Unique in its clarity, examples and range, Physical Mathematics explains as simply as possible the mathematics that graduate students and professional physicists need in their courses and research. The author illustrates the mathematics with numerous physical examples drawn from contemporary research. In addition to basic subjects such as linear algebra, Fourier analysis, complex variables, differential equations and Bessel functions, this textbook covers topics such as the singular-value decomposition, Lie algebras, the tensors and forms of general relativity, the central limit theorem and Kolmogorov test of statistics, the Monte Carlo methods of experimental and theoretical physics, the renormalization group of condensed-matter physics and the functional derivatives and Feynman path integrals of quantum field theory.
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Fourier and Laplace transforms
Probability and statistics
Monte Carlo methods
Chaos and fractals
Tensors and local symmetries
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4-vector adjoint algebra analytic antisymmetric basis vectors Bessel functions boundary conditions coefﬁcients commutation relations complex number constant contour integral converges coordinates covariant deﬁned deﬁnition delta function density derivative differential equation differential operator dimensions Dirac eigenfunctions eigenstates eigenvalue eigenvectors elements energy Example exponential exterior derivative factor ﬁnd ﬁnite ﬁrst ﬁrst-order ﬂat Fourier series Fourier transform Friedmann equation function f gauge gaussian ghost contour Green’s function group G half-plane hermitian homogeneous identity inﬁnite inhomogeneous inner product integral formula interval invariant inverse Legendre linear combination linearly independent Lorentz group Lorentz transformation maps matrix metric tensor momentum multiplied notation orthogonal orthonormal particle Pn(x polynomials potential probability distribution quantum representation rotation satisﬁes satisfy show exercise singular solution spherical square symmetric tensor theorem tiny unitary unitary matrix vanishes variables vector space zero