Functions Of Several Variables

FUNCTIONS OF SEVERAL VARIABLES treats selected topics from the calculus of several variables that provide the undergraduate background for graduate courses in differential geometry and complex variables. It is designed for students in the third or fourth-year analysis program who have completed the standard freshman-sophomore calculus sequence plus an introduction to linear algebra. The topics included in this textbook were selected according to two objectives: they cover the notions usually called "vector analysis" and they are concepts that can be easily generalized to differential manifolds in a relatively coordinate-free manner. Except for the fundamental existence and uniqueness theorem for ordinary differential equations, the results used in this book, are proved in the body of the text. Professor Woll's treatment of the theory of functions of several variables can be considered in three general sections. In the first two chapters a theoretical foundation is laid for the material developed in the later chapters. Chapters Three, Four, and Five are basically manipulative, covering such notions as a tangent vector at p ∈ R^n, covectors at p, multilinear algebra, and differential forms on R^n. The last section, Chapters Six and Seven, is again somewhat theoretical. Chapter Six treats the concept of a flow with velocity field X and the related derivations on vector fields and differential forms. In Chapter Seven Professor Woll shows how the notation and ideas developed in the preceding chapters can be used in the theory of functions of a complex variable.