthogonality, 129

real, 128

an operator, 217

entire functions, 247

Euler's linear equation, 83

Fourier series, 144

convergence and Dirichlet conditions, 150

diÿerentiation, 157

Euler±Fourier formulas, 145

exponential form of Fourier series, 156

Gibbs phenomenon, 150

half-range Fourier series, 151

integration, 157

interval, change of, 152

orthogonality, 162

Parseval's identity, 153

vibrtating strings, 157

Fourier transform, 164

convolution theorem, 188

delta function derivation, 183

Fourier integral, 164

Fourier series (contd)

Fourier sine and cosine transforms, 172

Green's function method, 192

head conduction, 179

Heisenberg's uncertainty principle, 173

Parseval's identity, 186

solution of integral equation, 421

transform of derivatives, 190

wave packets and group velocity, 174

Fredholm integral equation, 413 see also integral

equations

Frobenius' method see series solution of

diÿerential equations

Frobenius±Fuch's theorem, 86

function

analytic,

entire,

harmonic, 247

function spaces, 226

gamma function, 94

gauge transformation, 411

Gauss' law, 391

Gauss' theorem, 37

generating function, for

associated Laguerre polynomials, 320

Bessel functions, 330

Hermite polynomials, 314

Laguerre polynomials, 317

Legendre polynomials, 301

Gibbs phenomenon, 150

gradient

cartesian, 20

curvilinear, 29

cylindrical, 33

spherical polar, 35

Gram±Schmidt orthogonalization, 209

Green's functions, 192,

construction of

one dimension, 192

three dimensions, 405

delta function, 193

Green's theorem, 43

in the plane, 44

group theory, 430

conjugate clsses, 440

cosets, 439

cyclic group, 433

rotation matrix, 234, 252

special unitary group, SU(2), 232

de®nitions, 430

dihedral groups, 446

generator, 451

homomorphism, 436

irreducible representations, 442

isomorphism, 435

multiplication table, 434

Lorentz group, 454

permutation group, 438

orthogonal group SO(3)

552

INDEX

group theory (contd)

symmetry group, 446

unitary group, 452

unitary unimodular group SUn

Hamilton±Jacobi equation, 364

Hamilton's principle and Lagrange equations of

motion, 355

Hankel functions, 328

Hankel transforms, 385

harmonic functions, 247

Helmholtz theorem, 44

Hermite equation, 311

Hermite polynomials, 312

generating function, 314

orthogonality, 314

recurrence relations, 313

hermitian matrices, 114

orthogonal eigenvectors, 129

real eigenvalues, 128

hermitian operator, 220

completeness of eigenfunctions, 221

eigenfunctions, orthogonal, 220

eigenvalues, real, 220

Hilbert space, 230

Hilbert-Schmidt method of solution, 421

homogeneous see linear equations

homomorphism, 436

indicial equation, 87

inertia, moment of, 135

in®nity see singularity, pole essential singularity

integral equations, 413

Abel's equation, 426

classical harmonic oscillator, 427

diÿerential equation±integral equation

transformation, 419

Fourier transform solution, 421

Fredholm equation, 413

Laplace transform solution, 420

Neumann series, 416

quantum harmonic oscillator, 427

Schmidt±Hilbert method, 421

separable kernel, 414

Volterra equation, 414

integral transforms, 384 see also Fourier

transform, Hankel transform, Laplace

transform, Mellin transform

Fourier, 164

Hankel, 385

Laplace, 372

Mellin, 385

integration, vector, 35

line integrals, 36

surface integrals, 36

interpolation, 1461

inverse operator, uniqueness of, 218

irreducible group representations, 442

Jacobian, 29

kernels of integral equations, 414

separable, 414

Kronecker delta, 6

mixed second-rank tensor, 53

Lagrangian, 355

Lagrangian multipliers, 354

Laguerre equation, 316

associated Laguerre equation, 320

Laguerre functions, 317

associated Laguerre polynomials, 320

generating function, 317

orthogonality, 319

Rodrigues' representation, 318

Laplace equation, 389

solutions of, 392

Laplace transform, 372

existence of, 373

integration of tranforms, 383

inverse transformation, 373

solution of integral equation, 420

the ®rst shifting theorem, 378

the second shifting theorem, 379

transform of derivatives, 382

Laplacian

cartesian, 24

cylindrical, 34

scalar, 24

spherical polar, 35

Laurent expansion, 274

Legendre equation, 296

associated Legendre equation, 307

series solution of Legendre equation, 296

Legendre functions, 299

associated Legendre functions, 308

generating function, 301

orthogonality, 304

recurrence relations, 302

Rodrigues' formula, 299

linear combination, 204

linear independence, 204

linear operator, 212

Lorentz gauge condition, 411

Lorentz group, 454

Lorentz transformation, 455

mapping, 239

matrices, 100

anti-hermitian, 114

commutator

de®nition, 100

diagonalization, 129

direct product, 139

eigenvalues and eigenvectors, 124

Hermitian, 114

nverse, 111

matrix multiplication, 103

moment of inertia, 135

orthogonal, 115

and unitary transformations, 121

553

INDEX

matrices (contd)

Pauli spin, 142

representation, 226

rotational, 117

similarity transformation, 122

symmetric and skew-symmetric, 109

trace, 121

transpose, 108

unitary, 116

Maxwell equations, 411

derivation of wave equation, 411

Mellin transforms, 385

metric tensor, 51

mixed tensor, 49

Morera's theorem, 259

multipoles, 248

Neumann functions, 327

Newton's root ®nding formula, 465

normal modes of vibrations 136

numerical methods, 459

roots of equations, 460

false position (linear interpolation), 461

graphic methods, 460

Newton's method, 464

integration, 466

rectangular rule, 455

Simpson's rule, 469

trapezoidal rule, 467

interpolation, 459

least-square ®t, 477

solutions of diÿerential equations, 469

Euler's rule, 470

Runge±Kutta method, 473

Taylor series method, 472

system of equations, 476

operators

adjoint, 212

angular momentum operator, 18

commuting, 225

del, 22

diÿerential operator D (d=dx), 78

linear, 212

orthonormal, 161, 207

oscillator,

damped, 80

integral equations for, 427

simple harmonic, 427

Parseval's identity, 153, 186

partial diÿerential equations, 387

linear second order, 388

elliptic, hyperbolic, parabolic, 388

Green functions, 404

Laplace transformation

separation of variables

Laplace's equation, 392, 395, 398

wave equation, 402

Pauli spin matrices, 142

phase of a complex number, 235

Poisson equation. 389

polar form of complex numbers, 234

poles, 248

probability theory , 481

combinations, 485

continuous distributions, 500

Gaussian, 502

Maxwell±Boltzmann, 503

de®nition of probability, 481

expectation and variance, 490

fundamental theorems, 486

probability distributions, 491

binomial, 491

Gaussian, 497

Poisson, 495

sample space, 482

power series, 269

solution of diÿerential equations, 85

projection operators, 222

pseudovectors, 8

quotient rule, 50

rank (order), of tensor, 49

of group, 430

Rayleigh±Ritz (variational) method, 359

recurrence relations

Bessel functions, 332

Hermite functions, 313

Laguerre functions, 318

Legendre functions, 302

residue theorem, 282

residues 279

calculus of residues, 280

Riccati equation, 98

Riemann surface, 241

Rodrigues' formula

Hermite polynomials, 313

Laguerre polynomials, 318

associated Laguerre polynomials, 320

Legendre polynomials, 299

associated Legendre polynomials, 308

root diagram, 238

rotation

groups SO2; SO3, 450

of coordinates, 11, 117

of vectors, 11±13

Runge±Kutta solution, 473

scalar, de®nition of, 1

scalar potential, 20, 390, 411

scalar product of vectors, 5

Schmidt orthogonalization see Gram±Schmidt

orthogonalization

Schro

dinger wave equation, 427

variational approach, 368

Schwarz±Cauchy inequality, 210

secular (characteristic) equation, 125

554

INDEX

series solution of diÿerential equations,

Bessel's equation, 322

Hermite's equation, 311

Laguerre's equation, 316

Legendre's equation, 296

associated Legendre's equation, 307

similarity transformation, 122

singularity, 86, 248

branch point, 240

diÿerential equation, 86

Laurent series, 274

on contour of integration, 290

special unitary group, SUn, 452

spherical polar coordinates, 34

step function, 380

Stirling's asymptotic formula for n!, 99

Stokes' theorem, 40

Sturm±Liouville equation, 340

subgroup, 439

summation convention (Einstein's), 48

symbolic software, 492

Taylor series of elementary functions, 272

tensor analysis, 47

associated tensor, 53

basic operations with tensors, 49

contravariant vector, 48

covariant diÿerentiation, 55

covariant vector, 48

tensor analysis (contd)

de®nition of second rank tensor, 49

geodesic in Riemannian space, 53

metric tensor, 51

quotient law, 50

symmetry±antisymmetry, 50

trace (matrix), 121

triple scalar product of vectors, 10

triple vector product of vectors, 11

uncertainty principle in quantum theory, 173

unit group element, 430

unit vectors

cartesian, 3

cylindrical, 32

spherical polar, 34

variational principles, see calculus of variations

vector and tensor analysis, 1±56

vector potential, 411

vector product of vectors, 7

vector space, 13, 199

Volterra integral equation. see integral equations

wave equation, 389

derivation from Maxwell's equations, 411

solution of, separation of variables, 402

wave packets, 174

group velocity, 174

555

INDEX