Entry requirements

Linear Algebra and Geometry I, Single Variable Calculus, or Series and Ordinary Differential Equations.

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 several variables;
• parametrise curves and surfaces;
• compute partial derivatives of elementary functions; and use partial derivatives to compute local and global extreme values - with and without constraints;
• give an account of basic concepts from topology and convergence in several dimensions;
• outline the definition of the multiple integral, compute multiple integrals and use multiple integrals to compute volumes etc.; as well as give an account of the concepts of line integral and surface integral and know how to compute such integrals;
• use the theorems of Green, Stokes and Gauss;
• express problems from relevant areas of applications in a mathematical form suitable for further analysis;
• 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. Continuity, Partial derivatives, differentiability, gradient, direction derivative, differential. Derivatives of higher order. The chain rule. The Jacobian. Taylor's formula. Implicit functions. Optimisation: local and global problems, problems with equality constraints. Topology in several dimensions: open, closed and compact sets. Uniform continuity. Multiple integrals, change of variables, improper integrals, applications of multiple integrals: volume, centres of mass, etc. Line integrals and surface integrals of scalar functions and vector fields. Divergence and curl. Identities for grad, div and curl. Green's, Stokes's and Gauss's theorems. Function sequences and function series, uniform convergence.

Instruction

Lectures, lessons and problem solving sessions.

Assessment

Written examination at the end of the course, or two written tests each of five credit points. Moreover, compulsory assignments may be given during the course in accordance with instructions at the beginning of the course.

Other regulations

The course cannot be included in passing degree together with the course Several Variable Calculus, limited version 1MA017 or Several Variable Calculus 1MA016.