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Unformatted text preview: 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 . . . . . . . . . . . . . . iii Bibliography . . . . . . . . . . . . . . . v 1 Basic Stuff . . . . . . . . . . . . . . . 1 Trigonometry Parametric Differentiation Gaussian Integrals erf and Gamma Differentiating Integrals Polar Coordinates Sketching Graphs 2 Infinite Series . . . . . . . . . . . . . 22 The Basics Deriving Taylor Series Convergence Series of Series Power series, two variables Stirling’s Approximation Useful Tricks Diffraction Checking Results 3 Complex Algebra . . . . . . . . . . . . 48 Complex Numbers Some Functions Applications of Euler’s Formula Series of cosines Logarithms Mapping 4 Differential Equations . . . . . . . . . . 62 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 . . . . . . . . . . . . . 92 Examples Computing Fourier Series Choice of Basis Musical Notes Periodically Forced ODE’s Return to Parseval Gibbs Phenomenon 6 Vector Spaces . . . . . . . . . . . . . 114 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 . . . . . . . . . 133 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 . . . . . . . . . 168 Partial Derivatives Chain Rule Differentials 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 i Rainbow 3D Visualization 9 Vector Calculus 1 . . . . . . . . . . . 202 Fluid Flow Vector Derivatives Computing the divergence Integral Representation of Curl The Gradient Shorter Cut for div and curl Identities for Vector Operators Applications to Gravity Gravitational Potential Index Notation More Complicated Potentials 10 Partial Differential Equations . . . . . . 231 The Heat Equation Separation of Variables Oscillating Temperatures Spatial Temperature Distributions Specified Heat Flow Electrostatics Cylindrical Coordinates 11 Numerical Analysis . . . . . . . . . . . 256 Interpolation Solving equations Differentiation Integration Differential Equations Fitting of Data Euclidean Fit Differentiating noisy data Partial Differential Equations...
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