Functions of Multiple Variables

These lecture notes give an elementary overview of calculus of functions (and vector-valued functions) of multiple variables. The notes contain a systematic construction of the theory (in the form of definitions and theorems with proofs). They start with preliminary definitions and continue with the analysis of topological properties, differential and integral calculus and end with the analysis of extrema and performing variable transformations in partial differential expressions. Apart from studying functions defined explicitly we analyze properties of functions defined implicitly by one or more generating equations. For the sake of completeness, several marginal branches of the theory are discussed, such as sequences and series of functions of multiple variables, Taylor expansions, and line and surface integration. The standalone chapter on introduction to partial differential equations introduces the method of so-called canonical transformations. The final section includes an exhaustive list of problems to practice the application of relevant theory. The formal part of the lecture notes is complimented by many comments, explanations, figures, and solved problems with the aim of helping the reader understand the topics at hand.