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 Riemann 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 difierential equations

Gronwall's lemma

The variation of constants formula

Determinants and eigenvalues

Fundamental matrices

Second order linear differential equations

2 Differentiability

The definition

The derivative

Directional derivatives

Partial derivatives

The Jacobi matrix

A differentiability criterion

The Riesz representation theorem

The gradient

Complex differentiability

3 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

4 Multilinear maps

Continuous multilinear maps

The canonical isomorphism

Symmetric multilinear maps

The derivative of multilinear maps

5 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-Lindel？f 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

Parametrization 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 Poincaré 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 Meromorphic 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