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Preface | |

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Describing the Universe | |

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A Universal Language | |

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Scalar and Vector Fields | |

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Curvilinear Coordinates | |

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The Helmholtz Theorem | |

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Vector Spaces | |

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Matrices | |

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Problems | |

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Complex Variables | |

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All About Numbers | |

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Functions of Complex Variables | |

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Complex Series | |

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Complex Numbers and Laplace's Equation | |

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Poles and Zeros | |

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The Residue Theorem | |

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Using the Residue Theorem | |

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Conformal Mapping | |

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The Gamma Function | |

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Problems | |

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Differential Equations | |

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Some Definitions | |

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Common Differential Equations Arising in Physics | |

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Solution of Linear, Ordinary Differential Equations | |

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Numerical Methods | |

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Partial Differential Equations: Separation of Variables | |

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Problems | |

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Fourier Series | |

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Fourier's Theorem | |

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Finding the Coefficients | |

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Fourier Sine and Cosine Series | |

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Use of Fourier Series to Solve Differential Equations | |

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Convergence of Fourier Series | |

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Problems | |

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Laplace Transforms | |

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Definition of the Laplace Transform | |

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Some Basic Properties of the Transform | |

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Use of the Laplace Transform to Solve a Differential Equation | |

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Some Additional Useful Tricks | |

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Convolution | |

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The General Inversion Procedure | |

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Some More Physics | |

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Problems | |

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Generalized Functions in Physics | |

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The Delta Function | |

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Developing a Theory of Distributions | |

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Properties of Distributions | |

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Sequences and Series | |

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Distributions in N Dimensions | |

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Describing Physical Quantities Using Delta Functions | |

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The Green's Function | |

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Problems | |

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Fourier Transforms | |

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Definition of the Fourier Transform | |

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Some Examples | |

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Properties of the Fourier Transform | |

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Causality | |

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Use of Fourier Transforms in the Solution of Partial Differential Equations | |

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Fourier Transforms and Power Spectra | |

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Sine and Cosine Transforms | |

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Problems | |

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Sturm-Liouville Theory | |

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The Sturm-Liouville Problem | |

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Use of Sturm-Liouville Theory in Physics | |

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Problems with Spherical Symmetry: Spherical Harmonics | |

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Problems with Cylindrical Symmetry: Bessel Functions | |

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Spherical Bessel Functions | |

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The Classical Orthogonal Polynomials | |

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Problems | |

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Optional Topics | |

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Tensors | |

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Cartesian Tensors | |

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Inner and Outer Products | |

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Pseudo-tensors and Cross Products | |

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General Tensor Calculus | |

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The Metric Tensor | |

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Contraction | |

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Basis Vectors and Basis Forms | |

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Derivatives | |

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Problems | |

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Group Theory | |

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Definition of a Group | |

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Examples of Groups | |

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Classes | |

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Subgroups | |

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Cyclic Groups | |

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Factor Groups and Direct Product Groups | |

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Isomorphism | |

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Representations | |

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Generators of Groups | |

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Lie Algebras | |

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Problems | |

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Green's Functions | |

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Division-of-Region Method | |

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Expansion in Eigenfunctions | |

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Transform Methods | |

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Extension to N Dimensions | |

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Inhomogeneous Boundary Conditions | |

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Green's Theorem | |

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The Green's Function for Poisson's Equation in a Bounded Region | |

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Problems | |

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Approximate Evaluation of Integrals | |

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The Method of Steepest Descent | |

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The Method of Stationary Phase | |

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Problems | |

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Calculus of Variations | |

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Integral Principles in Physics | |

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The Euler Equation | |

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Variation Subject to Constraints | |

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Extension to Functions of More Than One Variable | |

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Problems | |

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Appendices | |

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Transformation Properties of the Vector Cross Product | |

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Proof of the Helmholtz Theorem | |

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Proof by Induction: The Cauchy Formula | |

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The Mean Value Theorem for Integrals | |

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The Gibbs Phenomenon | |

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The Laplace Transform and Convolution | |

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Proof That P[superscript m subscript l]([mu]) = (-1)[superscript m](1 - [mu superscript 2])[superscript m/2]d[superscript m]/d[mu superscript m]P[subscript l]([mu]) | |

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Proof of the Relation [function of superscript infinity subscript 0] [rho]J[subscript m](k[rho])J[subscript m](k'[rho])d[rho] = 1/k[delta](k - k') | |

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The Error Function | |

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Classification of Partial Differential Equations | |

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The Tangent Function: A Detailed Investigation of Series Expansions | |

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Bibliography | |

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Index | |