Mathematical Methods for Physicists: A concise introduction - Site Map

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INTEGRATION AND DIFFERENTIATION OF A FOURIER SERIES constructed directly from the given function. Furthermore, in the complex representation de®ned by Eqs. (4.13a) and (4.13b), a certain symmetry between the expressions for a function and for its Fourier coecients is evident. In fact the expressions (4.13a) and (4.13b) are of essentially the same structure, as the following correlation reveals: X x L; f …x† c n c…n†; e inx=L e inx=L 1 ; … † 1 Z L …†dx: 2L L nˆ1 This duality is worthy of note, and as our development proceeds to the Fourier integral, it will become more striking and fundamental. Integration and di€erentiation of a Fourier series The Fourier series of a function f …x† may always be integrated term-by-term to give a new series which converges to the integral of f …x†. Iff …x† is a continuous function of x for all x, and is periodic (of period 2) outside the interval
FOURIER SERIES AND INTEGRALS Figure 4.9. A vibrating string. We limit the displacement to small values, so that we may set cos ˆ 1 2 =2; sin tan ˆ dy=dx; then F y ˆ AP dy dx x‡dx dy dx x ˆ AP d2 y dx 2 dx: Using Newton's second law, the equation of motion of transverse vibration of the element becomes Adx @2 y @t 2 ˆ AP @2 y @x 2 dx; or @ 2 y @x 2 ˆ 1 @ 2 y v 2 @t 2 ; v ˆ p P= : Thus the transverse displacement of the string satis®es the partial di€erential wave equation @ 2 y @x 2 ˆ 1 @ 2 y v 2 ; 0 < x < L; t > 0 …4:14† 2 @t with the following boundary conditions: y…0; t† ˆy…L; t† ˆ0; @y=@t ˆ 0; y…x; 0† ˆf …x†; where f …x† describes the initial shape (position) of the string, and v is the velocity of propagation of the wave along the string. Solution of the wave equation To solve this boundary value problem, let us try the method of separation variables: y…x; t† ˆX…x†T…t†: …4:15† Substituting this into Eq. (4.14) yields …1=X†…d 2 X=dx 2 †ˆ…1=v 2 T†…d 2 T=dt 2 †. 158
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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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Page 127 and 128: MATRIX ALGEBRA ~B ~ A ˆ ~I. Multip
Page 129 and 130: MATRIX ALGEBRA but so that ~A 1 …
Page 131 and 132: MATRIX ALGEBRA But if ~ A is an in
Page 133 and 134: MATRIX ALGEBRA Taking the dot produ
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Page 139 and 140: MATRIX ALGEBRA Then ~A 0 R 0 ˆ…
Page 141 and 142: MATRIX ALGEBRA Once the eigenvalues
Page 143 and 144: MATRIX ALGEBRA Example 3.15 Show th
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Page 147 and 148: MATRIX ALGEBRA Since 0 1 1 B @ 1 0
Page 149 and 150: MATRIX ALGEBRA We now proceed to pr
Page 151 and 152: MATRIX ALGEBRA distance between any
Page 153 and 154: MATRIX ALGEBRA This matrix equation
Page 155 and 156: ~A ~B ˆ a11 B ~ ! a 12 ~B ˆ a 21
Page 157 and 158: MATRIX ALGEBRA 3.18 If A ~ ~B ˆ 0,
Page 159 and 160: 4 Fourier series and integrals Four
Page 161 and 162: FOURIER SERIES AND INTEGRALS Fourie
Page 163 and 164: FOURIER SERIES AND INTEGRALS Soluti
Page 165 and 166: FOURIER SERIES AND INTEGRALS Figure
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Page 169 and 170: FOURIER SERIES AND INTEGRALS Parsev
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Page 189 and 190: FOURIER SERIES AND INTEGRALS We sho
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Page 193 and 194: FOURIER SERIES AND INTEGRALS contin
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Page 215 and 216: LINEAR VECTOR SPACES Figure 5.1. A
Page 217 and 218: LINEAR VECTOR SPACES We now proceed
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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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