Calculus III

Aim

This course is a continuation of the first year course in single variable calculus. Consequently, the aims are similar: to give students an understanding of mathematical concepts and familiarity with mathematical methods of analysis. Here, these aims relate to the treatment of functions of several variables which arise in all branches of physics and engineering. Students will be expected to be able to do the following after completing this course:

• handle multivariable functions, e.g. to be able to determine limits,
• decide if a function is continuous and differentiable, determine partial derivatives and use the chain rule for transforming and solving partial differential equations
• solve global and local maximum and minimum problems, with and without constaints.
• quote and explain definitions of concepts like limit, contiuity, partial derivative, differentiability, gradient, tangent plane, multiple integrals.
• explain and use central theorems like the implicit function theorem, calculate double and triple integrals, derive expressions for area and volumes using multiple integrals.

Content

Functions of several variables. Limits and continuity. Partial derivative, the gradient, directional derivative and differential. Tangent plane and linearization. The chain rule. Taylor's formula. Vector-valued functions, the Jacobian matrix and the Jacobian. Implicit differentiation and implicit functions. Local and global maxima and minima. Finding of maxima and minima with and without constraints. Double and triple integrals. Iterated integrals. Change of variables. Space curves.