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El. knyga: Mathematical Methods for Geophysics and Space Physics

  • Formatas: 272 pages
  • Išleidimo metai: 03-May-2016
  • Leidėjas: Princeton University Press
  • Kalba: eng
  • ISBN-13: 9781400882823
Kitos knygos pagal šią temą:
Mathematical Methods for Geophysics and Space PhysicsDidesnis paveikslėlis
  • Formatas: 272 pages
  • Išleidimo metai: 03-May-2016
  • Leidėjas: Princeton University Press
  • Kalba: eng
  • ISBN-13: 9781400882823
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Graduate students in the natural sciences--including not only geophysics and space physics but also atmospheric and planetary physics, ocean sciences, and astronomy--need a broad-based mathematical toolbox to facilitate their research. In addition, they need to survey a wider array of mathematical methods that, while outside their particular areas of expertise, are important in related ones. While it is unrealistic to expect them to develop an encyclopedic knowledge of all the methods that are out there, they need to know how and where to obtain reliable and effective insights into these broader areas. Here at last is a graduate textbook that provides these students with the mathematical skills they need to succeed in today's highly interdisciplinary research environment.


This authoritative and accessible book covers everything from the elements of vector and tensor analysis to ordinary differential equations, special functions, and chaos and fractals. Other topics include integral transforms, complex analysis, and inverse theory; partial differential equations of mathematical geophysics; probability, statistics, and computational methods; and much more.


Proven in the classroom, Mathematical Methods for Geophysics and Space Physics features numerous exercises throughout as well as suggestions for further reading.



  • Provides an authoritative and accessible introduction to the subject
  • Covers vector and tensor analysis, ordinary differential equations, integrals and approximations, Fourier transforms, diffusion and dispersion, sound waves and perturbation theory, randomness in data, and a host of other topics
  • Features numerous exercises throughout
  • Ideal for students and researchers alike
  • An online illustration package is available to professors

Recenzijos

"It is to be hoped that . . . generations of geophysicists will derive great benefit from this book."---K. Alan Shore, Contemporary Physics

Preface ix
1 Mathematical Preliminaries
1(22)
1.1 Vectors, Indicial Notation, and Vector Operators
1(5)
1.2 Cylindrical and Spherical Geometry
6(4)
1.3 Theorems of Gauss, Green, and Stokes
10(1)
1.4 Rotation and Matrix Representation
11(4)
1.5 Tensors, Eigenvalues, and Eigenvectors
15(4)
1.6 Ramp, Heaviside, and Dirac δ Functions
19(1)
1.7 Exercises
20(3)
2 Ordinary Differential Equations
23(73)
2.1 Linear First-Order Ordinary Differential Equations
25(5)
2.2 Second-Order Ordinary Differential Equations
30(22)
2.2.1 Linear Second-Order Differential Equations
33(1)
2.2.2 Green's Functions
34(4)
2.2.3 LRC Circuits and Visco-Elastic Solids
38(1)
2.2.4 Driven Oscillators, Resonance, and Variation of Constants
39(4)
2.2.5 JWKB Method, Riccati Equation, and Adiabatic Invariants
43(4)
2.2.6 Nonlinearity and Perturbation Theory
47(5)
2.3 Special Functions, Laplacians, and Separation of Variables
52(17)
2.3.1 Cartesian Coordinates and Separation of Variables
53(1)
2.3.2 Polar and Cylindrical Coordinates and Separation of Variables; Bessel and Generating Functions
54(5)
2.3.3 Spherical Coordinates and Separation of Variables; Green's and Generating Function; Spherical Harmonics
59(10)
2.4 Nonlinear Ordinary Differential Equations
69(20)
2.4.1 Bullard's Homopolar Dynamo
69(2)
2.4.2 Poincare-Bendixson Theorem and the Van der Pol Oscillator
71(3)
2.4.3 Lorenz Attractor, Perturbation Theory, and Chaos
74(4)
2.4.4 Fractals
78(4)
2.4.5 Maps and Period Doubling
82(7)
2.5 Exercises
89(7)
3 Evaluation of Integrals and Integral Transform Methods
96(55)
3.1 Integration Methods, Approximations, and Special Cases
97(7)
3.1.1 Elementary Methods and Asymptotic Methods
97(4)
3.1.2 Steepest Descent Methods
101(2)
3.1.3 Special Problems in Geophysics; Elliptic Integrals
103(1)
3.2 Complex Analysis and Elementary Contour Integration
104(9)
3.3 Fourier Transforms and Analysis Methods
113(21)
3.3.1 Fourier Series, Transforms, and Convolutions
113(2)
3.3.2 Illustrative Examples of Fourier Transform Pairs
115(4)
3.3.3 Multidimensional and Other Fourier Transform Pairs
119(7)
3.3.4 Sampling Theorem, Aliasing, and Approximation Methods
126(5)
3.3.5 Fast Fourier Transform
131(3)
3.4 Inverse Theory, Calculus of Variations, and Integral Equations
134(12)
3.4.1 Linear Inverse Theory
135(1)
3.4.2 Abel Transform
136(2)
3.4.3 Radon Transform
138(1)
3.4.4 Calculus of Variations
139(1)
3.4.5 Herglotz-Wiechert Travel-Time Transform
140(6)
3.5 Exercises
146(5)
4 Partial Differential Equations of Mathematical Geophysics
151(51)
4.1 Introduction to Partial Differential Equations
151(13)
4.1.1 Classification of Partial Differential Equations and Boundary Condition Types
151(4)
4.1.2 Wave Equation in One Dimension
155(4)
4.1.3 Elements of Fluid Flow
159(5)
4.2 Three-Dimensional Applications
164(8)
4.2.1 Diffusion Equation in Three Dimensions
165(1)
4.2.2 Wave Equation in Three Dimensions
166(4)
4.2.3 Gravitational Potential and Green's Function Methods
170(2)
4.3 Diffusion, Dispersion, Perturbation Methods, and Nonlinearity
172(21)
4.3.1 Diffusion and Dispersion
172(8)
4.3.2 Sound Waves and Perturbation Theory
180(2)
4.3.3 Burgers's Equation and Solitary Waves
182(2)
4.3.4 Korteweg-de Vries Equation and Solitons
184(7)
4.3.5 Self-Similarity, Scaling, and Kolmogorov Turbulence
191(2)
4.4 Exercises
193(9)
5 Probability, Statistics, and Computational Methods
202(39)
5.1 Binomial, Poisson, and Gaussian Distributions
203(11)
5.1.1 Binomial Distribution
208(1)
5.1.2 Poisson Distribution
209(2)
5.1.3 Normal Distribution
211(3)
5.2 Central Limit Theorem
214(3)
5.3 Randomness in Data and in Simulations
217(4)
5.3.1 Regression and Fitting of Experimental Data
217(2)
5.3.2 Random Number Generation and Monte Carlo Simulation
219(2)
5.4 Computational Geophysics
221(17)
5.4.1 Computation, Round-off Error, and Seminumerical Algorithms
221(2)
5.4.2 Roots of Equations
223(3)
5.4.3 Numerical Solution of Ordinary Differential Equations
226(7)
5.4.4 General Issues in the Numerical Solution of Partial Differential Equations
233(1)
5.4.5 Numerical Solution of Parabolic Partial Differential Equations
234(2)
5.4.6 Numerical Solution of Hyperbolic Partial Differential Equations
236(2)
5.5 Exercises
238(3)
References 241(6)
Index 247
William I. Newman is professor in the Department of Earth, Planetary, and Space Sciences, the Department of Physics and Astronomy, and the Department of Mathematics at the University of California, Los Angeles. He is the author of Continuum Mechanics in the Earth Sciences.
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