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.

Foreword

Chapter 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 integral

Chapter 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 multipliers

Chapter 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 integrals

References

Index