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| Vectors and fields in space | |
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| Concepts of space | |
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| Vectors in space | |
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| Permutation symbols | |
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| Vector differentiation of a scalar field | |
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| Vector differentiation of a vector field | |
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| Path-dependent scalar and vector integrations | |
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| Flux, divergence and Gauss's theorem | |
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| Circulation, curl and Stokes's theorem | |
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| Helmholtz's theorem | |
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| Orthogonal curvilinear coordinate systems | |
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| Vector differential operators in orthogonal curvilinear coordinate systems | |
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| Tables of mathematical formulas | |
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| Transformations, matrices and operators | |
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| Transformations and the laws of physics | |
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| Rotations in space: Matrices | |
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| Determinant and matrix inversion | |
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| Homogeneous equations | |
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| The matrix eigenvalue problem | |
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| Generalized matrix eigenvalue problems | |
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| Eigenvalues and eigenvectors of Hermitian matrices | |
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| The wave equation | |
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| Displacement in time and translation in space: Infinitesimal generators | |
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| Rotation operators | |
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| Matrix groups | |
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| Tables of mathematical formulas | |
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| Relativistic square-root spaces* | |
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| Introduction | |
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| Special relativity and Lorentz transformations | |
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| Relativistic kinematics and the mass-energy equivalence | |
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| Quaternions | |
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| Dirac equation, spinors and matrices | |
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| Symmetries of the Dirac equation* | |
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| Weyl and Majorana spinors, symmetry violations* | |
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| Lorentz group | |
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| Cartan spinors and spin transformations in square-root space | |
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| Dyadics | |
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| Cartesian tensors | |
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| Tensor analysis | |
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| Tables of mathematical formulas | |
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| Fourier series and Fourier transforms | |
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| Wave-particle duality: Quantum mechanics | |
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| Fourier series | |
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| Fourier coefficients and Fourier-series representation | |
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| Complex Fourier series and the Dirac � function | |
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| Fourier transform | |
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| Green function and convolution | |
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| Heisenberg's uncertainty principle | |
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| Conjugate variables and operators in wave mechanics | |
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| Generalized Fourier series and Legendre polynomials | |
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| Orthogonal functions and orthogonal polynomials | |
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| Mean-square error and mean-square convergence | |
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| Convergence of Fourier series | |
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| Maxwell equations in Fourier spaces | |
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| 3D Fourier transforms: Helmholtz decomposition theorem | |
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| Short table of Fourier cosine series | |
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| Short table of Fourier sine series | |
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| Short table of Fourier transforms | |
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| Short table of 3D and 4D Fourier transforms | |
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| Tables of mathematical formulas | |
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| Differential equations in physics | |
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| Introduction | |
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| Linear differential equations | |
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| First-order differential equations | |
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| Second-order linear differential equations | |
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| The second homogeneous solution and an inhomogeneous solution | |
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| Green functions | |
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| Series solution of the homogeneous second-order linear differential equation | |
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| Differential eigenvalue equations and orthogonal functions | |
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| Partial differential equations of physics | |
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| Separation of variables and eigenfunction expansions | |
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| Boundary and initial conditions | |
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| Separation of variables for the Laplacian | |
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| Green functions for partial differential equations | |
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| Tables of mathematical formulas | |
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| Nonlinear systems* | |
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| Introduction | |
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| Nonlinear instabilities | |
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| Logistic map and chaos | |
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| Strange attractor | |
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| Driven dissipative linear pendula | |
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| Chaos in parametrically driven dissipative nonlinear pendula | |
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| Solitons | |
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| Traveling kinks | |
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| Nonlinear superposition of solitons | |
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| More general methods for multi-solitons* | |
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| Tables of mathematical formulas | |
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| Special functions | |
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| Introduction | |
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| Generating function for Legendre polynomials | |
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| Hermite polynomials and the quantum oscillator | |
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| Orthogonal polynomials | |
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| Classical orthogonal polynomials* | |
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| Associated Legendre polynomials and spherical harmonics | |
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| Bessel functions | |
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| Sturm-Liouville equation and eigenfunction expansions | |
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| Tables of mathematical formulas | |
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| Functions of a complex variable | |
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| Introduction | |
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| Functions of a complex variable | |
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| Multivalued functions and Riemann surfaces | |
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| Complex differentiation: Analytic functions and singularities | |
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| Complex integration: Cauchy integral theorem and integral formula | |
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| Harmonic functions in the plane | |
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| Taylor series and analytic continuation | |
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| Laurent series | |
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| Residues | |
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| Complex integration: Calculus of residues | |
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| Poles on the contour and Green functions | |
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| Laplace transform | |
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| Inverse Laplace transform | |
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| Construction of functions and dispersion relations | |
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| Asymptotic expansions* | |
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| Tables of mathematical formulas | |
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| Tutorials | |
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| Complex algebra | |
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| Vectors | |
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| Simple and partial differentiations | |
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| Simple and multiple integrals | |
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| Matrices and determinants | |
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| Infinite series | |
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| Exponential functions | |
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| Mathematica and other computer algebra systems | |
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| Computer algebra (CA) with Mathematica | |
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| Introduction to CA | |
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| Equation solvers | |
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| Drawing figures and graphs | |
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| Number-intensive calculations | |
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| Resources for students | |
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| Bibliography | |
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| Name index | |
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| Subject index | |