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The topology of the calculus of variations in the large
Liusternik, L. A.(Lazar Aronovich)
اطلاعات کتابشناختی
The topology of the calculus of variations in the large
Author :
Liusternik, L. A.(Lazar Aronovich)
Publisher :
American Mathematical Society,
Pub. Year :
1966
Subjects :
Calculus of variations. Topology.
Call Number :
QA 316 .L713
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Cover
(1)
Half-title
(3)
Title
(5)
Copyright
(6)
Dedication
(7)
Contents
(9)
Preface
(17)
1 Vector algebra
(21)
1.1 Preliminaries
(21)
1.2 Coordinate system invariance
(24)
1.3 Vector multiplication
(29)
1.3.1 Multiplication by a scalar
(29)
1.3.2 Scalar or dot product
(29)
1.3.3 Vector or cross product
(30)
1.4 Useful products of vectors
(32)
1.4.1 Triple scalar product
(32)
1.4.2 Triple vector product
(33)
1.5 Linear vector spaces
(33)
1.6 Focus: periodic media and reciprocal lattice vectors
(37)
1.7 Additional reading
(44)
1.8 Exercises
(44)
2 Vector calculus
(48)
2.1 Introduction
(48)
2.2 Vector integration
(49)
2.2.1 Path integrals
(49)
2.2.2 Surface integrals
(51)
2.3 The gradient, V
(55)
2.4 Divergence, V
(57)
2.5 The curl, Vx
(61)
2.6 Further applications of V
(63)
2.6.1…
(63)
2.6.2…
(63)
2.6.3…
(64)
2.6.4 Laplacian,…
(64)
2.7 Gauss' theorem (divergence theorem)
(65)
2.8 Stokes' theorem
(67)
2.9 Potential theory
(68)
2.10 Focus: Maxwell's equations in integral and differential form
(71)
2.10.1 Gauss’ law and “no magnetic monopoles”
(72)
2.10.2 Faraday’s law and Ampère’s law
(73)
2.11 Focus: gauge freedom in Maxwell’s equations
(77)
2.11 Focus: gauge freedom in Maxwell's equations
(77)
2.12 Additional reading
(80)
2.13 Exercises
(80)
3 Vector calculus in curvilinear coordinate systems
(84)
3.1 Introduction: systems with different symmetries
(84)
3.2 General orthogonal coordinate systems
(85)
3.3 Vector operators in curvilinear coordinates
(89)
3.3.1 The gradient
(89)
3.3.2 The divergence
(90)
3.3.3 The curl
(91)
3.4 Cylindrical coordinates
(93)
3.5 Spherical coordinates
(96)
3.6 Exercises
(99)
4 Matrices and linear algebra
(103)
4.1 Introduction: Polarization and Jones vectors
(103)
4.1.1 Linear polarization to circular polarization
(107)
4.1.2 Rotation of polarization
(107)
4.1.3 Optical attenuator
(107)
4.2 Matrix algebra
(108)
4.3 Systems of equations, determinants, and inverses
(113)
4.3.1 Gaussian elimination
(119)
4.3.2 Gauss–Jordan matrix inversion
(120)
4.4 Orthogonal matrices
(122)
4.5 Hermitian matrices and unitary matrices
(125)
4.6 Diagonalization of matrices, eigenvectors, and eigenvalues
(127)
4.6.1 Hermitian matrices, normal matrices, and eigenvalues
(132)
4.7 Gram--Schmidt orthonormalization
(135)
4.8 Orthonormal vectors and basis vectors
(138)
4.9 Functions of matrices
(140)
4.10 Focus: matrix methods for geometrical optics
(140)
4.10.1 Introduction
(140)
4.10.2 Translation and refraction in matrix form
(143)
4.10.3 Lenses
(145)
4.10.4 Conjugate matrix
(148)
4.10.5 Imaging by a thin lens
(150)
4.10.6 Lens with a dielectric plate
(151)
4.10.7 Telescope
(152)
4.11 Additional reading
(153)
4.12 Exercises
(153)
5 Advanced matrix techniques and tensors
(159)
5.1 Introduction: Foldy–Lax scattering theory
(159)
5.2 Advanced matrix terminology
(162)
5.3 Left--right eigenvalues and biorthogonality
(163)
5.4 Singular value decomposition
(166)
5.5 Other matrix manipulations
(173)
5.5.1 Gaussian elimination with pivoting
(173)
5.5.2 LU decomposition
(175)
5.6 Tensors
(179)
5.6.1 Einstein summation convention
(179)
5.6.2 Covariant and contravariant vectors and tensors
(180)
5.6.3 Tensor addition and tensor products
(182)
5.6.4 The metric tensor
(184)
5.6.5 Pseudotensors
(188)
5.6.6 Tensors in curvilinear coordinates
(189)
5.6.7 The covariant derivative
(192)
5.7 Additional reading
(194)
5.8 Exercises
(194)
6 Distributions
(197)
6.1 Introduction: Gauss' law and the Poisson equation
(197)
6.2 Introduction to delta functions
(201)
6.3 Calculus of delta functions
(204)
6.3.1 Shifting
(204)
6.3.2 Scaling
(204)
6.3.3 Composition
(204)
6.3.4 Derivatives
(205)
6.4 Other representations of the delta function
(205)
6.5 Heaviside step function
(207)
6.6 Delta functions of more than one variable
(208)
6.7 Additional reading
(212)
6.8 Exercises
(212)
7 Infinite series
(215)
7.1 Introduction: the Fabry--Perot interferometer
(215)
7.2 Sequences and series
(218)
7.3 Series convergence
(221)
7.4 Series of functions
(230)
7.5 Taylor series
(233)
7.6 Taylor series in more than one variable
(238)
7.7 Power series
(240)
7.8 Focus: convergence of the Born series
(241)
7.9 Additional reading
(246)
7.10 Exercises
(246)
8 Fourier series
(250)
8.1 Introduction: diffraction gratings
(250)
8.2 Real-valued Fourier series
(253)
8.3 Examples
(256)
8.4 Integration range of the Fourier series
(259)
8.5 Complex-valued Fourier series
(259)
8.6 Properties of Fourier series
(260)
8.6.1 Symmetric series
(260)
8.6.2 Existence, completeness, and closure relations
(262)
8.7 Gibbs phenomenon and convergence in the mean
(263)
8.8 Focus: X-ray diffraction from crystals
(266)
8.9 Additional reading
(269)
8.10 Exercises
(269)
9 Complex analysis
(272)
9.1 Introduction: electric potential in an infinite cylinder
(272)
9.2 Complex algebra
(274)
9.3 Functions of a complex variable
(278)
9.4 Complex derivatives and analyticity
(281)
9.5 Complex integration and Cauchy's integral theorem
(285)
9.6 Cauchy's integral formula
(289)
9.7 Taylor series
(291)
9.8 Laurent series
(293)
9.9 Classification of isolated singularities
(296)
9.10 Branch points and Riemann surfaces
(298)
9.11 Residue theorem
(305)
9.12 Evaluation of definite integrals
(308)
9.12.1 Definite integrals…
(308)
9.12.2 Definite integrals…
(309)
9.12.3 Definite integrals…
(312)
9.12.4 Other integrals
(314)
9.13 Cauchy principal value
(317)
9.14 Focus: Kramers--Kronig relations
(319)
9.15 Focus: optical vortices
(322)
9.16 Additional reading
(328)
9.17 Exercises
(328)
10 Advanced complex analysis
(332)
10.1 Introduction
(332)
10.2 Analytic continuation
(332)
10.3 Stereographic projection
(336)
10.4 Conformal mapping
(345)
10.5 Significant theorems in complex analysis
(352)
10.5.1 Liouville’s theorem
(353)
10.5.2 Morera’s theorem
(353)
10.5.3 Maximum modulus principle
(355)
10.5.4 Arguments, zeros, and Rouché’s theorem
(356)
10.6 Focus: analytic properties of wavefields
(360)
10.7 Focus: optical cloaking and transformation optics
(365)
10.8 Exercises
(368)
11 Fourier transforms
(370)
11.1 Introduction: Fraunhofer diffraction
(370)
11.2 The Fourier transform and its inverse
(372)
11.3 Examples of Fourier transforms
(374)
11.4 Mathematical properties of the Fourier transform
(378)
11.4.1 Existence conditions for Fourier transforms
(379)
11.4.2 The Fourier inverse and delta functions
(381)
11.4.3 Other forms of the Fourier transform
(382)
11.4.4 Linearity of the Fourier transform
(383)
11.4.5 Conjugation
(383)
11.4.6 Time shift
(383)
11.4.7 Frequency shift
(384)
11.4.8 Scaling property
(384)
11.4.9 Differentiation in the time domain
(385)
11.4.10 Differentiation in the frequency domain
(385)
11.5 Physical properties of the Fourier transform
(385)
11.5.1 The convolution theorem
(386)
11.5.2 Parseval’s theorem and Plancherel’s identity
(388)
11.5.3 Uncertainty relations
(389)
11.6 Eigenfunctions of the Fourier operator
(392)
11.7 Higher-dimensional transforms
(393)
11.8 Focus: spatial filtering
(395)
11.9 Focus: angular spectrum representation
(397)
11.9.1 Angular spectrum representation in a half-space
(398)
11.9.2 The Weyl representation of a spherical wave
(400)
11.10 Additional reading
(402)
11.11 Exercises
(403)
12 Other integral transforms
(406)
12.1 Introduction: the Fresnel transform
(406)
12.1.1 Shift
(407)
12.1.2 Scaling property
(407)
12.1.3 Differentiation
(408)
12.1.4 Inverse Fresnel transform
(408)
12.1.5 Convolution
(409)
12.1.6 Plancherel’s identity
(410)
12.1.7 Observations
(410)
12.2 Linear canonical transforms
(411)
12.2.1 Inversion
(413)
12.2.2 Plancherel’s identity
(413)
12.2.3 Composition
(413)
12.2.4 Identity limit
(414)
12.2.5 Observations
(415)
12.3 The Laplace transform
(415)
12.3.1 Existence of the Laplace transform
(416)
12.3.2 Linearity of the Laplace transform
(418)
12.3.3 Time shift
(418)
12.3.4 Scaling property
(418)
12.3.5 Differentiation in the time domain
(418)
12.3.6 Inverse transform
(419)
12.4 Fractional Fourier transform
(420)
12.5 Mixed domain transforms
(422)
12.6 The wavelet transform
(426)
12.7 The Wigner transform
(429)
12.8 Focus: the Radon transform and computed axial tomography (CAT)
(430)
12.9 Additional reading
(436)
12.10 Exercises
(436)
13 Discrete transforms
(439)
13.1 Introduction: the sampling theorem
(439)
13.2 Sampling and the Poisson sum formula
(443)
13.3 The discrete Fourier transform
(447)
13.4 Properties of the DFT
(450)
13.4.1 Nyquist frequency
(450)
13.4.2 Shift in the n-domain
(451)
13.4.3 Shift in the k-domain
(452)
13.5 Convolution
(452)
13.6 Fast Fourier transform
(453)
13.7 The z-transform
(457)
13.7.1 Relation to Laplace and Fourier transform
(459)
13.7.2 Convolution theorem
(462)
13.7.3 Time shift
(463)
13.7.4 Initial and final values
(463)
13.8 Focus: z-transforms in the numerical solution of Maxwell's equations
(465)
13.9 Focus: the Talbot effect
(469)
13.10 Exercises
(476)
14 Ordinary differential equations
(478)
14.1 Introduction: the classic ODEs
(478)
14.2 Classification of ODEs
(479)
14.3 Ordinary differential equations and phase space
(480)
14.4 First-order ODEs
(489)
14.4.1 Separable equation
(489)
14.4.2 Exact equation
(491)
14.4.3 Linear equation
(493)
14.5 Second-order ODEs with constant coefficients
(494)
14.6 The Wronskian and associated strategies
(496)
14.7 Variation of parameters
(498)
14.8 Series solutions
(500)
14.9 Singularities, complex analysis, and general Frobenius solutions
(501)
14.10 Integral transform solutions
(505)
14.11 Systems of differential equations
(506)
14.12 Numerical analysis of differential equations
(508)
14.12.1 Euler’s method
(510)
14.12.2 Trapezoidal method and implicit methods
(512)
14.12.3 Runge–Kutta methods
(514)
14.12.4 Higher-order equations and stiff equations
(517)
14.12.5 Observations
(520)
14.13 Additional reading
(521)
14.14 Exercises
(521)
15 Partial differential equations
(525)
15.1 Introduction: propagation in a rectangular waveguide
(525)
15.2 Classification of second-order linear PDEs
(528)
15.2.1 Classification of second-order PDEs in two variables with constant coefficients
(529)
15.2.2 Classification of general second-order PDEs
(532)
15.2.3 Auxiliary conditions for second-order PDEs, existence, and uniqueness
(535)
15.3 Separation of variables
(537)
15.4 Hyperbolic equations
(539)
15.5 Elliptic equations
(545)
15.6 Parabolic equations
(550)
15.7 Solutions by integral transforms
(554)
15.8 Inhomogeneous problems and eigenfunction solutions
(558)
15.9 Infinite domains the d'Alembert solution
(559)
15.10 Method of images
(564)
15.11 Additional reading
(565)
15.12 Exercises
(565)
16 Bessel functions
(570)
16.1 Introduction: propagation in a circular waveguide
(570)
16.2 Bessel's equation and series solutions
(572)
16.3 The generating function
(575)
16.4 Recurrence relations
(577)
16.5 Integral representations
(580)
16.6 Hankel functions
(584)
16.7 Modified Bessel functions
(585)
16.8 Asymptotic behavior of Bessel functions
(586)
16.9 Zeros of Bessel functions
(587)
16.10 Orthogonality relations
(589)
16.11 Bessel functions of fractional order
(592)
16.12 Addition theorems, sum theorems, and product relations
(596)
16.13 Focus: nondiffracting beams
(599)
16.14 Additional reading
(602)
16.15 Exercises
(602)
17 Legendre functions and spherical harmonics
(605)
17.1 Introduction: Laplace's equation in spherical coordinates
(605)
17.2 Series solution of the Legendre equation
(607)
17.3 Generating function
(609)
17.4 Recurrence relations
(610)
17.5 Integral formulas
(612)
17.6 Orthogonality
(614)
17.7 Associated Legendre functions
(617)
17.8 Spherical harmonics
(622)
17.9 Spherical harmonic addition theorem
(625)
17.10 Solution of PDEs in spherical coordinates
(628)
17.11 Gegenbauer polynomials
(630)
17.12 Focus: multipole expansion for static electric fields
(631)
17.13 Focus: vector spherical harmonics and radiation fields
(634)
17.14 Exercises
(638)
18 Orthogonal functions
(642)
18.1 Introduction: Sturm--Liouville equations
(642)
18.2 Hermite polynomials
(647)
18.2.1 Series solution
(647)
18.2.2 Generating function
(649)
18.2.3 Recurrence relations
(649)
18.2.4 Orthogonality
(650)
18.2.5 The quantum harmonic oscillator
(651)
18.2.6 Hermite–Gauss laser beams
(653)
18.3 Laguerre functions
(661)
18.3.1 Series solution
(661)
18.3.2 Generating function
(662)
18.3.3 Recurrence relations
(663)
18.3.4 Associated Laguerre functions
(664)
18.3.5 Wavefunction of the hydrogen atom
(665)
18.3.6 Laguerre–Gauss laser beams
(667)
18.4 Chebyshev polynomials
(670)
18.4.1 Polynomials of the second kind
(670)
18.4.2 Polynomials of the first kind
(672)
18.5 Jacobi polynomials
(674)
18.6 Focus: Zernike polynomials
(675)
18.7 Additional reading
(682)
18.8 Exercises
(682)
19 Green's functions
(685)
19.1 Introduction: the Huygens--Fresnel integral
(685)
19.2 Inhomogeneous Sturm--Liouville equations
(689)
19.3 Properties of Green's functions
(694)
19.4 Green's functions of second-order PDEs
(696)
19.4.1 Elliptic equation: Poisson’s equation
(696)
19.4.2 Elliptic equation: Helmholtz equation
(697)
19.4.3 Hyperbolic equation: wave equation
(700)
19.4.4 Parabolic equation: diffusion equation
(702)
19.5 Method of images
(705)
19.6 Modal expansion of Green's functions
(709)
19.7 Integral equations
(713)
19.7.1 Types of linear integral equation
(713)
19.7.2 Solution of basic linear integral equations
(715)
19.7.3 Liouville–Neumann series
(716)
19.7.4 Hermitian kernels and Mercer’s theorem
(719)
19.8 Focus: Rayleigh--Sommerfeld diffraction
(721)
19.9 Focus: dyadic Green's function for Maxwell's equations
(724)
19.10 Focus: Scattering theory and the Born series
(729)
19.11 Exercises
(732)
20 The calculus of variations
(735)
20.1 Introduction: principle of Fermat
(735)
20.2 Extrema of functions and functionals
(738)
20.3 Euler's equation
(741)
20.4 Second form of Euler's equation
(747)
20.5 Calculus of variations with several dependent variables
(750)
20.6 Calculus of variations with several independent variables
(752)
20.7 Euler's equation with auxiliary conditions: Lagrange multipliers
(754)
20.8 Hamiltonian dynamics
(759)
20.9 Focus: aperture apodization
(762)
20.10 Additional reading
(765)
20.11 Exercises
(765)
21 Asymptotic techniques
(768)
21.1 Introduction: foundations of geometrical optics
(768)
21.2 Definition of an asymptotic series
(773)
21.3 Asymptotic behavior of integrals
(776)
21.4 Method of stationary phase
(783)
21.5 Method of steepest descents
(786)
21.6 Method of stationary phase for double integrals
(791)
21.7 Additional reading
(792)
21.8 Exercises
(793)
Appendix A: The gamma function
(795)
A.1 Definition
(795)
A.2 Basic properties
(796)
A.3 Stirling's formula
(798)
A.4 Beta function
(799)
A.5 Useful integrals
(800)
Appendix B: Hypergeometric functions
(803)
B.1 Hypergeometric function
(804)
B.2 Confluent hypergeometric function
(805)
B.3 Integral representations
(805)
References
(807)
Index
(813)
