Mathematical_methods-two

mathematical_methods-two
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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 Stirlings Approximation Useful Tricks Diffraction Checking Results 3 Complex Algebra . . . . . . . . . . . . 48 Complex Numbers Some Functions Applications of Eulers Formula Series of cosines Logarithms Mapping 4 Differential Equations . . . . . . . . . . 62 Linear Constant-Coefficient Forced Oscillations Series Solutions Some General Methods Trigonometry via ODEs Greens Functions Separation of Variables Circuits Simultaneous Equations Simultaneous ODEs Legendres Equation 5 Fourier Series . . . . . . . . . . . . . 92 Examples Computing Fourier Series Choice of Basis Musical Notes Periodically Forced ODEs 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?...
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