Syllabus for Several Variable Calculus, Limited Version

Learning outcomes

In order to pass the course (grade 3) the student should be able to

• give an account of the concepts of limit, continuity, partial derivative, gradient and differentiability for functions of severable variables;
• parametrise curves and surfaces;
• compute partial derivatives of elementary functions;
• use partial derivatives to compute local and global extreme values - with and without constrains;
• outline the definition of the multiple integral, compute multiple integrals and use multiple integrals to compute volumes, centres of gravity, etc.;
• compute line integrals of vector fields in the plane;
• exemplify and interpret important concepts in specific cases;
• formulate important results and theorems covered by the course;
• express problems from relevant areas of applications in a mathematical form suitable for further analysis;
• use the theory, methods and techniques of the course to solve mathematical problems;
• present mathematical arguments to others.

Content

Polar, cylindrical and spherical coordinates. Parameterisations of curves and surfaces.

Level curves and level surfaces. Arc length. Scalar and vector valued functions of several variables. Partial derivatives, differentiability, gradient, direction derivative, differential. Derivatives of higher order. The chain rule. Taylor’s formula. Optimisation: local and global problems, problems with equality constraints. Multiple integrals, change of variables especially polar coordinates, improper integrals, applications of multiple integrals: volume, centres of mass, etc. Line integrals of vector fields. Green's theorem in the plane.

Instruction

Lectures and problem solving sessions. Team-working may occur.

Assessment

Written examination at the end of the course. Moreover, compulsory assignments may be given during the course.