Advanced Mathematics for Applications
The partial differential equations that govern scalar and vector fields are the very language used to model a variety of phenomena in solid mechanics, fluid flow, acoustics, heat transfer, electromagnetism and many others. A knowledge of the main equations and of the methods for analyzing them is therefore essential to every working physical scientist and engineer. Andrea Prosperetti draws on many years' research experience to produce a guide to a wide variety of methods, ranging from classical Fourier-type series through to the theory of distributions and basic functional analysis. Theorems are stated precisely and their meaning explained, though proofs are mostly only sketched, with comments and examples being given more prominence. The book structure does not require sequential reading: each chapter is self-contained and users can fashion their own path through the material. Topics are first introduced in the context of applications, and later complemented by a more thorough presentation.
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Part I Applications
Part III Essential Tools
Part IV Some Advanced Tools
analytic function applied arbitrary argument asymptotic axis Bessel functions boundary conditions bounded operator Cauchy chapter coefﬁcients compact complex plane consider constant contour convergence coordinates corresponding cosine deﬁned deﬁnition derivative differential equation distributions domain eigenfunctions eigenvalues eigenvectors elements equal example exists expansion expression ﬁeld ﬁnd ﬁnite ﬁnite-dimensional ﬁrst formula Fourier series Fourier transform function f given Green’s function Hilbert space homogeneous inﬁnite integral interval inverse L2 norm Laplace equation Laplace transform Legendre limit linear mapping matrix multiplication non-zero norm orthogonal orthonormal point-wise polynomials power series problem properties relation result Riemann satisﬁed satisfy scalar product self-adjoint sense sequence sgnx sine singular solution solve speciﬁc spectrum spherical subspace sufﬁcient summation Table term test functions theorem uniform convergence vanish variable vector wave zero