Mathematics for the Physical Sciences

This book is intended to furnish a bridge from courses in general physics to the intermediate-level courses in classical mechanics, electrodynamics, and quantum mechanics. It emphasizes the use of physical concepts to illustrate and clarify the mathematical methods.The book begins with a short review of some topics from general physics that then provide the physical contexts for the later discussions. Thus, for example, the concept of magnetic flux serves to give physical meaning to the integral theorems of vector calculus; the a conducting sphere in an electric field, a vibrating drum head, the harmonic oscillator, and a particle in a box illustrate the discussion of differential equations; and coupled oscillators and the principal axes of a rotating rigid body supply the physical context for the discussion of matrices.Problems at the end of each chapter furnish the student with experience in applying the mathematics and illustrative exercises throughout give guidance. Many of the exercises call for graphical representations, and some are particularly amenable to the use of numerical methods, but the treatment avoids the implication that computers are necessary to solve the problems.

Table of Contents:
1 A Review.- 1.1 Electrostatics.- 1.2 Electric Current.- 1.3 Magnetic Flux.- 1.4 Simple Harmonic Motion.- 1.5 A Rigid Rotator.- 1.6 Exercises.- 2 Vectors.- 2.1 Representations of Vectors.- 2.2 The Scalar Product of Two Vectors.- 2.3 The Vector Product of Two Vectors.- 2.4 Exercises.- 3 Vector Calculus.- 3.1 Partial Derivatives.- 3.2 A Vector Differential Operator.- 3.3 Components of the Gradient.- 3.4 Flux.- 3.5 Exercises.- 4 Complex Numbers.- 4.1 Why Study Complex Numbers?.- 4.2 Roots of a Complex Number.- 4.3 Exercises.- 5 Differential Equations.- 5.1 Infinite Series.- 5.2 Analytic Functions.- 5.3 The Classical Harmonic Oscillator.- 5.4 Boundary Conditions.- 5.5 Polynomial Solutions.- 5.6 Elementary Functions.- 5.7 Singularities.- 5.8 Exercises.- 6 Partial Differential Equations.- 6.1 The Method of Separation of Variables.- 6.2 The Quantum Harmonic Oscillator.- 6.3 A Conducting Sphere in an Electric Field.- 6.4 The Schrödinger Equation for a Central Field.- 6.5 Exercises.- 7 Eigenvalue Problems.- 7.1 Boundary Value Problems.- 7.2 A Vibrating Drumhead.- 7.3 A Particle in a One-Dimensional Box.- 7.4 Exercises.- 8 Orthogonal Functions.- 8.1 The Failure of Classical Physics.- 8.2 Observables and Their Measurement.- 8.3 Mathematical Operators.- 8.4 Eigenvalue Equations.- 8.5 The Quantum Harmonic Oscillator.- 8.6 Sturm-Liouville Theory.- 8.7 The Dirac Delta Function.- 8.8 Fourier Integrals.- 8.9 Fourier Series.- 8.10 Periodic Functions.- 8.11 Exercises.- 9 Matrix Formulation of the Eigenvalue Problem.- 9.1 Reformulating the Eigenvalue Problem.- 9.2 Systems of Linear Equations.- 9.3 Back to the Eigenvalue Problem.- 9.4 Coupled Harmonic Oscillators.- 9.5 A Rotating Rigid Body.- 9.6 Exercises.- 10 Variational Principles.- 10.1 Fermat’s Principle.- 10.2 Another VariationalCalculation.- 10.3 The Euler-Lagrange Equation.- 10.4 Exercises.- Appendix A Vector Relations.- A.1 Vector Identities.- A.2 Integral Theorems.- A.3 The Functions of Vector Calculus.- Appendix B Fundamental Equations of Physics.- B.1 Poisson’s Equation.- B.2 Laplace’s Equation.- B.3 Maxwell’s Equations.- B.4 Time-Dependent Schrödinger Equation.- Appendix C Some Useful Integrals and Sums.- C.1 Integrals.- C.2 Sums.- Appendix D Algebraic Equations.- D.1 Quadratic Equation.- D.2 Cubic Equation.- References.

Review :

From the reviews:

The book is written nicely and can be very helpful for the students of physics who need mathematical background tailored especially for their needs.

--Zentralblatt Math

"Here is a bridge from courses in general physics to the intermediate-level courses in classical mechanics, electrodynamics and quantum mechanics. The author bases the mathematical discussions on specific physical problems to provide a basis for developing mathematical intuition. The text concludes with a brief discussion of variational methods and the Euler-Lagrange equation." (Meteorology and Atmospheric Physics, Vol. 84 (1-2), 2003)

"The purpose of this textbook is to collect under a single cover mathematics required for mastering intermediate-level courses in classical mechanics, electricity and magnetism, and quantum mechanics. … A set of exercises is provided for each chapter giving the student an excellent opportunity to apply mathematical apparatus for the study of physical systems. … The book is written nicely and can be very helpful for the students of physics who need mathematical background tailored especially for their needs." (Yuri V. Rogovchenko, Zentralblatt MATH, Vol. 986, 2002)