Derivatives and Integrals of Multivariable Functions
The book is intended to be a text for a one-semester course on calculus of several variables. The author presupposes that the reader is familiar with topology of Euclidean spaces, properties of continuous mappings, theory of limits, and basic linear algebra. The theory of derivatives covers the chain rule, the mean value theorem, the inverse and implicit function theorems, the connection of extreme values and second derivatives, and the theorem on Lagrange’s multipliers (a necessary condition for the case of several constrains). Also some information on curves, surfaces and tangent planes is given, but the proof of the Lagrange theorem does not use these geometrical notions. The theory of the Riemann integral for functions of several variables includes versions of Fubini’s theorem and the change of variable theorem. The theory of line and surface integrals ends with versions of Green’s theorem, classical Stokes’s theorem and the divergence theorem. All sections contain exercises and their solutions (68 pp.) are presented at the end of the book. The textbook is written in a conversational non-formal style but formulations of theorems and proofs are essentially quite rigorous. An exception is the theory of curves and surfaces where, for example, independence of tangent planes and surface integrals on parameterizations is not proved.