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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 ï¿½ 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 | |