COURSE AIMS AND OBJECTIVES: The goal of the course is to introduce the structure of the Euclidean space Rn, continuity and differentiability of functions of several variables and the basic theorems of differential calculus.
COURSE DESCRIPTION AND SYLLABUS:
1. Structure of Rn. Rn as a vector space and a metric space. Open and closed sets. Accumulation points, closure and boundary of sets. Sequences. Convergence. Bolzano - Weierstrass theorem. Sequential characterization of closed sets. Cauchy property for sequences. Completeness of Rn Banach fix point theorem. Compactness. Characterization of compactness in Rn. Lebesgue number. (4 weeks)
2. Continuity and limit. Real and vector functions of one and several variables. Continuity. Limit. Heine's characterization of continuity. Uniform continuity. Lipschitz functions. Continuous functions on compact sets. Paths and curves in Rn. Path connected sets. Connected sets. Surfaces in R3. (3 weeks)
3. Differential and derivatives. Linear approximation of a function - differential. Partial derivatives. Gradient. Jacobi matrix. Basic properties. Applications in geometry (tangent to a curve in Rn, tangent plane, ...). Higher order differentiability. Schwarz's theorem. The mean value theorem. Taylor's theorem. Minima and maxima. Implicit function theorem. Inverse function theorem. Consequences and applications. (6 weeks)
Basic notions are introduced and explained on examples at lectures. Specific problems and techniques for their solving are explained at tutorials.