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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
