Multivariable calculus

Aim

To give the basic knowledge about concepts and methods in analysis of several variables which is used in technical courses. To pass this course students will need to be able to * formulate and understand definitions of the following concepts: topological types of sets, a function of several variables, a limit, continuity, partial derivatives, extreme points and values, multiple integrals. * formulate, explain and apply the following theorems: the max-min theorem for continuous functions on compact sets, the chain rule, the Taylor formula, the classifying of critical points via quadratic forms, the theorem about local extreme points under one or two conditions, the change of variables in multiple integrals. * do calculations with limits and continuity, apply the chain rule to solve partial differential equations. * understand the geometric meaning of gradient * find equations of the tangent plane * carry out investigations of local and global max and min. * compute multiple integrals by iteration. * compute multiple integrals with the help of change of variables (in particular, the polar and spherical coordinates).

Content

Functions of several variables, limits, continuity. Partial derivatives, chain rule, gradient. Taylor formula, local extreme points and values, quadratic forms. Max and min values, optimization on compact and non-compact sets. Optimization under conditions, Multiple integrals. Change of variables in multiple integrals.