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
