Guided Tour of Mathematical Methods For the Physical Sciences

ISBN-10: 0521542618
ISBN-13: 9780521542616
Edition: 2nd 2004 (Revised)
Authors: Roel Snieder
List price: $64.99
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Description: Mathematical methods are essential tools for all physical scientists. This second edition provides a comprehensive tour of the mathematical knowledge and techniques that are needed by students in this area. In contrast to more traditional textbooks,  More...

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Book details

List price: $64.99
Edition: 2nd
Copyright year: 2004
Publisher: Cambridge University Press
Publication date: 4/9/2009
Binding: Paperback
Pages: 524
Size: 6.75" wide x 9.75" long x 1.00" tall
Weight: 1.892
Language: English

Mathematical methods are essential tools for all physical scientists. This second edition provides a comprehensive tour of the mathematical knowledge and techniques that are needed by students in this area. In contrast to more traditional textbooks, all the material is presented in the form of problems. Within these problems the basic mathematical theory and its physical applications are well integrated. The mathematical insights that the student acquires are therefore driven by their physical insight. Topics that are covered include vector calculus, linear algebra, Fourier analysis, scale analysis, complex integration, Greens functions, normal modes, tensor calculus, and perturbation theory. The second edition contains new chapters on dimensional analysis, variational calculus, and the asymptotic evaluation of integrals. This book can be used by undergraduates, and lower-level graduate students in the physical sciences. It can serve as a stand-alone text, or as a source of problems and examples to complement other textbooks.

Roel Snieder holds the Keck Foundation Endowed Chair of Basic Exploration Science at the Colorado School of Mines. From 1997 to 2000, he served as Dean of the Faculty of Earth Sciences at the University of Utrecht. Snieder has served on the editorial boards of Geophysical Journal International, Inverse Problems, Reviews of Geophysics, and the European Journal of Physics. In 2000, he was elected Fellow of the American Geophysical Union. He is co-author of the textbook The Art of Being a Scientist: A Guide for Graduate Students and their Mentors (Cambridge University Press, 2009). From 2003 to 2011, he was a member of the Earth Science Council of the US Department of Energy. In 2008, Snieder worked for the Global Climate and Energy Project at Stanford University on outreach and education on global energy. That same year, he was a founding member of the humanitarian organization Geoscientists Without Borders, where he served as chair until 2013. In 2011, he was elected Honorary Member of the Society of Exploration Geophysicists.

Introduction
Power series
The Taylor series
Growth of the Earth by cosmic dust
The bouncing ball
Reflection and transmission by a stack of layers
Spherical and cylindrical coordinates
Introducing spherical coordinates
Changing coordinate systems
The acceleration in spherical coordinates
Volume integration in spherical coordinates
Cylindrical coordinates
The gradient
Properties of the gradient vector
The pressure force
Differentiation and integration
Newton's law from energy conservation
Total and partial time derivatives
The gradient in spherical coordinates
The divergence of a vector field
The flux of a vector field
Introduction of the divergence
Sources and sinks
The divergence in cylindrical coordinates
Is life possible in a five-dimensional world?
The curl of a vector field
Introduction of the curl
What is the curl of the vector field?
The first source of vorticity: rigid rotation
The second source of vorticity: shear
The magnetic field induced by a straight current
Spherical coordinates and cylindrical coordinates
The theorem of Gauss
Statement of Gauss's law
The gravitational field of a spherically symmetric mass
A representation theorem for acoustic waves
Flowing probability
The theorem of Stokes
Statement of Stokes's law
Stokes's theorem from the theorem of Gauss
The magnetic field of a current in a straight wire
Magnetic induction and Lenz's law
The Aharonov--Bohm effect
Wingtips vortices
The Laplacian
The curvature of a function
The shortest distance between two points
The shape of a soap film
Sources of curvature
The instability of matter
Where does lightning start?
The Laplacian in spherical and cylindrical coordinates
Averaging integrals for harmonic functions
Conservation laws
The general form of conservation laws
The continuity equation
Conservation of momentum and energy
The heat equation
The explosion of a nuclear bomb
Viscosity and the Navier--Stokes equation
Quantum mechanics and hydrodynamics
Scale analysis
The vortex in a bathtub
Three ways to estimate a derivative
The advective terms in the equation of motion
Geometric ray theory
Is the Earth's mantle convecting?
Making an equation dimensionless
Linear algebra
Projections and the completeness relation
A projection on vectors that are not orthogonal
The Coriolis force and centrifugal force
The eigenvalue decomposition of a square matrix
Computing a function of a matrix
The normal modes of a vibrating system
Singular value decomposition
The Householder transformation
The Dirac delta function
Introduction of the delta function
Properties of the delta function
The delta function of a function
The delta function in more dimensions
The delta function on the sphere
The self energy of the electron
Fourier analysis
The real Fourier series on a finite interval
The complex Fourier series on a finite interval
The Fourier transform on an infinite interval
The Fourier transform and the delta function
Changing the sign and scale factor
The convolution and correlation of two signals
Linear filters and the convolution theorem
The dereverberation filter
Design of frequency filters
Linear filters and linear algebra
Analytic functions
The theorem of Cauchy--Riemann
The electric potential
Fluid flow and analytic functions
Complex integration
Nonanalytic functions
The residue theorem
Solving integrals without knowing the primitive function
Response of a particle in syrup
Green's functions: principles
The girl on a swing
You have seen Green's functions before!
The Green's function as impulse response
The Green's function for a general problem
Radiogenic heating and the Earth's temperature
Nonlinear systems and Green's functions
Green's functions: examples
The heat equation in N dimensions
The Schrodinger equation with an impulsive source
The Helmholtz equation in one, two and three dimensions
The wave equation in one, two and three dimensions
Normal modes
The normal modes of a string
The normal modes of drum
The normal modes of a sphere
Normal modes and orthogonality relations
Bessel functions behave as decaying cosines
Legendre functions behave as decaying cosines
Normal modes and the Green's function
Guided waves in a low-velocity channel
Leaky modes
Radiation damping
Potential theory
The Green's function of the gravitational potential
Upward continuation in a flat geometry
Upward continuation in a flat geometry in three dimensions
The gravity field of the Earth
Dipoles, quadrupoles and general relativity
The multipole expansion
The quadrupole field of the Earth
The fifth force
Cartesian tensors
Coordinate transforms
Unitary matrices
Shear or dilatation?
The summation convention
Matrices and coordinate transforms
Definition of a tensor
Not every vector is a tensor
The products of tensors
Deformation and rotation again
The stress tensor
Why pressure in a fluid is isotropic
Special relativity
Perturbation theory
Regular perturbation theory
The Born approximation
Linear travel time tomography
Limits on perturbation theory
The WKB approximation
The need for consistency
Singular perturbation theory
Epilogue, on power and knowledge
References
Index

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