Beschreibung
This self-contained work is an introductory presentation of basic ideas, structures, and results of differential and integral calculus for functions of several variables.The wide range of topics covered include: differential calculus of several variables, including differential calculus of Banach spaces, the relevant results of Lebesgue integration theory, differential forms on curves, a general introduction to holomorphic functions, including singularities and residues, surfaces and level sets, and systems and stability of ordinary differential equations. An appendix highlights important mathematicians and other scientists whose contributions have made a great impact on the development of theories in analysis.Mathematical Analysis: An Introduction to Functions of Several Variables motivates the study of the analysis of several variables with examples, observations, exercises, and illustrations. It may be used in the classroom setting or for self-study by advanced undergraduate and graduate students and as a valuable reference for researchers in mathematics, physics, and engineering.Other books recently published by the authors include: Mathematical Analysis: Functions of One Variable, Mathematical Analysis: Approximation and Discrete Processes, and Mathematical Analysis: Linear and Metric Structures and Continuity, all of which provide the reader with a strong foundation in modern-day analysis.
Inhaltsverzeichnis
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 Poincare-Bendixson.-Exercises.-Appendix A: Mathematicians and other scientists.-References.-Index
