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LINEAR OPERATORS E 2 into E 3 . In particular, if jvi ˆ…1; 1†, then the image of jvi under T ~ is T ~ …jvi† ˆ …1; 2; 0†. It is easy to see that the transformation is linear. If jui ˆ…x 1 ; y 1 † and jvi ˆ…x 2 ; y 2 †, then so that T ~ jui‡jvi ˆ…x 1 ‡ x 2 ; y 1 ‡ y 2 †; … jui‡ ji v † ˆ … x 1 ‡ x 2 ; …x 1 ‡ x 2 †‡…y 1 ‡ y 2 †; …x 1 ‡ x 2 †…y 1 ‡ y 2 †† and if k is a scalar, then T ~ ˆ … x 1 ; x 1 ‡ y 1 ; x 1 y 1 †‡… x 2 ; x 2 ‡ y 2 ; x 2 y 2 † ˆ T … jui†‡T … ji v † ~ ~ … ku j i† ˆ … kx 1 ; kx 1 ‡ ky 1 ; kx 1 ky 1 † ˆ kx … 1 ; x 1 ‡ y 1 ; x 1 y 1 † ˆ kT … ji u †: ~ Thus T is a linear transformation. ~ If T maps the vector space onto itself (T: V ! V), then it is called a linear ~ ~ operator on V.InE 3 a rotation of the entire space about a ®xed axis is an example of an operation that maps the space onto itself. We saw in Chapter 3 that rotation can be represented by a matrix with elements ij …i; j ˆ 1; 2; 3†; ifx 1 ; x 2 ; x 3 are the components of an arbitrary vector in E 3 before the transformation and x1; 0 x2; 0 x3 0 the components of the transformed vector, then x1 0 9 ˆ 11 x 1 ‡ 12 x 2 ‡ 13 x 3 ; >= x2 0 ˆ 21 x 1 ‡ 22 x 2 ‡ 23 x 3 ; …5:15† >; x3 0 ˆ 31 x 1 ‡ 32 x 2 ‡ 33 x 3 : In matrix form we have ~x 0 ˆ ~…†~x; …5:16† where is the angle of rotation, and 0 x 0 1 0 1 0 1 1 x 1 11 12 13 ~x 0 B ˆ @ x2 0 C B A; ~x ˆ @ x C 2 A; and …† ~ B ˆ 21 22 C @ 23 A: x 3 31 32 33 x 0 3 In particular, if the rotation is carried out about x 3 -axis, …† ~ has the following form: 0 cos sin 1 0 B ~…† ˆ@ sin cos C 0 A: 0 0 1 213
LINEAR VECTOR SPACES Eq. (5.16) determines the vector x 0 if the vector x is given, and …† ~ is the operator (matrix representation of the rotation operator) which turns x into x 0 . Loosely speaking, an operator is any mathematical entity which operates on any vector in V and turns it into another vector in V. Abstractly, an operator L is ~ a mapping that assigns to a vector jvi in a linear vector space V another vector jui in V: jui ˆL jvi. The set of vectors jvi for which the mapping is de®ned, that is, ~ the set of vectors jvi for which L jvi has meaning, is called the domain of L . The ~ ~ set of vectors jui in the domain expressible as jui ˆL jvi is called the range of the ~ operator. An operator L is linear if the mapping is such that for any vectors ~ jui; jwi in the domain of L and for arbitrary scalars , , the vector ~ jui‡jwi is in the domain of L and ~ L ~ …jui‡jwi† ˆ L ~ jui‡L ~ jwi: A linear operator is bounded if its domain is the entire space V and if there exists a single constant C such that jL ~ jvij < Cjjvij for all jvi in V. We shall consider linear bounded operators only. Matrix representation of operators Linear bounded operators may be represented by matrix. The matrix will have a ®nite or an in®nite number of rows according to whether the dimension of V is ®nite or in®nite. To show this, let j1i; j2i; ... be an orthonormal basis in V; then every vector j'i in V may be written in the form Since L ~ ji is also in V, we may write But so j'i ˆ 1 j1i‡ 2 j2i‡: L ~ j'i ˆ 1 j1i‡ 2 j2i‡: L ~ j'i ˆ 1 L ~ j1i‡ 2 L ~ j2i‡; 1 j1i‡ 2 j2i‡ˆ 1 L ~ j1i‡ 2 L ~ j2i‡: Taking the inner product of both sides with h1j we obtain 1 ˆh1jL ~ j1i 1 ‡h1jL ~ j2i 2 ˆ 11 1 ‡ 12 2 ‡; 214
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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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Page 215 and 216: LINEAR VECTOR SPACES Figure 5.1. A
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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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