分析（第2卷）

出版时间：2012-9  出版社：Herbert Amann、 Joachim Escher 世界图书出版公司 (2012-09出版)  
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内容概要

数学

作者简介

作者:(德)阿莫恩

书籍目录

ForewordChapter Ⅵ  Integral calculus in one variable  1  Jump continuous functions  Staircase and jump continuous functions  A characterization of jump continuous functions  The Banach space of jump continuous functions  2  Continuous extensions  The extension of uniformly continuous functions  Bounded linear operators  The continuous extension of bounded linear operators  3  The Cauchy-Riemann Integral  The integral of staircase functions  The integral of jump continuous functions  Riemann sums  4  Properties of integrals  Integration of sequences of functions  The oriented integral  Positivity and monotony of integrals  Componentwise integration  The first fundamental theorem of calculus  The indefinite integral  The mean value theorem for integrals  5  The technique of integration  Variable substitution  Integration by parts  The integrals of rational functions  6  Sums and integrals  The Bernoulli numbers  Recursion formulas  The Bernoulli polynomials  The Euler-Maclaurin sum formula  Power sums  Asymptotic equivalence  The Biemann ζ function  The trapezoid rule  7  Fourier series  The L2 scalar product  Approximating in the quadratic mean  Orthonormal systems  Integrating periodic functions  Fourier coefficients  Classical Fourier series  Bessel's inequality  Complete orthonormal systems  Piecewise continuously differentiable functions  Uniform convergence  8  Improper integrals  Admissible functions  Improper integrals  The integral comparison test for series  Absolutely convergent integrals  The majorant criterion  9  The gamma function  Euler's integral representation  The gamma function on C\(-N)  Gauss's representation formula  The reflection formula  The logarithmic convexity of the gamma function  Stirling's formula  The Euler beta integralChapter Ⅶ  Multivariable differential calculus  1  Continuous linear maps  The completeness of/L(E, F)  Finite-dimensional Banach spaces  Matrix representations  The exponential map  Linear differential equations  Gronwall's lemma  The variation of constants formula  Determinants and eigenvalues  Fundamental matrices  Second order linear differential equations  Differentiability  The definition  The derivative  Directional derivatives  Partial derivatives  The Jacobi matrix  A differentiability criterion  The Riesz representation theorem  The gradient  Complex differentiability  Multivariable differentiation rules  Linearity  The chain rule  The product rule  The mean value theorem  The differentiability of limits of sequences of functions  Necessary condition for local extrema  Multilinear maps  Continuous multilinear maps  The canonical isomorphism  Symmetric multilinear maps  The derivative of multilinear maps  Higher derivatives  Definitions  Higher order partial derivatives  The chain rule  Taylor's formula  Functions of m variables  Sufficient criterion for local extrema  6  Nemytskii operators and the calculus of variations  Nemytskii operators  The continuity of Nemytskii operators  The differentiability of Nemytskii operators  The differentiability of parameter-dependent integrals  Variational problems  The Euler-Lagrange equation  Classical mechanics  7  Inverse maps  The derivative of the inverse of linear maps  The inverse function theorem  Diffeomorphisms  The solvability of nonlinear systems of equations  8  Implicit functions  Differentiable maps on product spaces  The implicit function theorem  Regular values  Ordinary differential equations  Separation of variables  Lipschitz continuity and uniqueness  The Picard-Lindelof theorem  9  Manifolds  Submanifolds of Rn  Graphs  The regular value theorem  The immersion theorem  Embeddings  Local charts and parametrizations  Change of charts  10  Tangents and normals  The tangential in Rn  The tangential space  Characterization of the tangential space  Differentiable maps  The differential and the gradient  Normals  Constrained extrema  Applications of Lagrange multipliersChapter Ⅷ  Line integrals  1  Curves and their lengths  The total variation  Rectifiable paths  Differentiable curves  Rectifiable curves  2  Curves in Rn  Unit tangent vectors  Paramctrization by arc length  Oriented bases  The Frenet n-frame  Curvature of plane curves  Identifying lines and circles  Instantaneous circles along curves  The vector product  The curvature and torsion of space curves  3  Pfaff forms  Vector fields and Pfaff forms  The canonical basis  Exact forms and gradient fields  The Poincare lemma  Dual operators  Transformation rules  Modules  4  Line integrals  The definition  Elementary properties  The fundamental theorem of line integrals  Simply connected sets  The homotopy invariance of line integrals  5  Holomorphic functions  Complex line integrals  Holomorphism  The Cauchy integral theorem  The orientation of circles  The Cauchy integral formula  Analytic functions  Liouville's theorem  The Fresnel integral  The maximum principle  Harmonic functions  Goursat's theorem  The Weierstrass convergence theorem  6  Meromorphie functions  The Laurent expansion  Removable singularities  Isolated singularities  Simple poles  The winding number  The continuity of the winding number  The generalized Cauchy integral theorem  The residue theorem  Fourier integralsReferencesIndex

编辑推荐

阿莫恩编著的《分析》内容介绍：As with the first, the second volume contains substantially more material than can be covered in a one-semester course. Such courses may omit many beautiful and well-grounded applications which connect broadly to many areas of mathematics.We of course hope that students will pursue this material independently; teachers may find it useful for undergraduate seminars.