Mathematical Methods for Physicists: A concise introduction - Site Map

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PROBLEMS 14.18 Given the density function f …x† ( f …x† ˆ kx2 0 < x < 3 0 otherwise : (a) ®nd the constant k; (b) compute P…1 < x < 2†; (c) ®nd the distribution function and use it to ®nd P…1 < x …2††: 505
Appendix 1 Preliminaries (review of fundamental concepts) This appendix is for those readers who need a review; a number of fundamental concepts or theorem will be reviewed without giving proofs or attempting to achieve completeness. We assume that the reader is already familiar with the classes of real numbers used in analysis. The set of positive integers (also known as natural numbers) 1, 2, ...; n admits the operations of addition without restriction, that is, they can be added (and therefore multiplied) together to give other positive integers. The set of integers 0, 1; 2; ...; n admits the operations of addition and subtraction among themselves. Rational numbers are numbers of the form p=q, where p and q are integers and q 6ˆ 0. Examples of rational numbers are 2/3, 10=7. This set admits the further property of division among its members. The set of irrational numbers includes all numbers which cannot be p expressed as the quotient of two integers. Examples of irrational numbers are 2 ; 3p p 11 ; and any number of the form n a=b , where a and b are integers which are perfect nth powers. The set of real numbers contains all the rationals and irrationals. The important property of the set of real numbers fxg is that it can be put into (1:1) correspondence with the set of points fPg of a line as indicated in Fig. A.1. The basic rules governing the combinations of real numbers are: commutative law: a ‡ b ˆ b ‡ a; a b ˆ b a; associative law: a ‡…b ‡ c† ˆ…a ‡ b†‡c; a …b c† ˆ…a b†c; distributive law: a …b ‡ c† ˆa b ‡ a c; index law a m a n ˆ a m‡n ; a m =a n ˆ a mn …a 6ˆ 0†; where a; b; c, are algebraic symbols for the real numbers. Problem A1.1 p Prove that 2 is an irrational number. 506
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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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Page 475 and 476: NUMERICAL METHODS is rather tedious
Page 477 and 478: NUMERICAL METHODS Figure 13.2. cour
Page 479 and 480: NUMERICAL METHODS and take g…x†
Page 481 and 482: NUMERICAL METHODS Solution: Here f
Page 483 and 484: NUMERICAL METHODS where y i ˆ f
Page 485 and 486: NUMERICAL METHODS The ®rst-order o
Page 487 and 488: NUMERICAL METHODS approximate value
Page 489 and 490: NUMERICAL METHODS Runge±Kutta meth
Page 491 and 492: NUMERICAL METHODS Equations of high
Page 493 and 494: NUMERICAL METHODS We now illustrate
Page 495 and 496: NUMERICAL METHODS 13.14. Find to th
Page 497 and 498: INTRODUCTION TO PROBABILITY THEORY
Page 519: INTRODUCTION TO PROBABILITY THEORY
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
Page 537 and 538: APPENDIX 1 PRELIMINARIES This follo
Page 539 and 540: APPENDIX 1 PRELIMINARIES Theorems o
Page 541 and 542: APPENDIX 1 PRELIMINARIES This is th
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 and 568: INDEX group theory (contd) symmetry
Page 569: INDEX series solution of di€erent
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