Mathematical Methods for Physicists: A concise introduction - Site Map

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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 di€erential 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 di€erential operator D (ˆd=dx), 78 linear, 212 orthonormal, 161, 207 oscillator, damped, 80 integral equations for, 427 simple harmonic, 427 Parseval's identity, 153, 186 partial di€erential 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 di€erential 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 SchroÈ dinger wave equation, 427 variational approach, 368 Schwarz±Cauchy inequality, 210 secular (characteristic) equation, 125 554
INDEX series solution of di€erential 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 di€erential 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 di€erentiation, 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
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Mathematical Methods for Physicists
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PUBLISHED BY CAMBRIDGE UNIVERSITY P
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Contents Preface xv 1 Vector and te
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CONTENTS 3 Matrix algebra 100 De®n
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CONTENTS The adjoint operators 219
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CONTENTS Sturm±Liouville systems 3
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CONTENTS The Runge±Kutta method 47
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Preface This book evolved from a se
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VECTOR AND TENSOR ANALYSIS Figure 1
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VECTOR AND TENSOR ANALYSIS Vector a
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VECTOR AND TENSOR ANALYSIS We can g
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VECTOR AND TENSOR ANALYSIS Using th
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VECTOR AND TENSOR ANALYSIS The trip
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VECTOR AND TENSOR ANALYSIS will be
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VECTOR AND TENSOR ANALYSIS orthonor
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VECTOR AND TENSOR ANALYSIS A(u) is
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VECTOR AND TENSOR ANALYSIS a circle
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VECTOR AND TENSOR ANALYSIS The vect
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VECTOR AND TENSOR ANALYSIS operate
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VECTOR AND TENSOR ANALYSIS We ®rst
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VECTOR AND TENSOR ANALYSIS system w
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VECTOR AND TENSOR ANALYSIS we have
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VECTOR AND TENSOR ANALYSIS with sim
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VECTOR AND TENSOR ANALYSIS where A
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VECTOR AND TENSOR ANALYSIS Z P2 P 1
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VECTOR AND TENSOR ANALYSIS name is
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VECTOR AND TENSOR ANALYSIS while th
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VECTOR AND TENSOR ANALYSIS Next we
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VECTOR AND TENSOR ANALYSIS This imp
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VECTOR AND TENSOR ANALYSIS Proof:
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VECTOR AND TENSOR ANALYSIS Here we
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VECTOR AND TENSOR ANALYSIS (2) Addi
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VECTOR AND TENSOR ANALYSIS A genera
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VECTOR AND TENSOR ANALYSIS two poin
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VECTOR AND TENSOR ANALYSIS which sh
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2 Ordinary di€erential equations
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ORDINARY DIFFERENTIAL EQUATIONS The
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ORDINARY DIFFERENTIAL EQUATIONS Sol
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ORDINARY DIFFERENTIAL EQUATIONS It
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ORDINARY DIFFERENTIAL EQUATIONS whe
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ORDINARY DIFFERENTIAL EQUATIONS Gen
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ORDINARY DIFFERENTIAL EQUATIONS and
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ORDINARY DIFFERENTIAL EQUATIONS (1)
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ORDINARY DIFFERENTIAL EQUATIONS Now
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ORDINARY DIFFERENTIAL EQUATIONS Obv
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ORDINARY DIFFERENTIAL EQUATIONS Sol
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ORDINARY DIFFERENTIAL EQUATIONS Ord
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ORDINARY DIFFERENTIAL EQUATIONS Cas
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ORDINARY DIFFERENTIAL EQUATIONS Int
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ORDINARY DIFFERENTIAL EQUATIONS of
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ORDINARY DIFFERENTIAL EQUATIONS and
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ORDINARY DIFFERENTIAL EQUATIONS Fig
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3 Matrix algebra As vector methods
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MATRIX ALGEBRA Four basic algebra o
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MATRIX ALGEBRA then ~A ~B ˆ ˆ 2
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MATRIX ALGEBRA Figure 3.1. Coordina
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MATRIX ALGEBRA in®nite sum of matr
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MATRIX ALGEBRA The product of two s
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MATRIX ALGEBRA ~B ~ A ˆ ~I. Multip
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MATRIX ALGEBRA but so that ~A 1 …
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MATRIX ALGEBRA But if ~ A is an in
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MATRIX ALGEBRA Taking the dot produ
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MATRIX ALGEBRA change the handednes
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MATRIX ALGEBRA preserves the norm o
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MATRIX ALGEBRA Then ~A 0 R 0 ˆ…
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MATRIX ALGEBRA Once the eigenvalues
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MATRIX ALGEBRA Example 3.15 Show th
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0 1 x 11 x 1i x 1n x 21 x 2i x
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MATRIX ALGEBRA Since 0 1 1 B @ 1 0
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MATRIX ALGEBRA We now proceed to pr
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MATRIX ALGEBRA distance between any
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MATRIX ALGEBRA This matrix equation
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~A ~B ˆ a11 B ~ ! a 12 ~B ˆ a 21
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MATRIX ALGEBRA 3.18 If A ~ ~B ˆ 0,
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4 Fourier series and integrals Four
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FOURIER SERIES AND INTEGRALS Fourie
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FOURIER SERIES AND INTEGRALS Soluti
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FOURIER SERIES AND INTEGRALS Figure
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FOURIER SERIES AND INTEGRALS but b
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FOURIER SERIES AND INTEGRALS Parsev
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FOURIER SERIES AND INTEGRALS the A
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FOURIER SERIES AND INTEGRALS Orthog
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FOURIER SERIES AND INTEGRALS Substi
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FOURIER SERIES AND INTEGRALS and th
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FOURIER SERIES AND INTEGRALS We sho
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FOURIER SERIES AND INTEGRALS and th
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FOURIER SERIES AND INTEGRALS contin
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FOURIER SERIES AND INTEGRALS amount
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FOURIER SERIES AND INTEGRALS Using
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FOURIER SERIES AND INTEGRALS It is
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FOURIER SERIES AND INTEGRALS Parsev
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FOURIER SERIES AND INTEGRALS This i
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FOURIER SERIES AND INTEGRALS Soluti
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FOURIER SERIES AND INTEGRALS The de
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FOURIER SERIES AND INTEGRALS By the
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FOURIER SERIES AND INTEGRALS 4.10 F
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FOURIER SERIES AND INTEGRALS 4.22 V
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LINEAR VECTOR SPACES Figure 5.1. A
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LINEAR VECTOR SPACES We now proceed
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LINEAR VECTOR SPACES Linear combina
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LINEAR VECTOR SPACES is a basis for
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LINEAR VECTOR SPACES then Eq. (5.10
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LINEAR VECTOR SPACES Step 3. Clearl
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LINEAR VECTOR SPACES then hujvi ˆh
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LINEAR VECTOR SPACES Eq. (5.16) det
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LINEAR VECTOR SPACES In general A ~
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LINEAR VECTOR SPACES The inverse of
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LINEAR VECTOR SPACES It is also eas
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LINEAR VECTOR SPACES or U ~ ‡ ˆ
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LINEAR VECTOR SPACES Change of basi
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LINEAR VECTOR SPACES Multiplying Eq
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LINEAR VECTOR SPACES For a vibratin
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LINEAR VECTOR SPACES the integratio
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LINEAR VECTOR SPACES (b) T…a† i
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FUNCTIONS OF A COMPLEX VARIABLE and
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FUNCTIONS OF A COMPLEX VARIABLE We
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FUNCTIONS OF A COMPLEX VARIABLE Fig
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FUNCTIONS OF A COMPLEX VARIABLE ver
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FUNCTIONS OF A COMPLEX VARIABLE z 0
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FUNCTIONS OF A COMPLEX VARIABLE If
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FUNCTIONS OF A COMPLEX VARIABLE How
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FUNCTIONS OF A COMPLEX VARIABLE Sim
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FUNCTIONS OF A COMPLEX VARIABLE Di
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FUNCTIONS OF A COMPLEX VARIABLE fun
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FUNCTIONS OF A COMPLEX VARIABLE Sin
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FUNCTIONS OF A COMPLEX VARIABLE pro
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FUNCTIONS OF A COMPLEX VARIABLE Alt
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FUNCTIONS OF A COMPLEX VARIABLE Exa
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FUNCTIONS OF A COMPLEX VARIABLE Exa
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FUNCTIONS OF A COMPLEX VARIABLE For
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FUNCTIONS OF A COMPLEX VARIABLE Sol
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FUNCTIONS OF A COMPLEX VARIABLE Rat
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FUNCTIONS OF A COMPLEX VARIABLE Not
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FUNCTIONS OF A COMPLEX VARIABLE The
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FUNCTIONS OF A COMPLEX VARIABLE (b)
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FUNCTIONS OF A COMPLEX VARIABLE to
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FUNCTIONS OF A COMPLEX VARIABLE Sol
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FUNCTIONS OF A COMPLEX VARIABLE the
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FUNCTIONS OF A COMPLEX VARIABLE Thu
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FUNCTIONS OF A COMPLEX VARIABLE If
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FUNCTIONS OF A COMPLEX VARIABLE the
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FUNCTIONS OF A COMPLEX VARIABLE onl
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FUNCTIONS OF A COMPLEX VARIABLE To
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FUNCTIONS OF A COMPLEX VARIABLE Tak
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FUNCTIONS OF A COMPLEX VARIABLE 6.2
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7 Special functions of mathematical
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SPECIAL FUNCTIONS OF MATHEMATICAL P
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THE CALCULUS OF VARIATIONS Figure 8
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THE CALCULUS OF VARIATIONS This is
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THE CALCULUS OF VARIATIONS p Lettin
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THE CALCULUS OF VARIATIONS provided
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THE CALCULUS OF VARIATIONS and Eq.
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THE CALCULUS OF VARIATIONS Example
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THE CALCULUS OF VARIATIONS must van
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THE CALCULUS OF VARIATIONS mechanic
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THE CALCULUS OF VARIATIONS Solution
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THE CALCULUS OF VARIATIONS The acti
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THE CALCULUS OF VARIATIONS is to ®
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THE CALCULUS OF VARIATIONS y…"; x
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9 The Laplace transformation The La
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THE LAPLACE TRANSFORMATION This est
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THE LAPLACE TRANSFORMATION or L‰x
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THE LAPLACE TRANSFORMATION Recall L
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THE LAPLACE TRANSFORMATION Let u ˆ
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THE LAPLACE TRANSFORMATION But f
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THE LAPLACE TRANSFORMATION but and
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THE LAPLACE TRANSFORMATION 9.6 Find
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PARTIAL DIFFERENTIAL EQUATIONS prob
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PARTIAL DIFFERENTIAL EQUATIONS If
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PARTIAL DIFFERENTIAL EQUATIONS In t
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PARTIAL DIFFERENTIAL EQUATIONS The
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PARTIAL DIFFERENTIAL EQUATIONS We a
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PARTIAL DIFFERENTIAL EQUATIONS Let
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PARTIAL DIFFERENTIAL EQUATIONS brin
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PARTIAL DIFFERENTIAL EQUATIONS We n
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PARTIAL DIFFERENTIAL EQUATIONS whic
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PARTIAL DIFFERENTIAL EQUATIONS to t
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PARTIAL DIFFERENTIAL EQUATIONS Subs
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PARTIAL DIFFERENTIAL EQUATIONS Now
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PARTIAL DIFFERENTIAL EQUATIONS Figu
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SIMPLE LINEAR INTEGRAL EQUATIONS f
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SIMPLE LINEAR INTEGRAL EQUATIONS Th
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SIMPLE LINEAR INTEGRAL EQUATIONS wh
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SIMPLE LINEAR INTEGRAL EQUATIONS in
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SIMPLE LINEAR INTEGRAL EQUATIONS A
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SIMPLE LINEAR INTEGRAL EQUATIONS no
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SIMPLE LINEAR INTEGRAL EQUATIONS Th
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SIMPLE LINEAR INTEGRAL EQUATIONS wh
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12 Elements of group theory Group t
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ELEMENTS OF GROUP THEORY The same s
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ELEMENTS OF GROUP THEORY Figure 12.
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ELEMENTS OF GROUP THEORY Table 12.3
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ELEMENTS OF GROUP THEORY Obviously
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ELEMENTS OF GROUP THEORY This can b
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ELEMENTS OF GROUP THEORY Group repr
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ELEMENTS OF GROUP THEORY (1) A matr
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ELEMENTS OF GROUP THEORY Equation (
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ELEMENTS OF GROUP THEORY Table 12.6
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ELEMENTS OF GROUP THEORY met. In qu
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ELEMENTS OF GROUP THEORY 0 1 1 0 0
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ELEMENTS OF GROUP THEORY Figure 12.
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ELEMENTS OF GROUP THEORY We can exp
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ELEMENTS OF GROUP THEORY Next, we a
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NUMERICAL METHODS is rather tedious
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NUMERICAL METHODS Figure 13.2. cour
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NUMERICAL METHODS and take g…x†
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NUMERICAL METHODS Solution: Here f
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NUMERICAL METHODS where y i ˆ f
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NUMERICAL METHODS The ®rst-order o
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NUMERICAL METHODS approximate value
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NUMERICAL METHODS Runge±Kutta meth
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NUMERICAL METHODS Equations of high
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NUMERICAL METHODS We now illustrate
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NUMERICAL METHODS 13.14. Find to th
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INTRODUCTION TO PROBABILITY THEORY
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Page 517 and 518: INTRODUCTION TO PROBABILITY THEORY
Page 519 and 520: INTRODUCTION TO PROBABILITY THEORY
Page 521 and 522: Appendix 1 Preliminaries (review of
Page 523 and 524: APPENDIX 1 PRELIMINARIES Problem A1
Page 527 and 528: APPENDIX 1 PRELIMINARIES Example A1
Page 529 and 530: APPENDIX 1 PRELIMINARIES Since the
Page 531 and 532: APPENDIX 1 PRELIMINARIES Figure A1.
Page 533 and 534: APPENDIX 1 PRELIMINARIES How do we
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Page 539 and 540: APPENDIX 1 PRELIMINARIES Theorems o
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Page 543 and 544: APPENDIX 1 PRELIMINARIES (d) Find t
Page 545 and 546: APPENDIX 1 PRELIMINARIES Divide the
Page 547 and 548: APPENDIX 1 PRELIMINARIES Example A1
Page 549 and 550: APPENDIX 1 PRELIMINARIES Di€erent
Page 551 and 552: APPENDIX 1 PRELIMINARIES derivative
Page 553 and 554: Appendix 2 Determinants The determi
Page 555 and 556: APPENDIX 2 DETERMINANTS where a 11
Page 557 and 558: APPENDIX 2 DETERMINANTS D ˆ a i1 C
Page 559 and 560: APPENDIX 2 DETERMINANTS Example A2.
Page 561 and 562: APPENDIX 2 DETERMINANTS Proof: Expa
Page 563 and 564: Appendix 3 Table of * F…x† ˆp
Page 565 and 566: Index Abel's integral equation, 426
Page 567: INDEX group theory (contd) symmetry
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