Mathematical Analysis, 2009 An Introduction to Functions of Several Variables

Preface.-Differential Calculus.-The differential calculus for scalar functions.-The differential calculus for vector-valued functions.-The theorems of the differential calculus.-Invertibility of map Rn to Rn.-Differential calculus in Banach spaces.-Exercises.-The Integral Calculus.-Lebesque’s integral.-Convergence theorems.-Mollifiers and approximation.-Integral calculus.-Measure and area.-The Gauss-Green formula.-Exercises.-Curves and Differential Forms.-Differential forms, fields, and work.-Conservative fields, exact forms, and potentials.-closed forms and irrotational fields.-Stokes formula in the plane.-Exercises.-Holomorphic functions.-Functions from C to C.-The fundamental theorem of calculus in C.-The fundamental theorems about holomorphic functions.-Examples of holomorphic functions.-Pointwise singularities of holomorphic functions.-Residues.-Further consequences of Cauchy formulas.-Maximum principle.-Schwarz lemma-Local properties.-Biholomorphisms.-Riemann’s theorem on conformal representations.-Harmonic functions and Riemann’s theorem.-Exercises.-Surfaces and level sets.-Surfaces and immersions.-Implicit functions.-Some applications.-The curvature of curves and surfaces.-Exercises.-Systems of Ordinary Differential Equations.-Linear equations.-Stability.-The theorem of Poincaré-Bendixson.-Exercises.-Appendix A: Mathematicians and other scientists.-References.-Index