Mathematical Methods for Physicists: A concise introduction - Site Map

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CONTENTS Transformation of an integral equation into a di€erential equation 419 Laplace transform solution 420 Fourier transform solution 421 The Schmidt±Hilbert method of solution 421 Relation between di€erential and integral equations 425 Use of integral equations 426 Abel's integral equation 426 Classical simple harmonic oscillator 427 Quantum simple harmonic oscillator 427 Problems 428 12 Elements of group theory 430 De®nition of a group (group axioms) 430 Cyclic groups 433 Group multiplication table 434 Isomorphic groups 435 Group of permutations and Cayley's theorem 438 Subgroups and cosets 439 Conjugate classes and invariant subgroups 440 Group representations 442 Some special groups 444 The symmetry group D 2 ; D 3 446 One-dimensional unitary group U…1† 449 Orthogonal groups SO…2† and SO…3† 450 The SU…n† groups 452 Homogeneous Lorentz group 454 Problems 457 13 Numerical methods 459 Interpolation 459 Finding roots of equations 460 Graphical methods 460 Method of linear interpolation (method of false position) 461 Newton's method 464 Numerical integration 466 The rectangular rule 466 The trapezoidal rule 467 Simpson's rule 469 Numerical solutions of di€erential equations 469 Euler's method 470 The three-term Taylor series method 472 xii
CONTENTS The Runge±Kutta method 473 Equations of higher order. System of equations 476 Least-squares ®t 477 Problems 478 14 Introduction to probability theory 481 A de®nition of probability 481 Sample space 482 Methods of counting 484 Permutations 484 Combinations 485 Fundamental probability theorems 486 Random variables and probability distributions 489 Random variables 489 Probability distributions 489 Expectation and variance 490 Special probability distributions 491 The binomial distribution 491 The Poisson distribution 495 The Gaussian (or normal) distribution 497 Continuous distributions 500 The Gaussian (or normal) distribution 502 The Maxwell±Boltzmann distribution 503 Problems 503 Appendix 1 Preliminaries (review of fundamental concepts) 506 Inequalities 507 Functions 508 Limits 510 In®nite series 511 Tests for convergence 513 Alternating series test 516 Absolute and conditional convergence 517 Series of functions and uniform convergence 520 Weistrass M test 521 Abel's test 522 Theorem on power series 524 Taylor's expansion 524 Higher derivatives and Leibnitz's formula for nth derivative of a product 528 Some important properties of de®nite integrals 529 xiii
Page 1 and 2: Mathematical Methods for Physicists
Page 3 and 4: PUBLISHED BY CAMBRIDGE UNIVERSITY P
Page 5 and 6: Contents Preface xv 1 Vector and te
Page 7 and 8: CONTENTS 3 Matrix algebra 100 De®n
Page 9 and 10: CONTENTS The adjoint operators 219
Page 11: CONTENTS Sturm±Liouville systems 3
Page 15 and 16: Preface This book evolved from a se
Page 17 and 18: VECTOR AND TENSOR ANALYSIS Figure 1
Page 19 and 20: VECTOR AND TENSOR ANALYSIS Vector a
Page 21 and 22: VECTOR AND TENSOR ANALYSIS We can g
Page 23 and 24: VECTOR AND TENSOR ANALYSIS Using th
Page 25 and 26: VECTOR AND TENSOR ANALYSIS The trip
Page 27 and 28: VECTOR AND TENSOR ANALYSIS will be
Page 29 and 30: VECTOR AND TENSOR ANALYSIS orthonor
Page 31 and 32: VECTOR AND TENSOR ANALYSIS A(u) is
Page 33 and 34: VECTOR AND TENSOR ANALYSIS Figure 1
Page 35 and 36: VECTOR AND TENSOR ANALYSIS a circle
Page 37 and 38: VECTOR AND TENSOR ANALYSIS The vect
Page 39 and 40: VECTOR AND TENSOR ANALYSIS operate
Page 41 and 42: VECTOR AND TENSOR ANALYSIS We ®rst
Page 43 and 44: VECTOR AND TENSOR ANALYSIS system w
Page 45 and 46: VECTOR AND TENSOR ANALYSIS we have
Page 47 and 48: VECTOR AND TENSOR ANALYSIS with sim
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Page 51 and 52: VECTOR AND TENSOR ANALYSIS Z P2 P 1
Page 53 and 54: VECTOR AND TENSOR ANALYSIS name is
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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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VECTOR AND TENSOR ANALYSIS Figure 1
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VECTOR AND TENSOR ANALYSIS Figure 1
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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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Appendix 1 Preliminaries (review of
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APPENDIX 1 PRELIMINARIES Problem A1
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APPENDIX 1 PRELIMINARIES Example A1
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APPENDIX 1 PRELIMINARIES Since the
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APPENDIX 1 PRELIMINARIES Figure A1.
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APPENDIX 1 PRELIMINARIES How do we
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APPENDIX 1 PRELIMINARIES This follo
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APPENDIX 1 PRELIMINARIES Theorems o
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APPENDIX 1 PRELIMINARIES This is th
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APPENDIX 1 PRELIMINARIES (d) Find t
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APPENDIX 1 PRELIMINARIES Divide the
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APPENDIX 1 PRELIMINARIES Example A1
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APPENDIX 1 PRELIMINARIES Di€erent
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APPENDIX 1 PRELIMINARIES derivative
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Appendix 2 Determinants The determi
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APPENDIX 2 DETERMINANTS where a 11
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APPENDIX 2 DETERMINANTS D ˆ a i1 C
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APPENDIX 2 DETERMINANTS Example A2.
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APPENDIX 2 DETERMINANTS Proof: Expa
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Appendix 3 Table of * F…x† ˆp
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Index Abel's integral equation, 426
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INDEX group theory (contd) symmetry
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INDEX series solution of di€erent
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