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Mathematical Tools for Physics

I wrote this text for a one semester course at the sophomore-junior level. Our experience with students taking our junior physics courses is that even if they’ve had the mathematical prerequisites, they usually need more experience using the mathematics to handle it efficiently and to possess usable intuition about the processes involved. If you’ve seen infinite series in a calculus course, you may have no idea that they’re good for anything. If you’ve taken a differential equations course, which of the scores of techniques that you’ve seen are really used a lot? The world is (at least) three dimensional so you clearly need to understand multiple. | Mathematical Tools for Physics by James Nearing Physics Department University of Miami jnearing@miami.edu www.physics.miami.edu nearing mathmethods Copyright 2003 James Nearing Permission to copy for individual or classroom use is granted. QA 37.2 Rev. Nov 2006 Contents Introduction. Bibliography . 1 Basic Stuff. Trigonometry Parametric Differentiation Gaussian Integrals erf and Gamma Differentiating Integrals Polar Coordinates Sketching Graphs 2 Infinite Series.23 The Basics Deriving Taylor Series Convergence Series of Series Power series two variables Stirling s Approximation Useful Tricks Diffraction Checking Results 3 Complex Algebra.50 Complex Numbers Some Functions Applications of Euler s Formula Series of cosines Logarithms Mapping 4 Differential Equations.65 Linear Constant-Coefficient Forced Oscillations Series Solutions Some General Methods Trigonometry via ODE s Green s Functions Separation of Variables Circuits Simultaneous Equations Simultaneous ODE s Legendre s Equation 5 Fourier Series.96 Examples Computing Fourier Series iii Choice of Basis Musical Notes Periodically Forced ODE s 1 Return to Parseval Gibbs Phenomenon 6 Vector Spaces. The Underlying Idea Axioms Examples of Vector Spaces Linear Independence Norms Scalar Product Bases and Scalar Products Gram-Schmidt Orthogonalization Cauchy-Schwartz inequality Infinite Dimensions 7 Operators and Matrices. The Idea of an Operator Definition of an Operator Examples of Operators Matrix Multiplication Inverses Areas Volumes Determinants Matrices as Operators Eigenvalues and Eigenvectors Change of Basis Summation Convention Can you Diagonalize a Matrix Eigenvalues and Google Special Operators 8 Multivariable Calculus. Partial Derivatives Differentials Chain Rule Geometric Interpretation Gradient Electrostatics Plane Polar Coordinates Cylindrical Spherical Coordinates Vectors Cylindrical Spherical Bases Gradient in other Coordinates Maxima Minima Saddles Lagrange Multipliers Solid Angle 120 141 178 i .