Several Variable Calculus

10 credits

Syllabus, Bachelor's level, 1MA016

A revised version of the syllabus is available.

Code: 1MA016
Education cycle: First cycle
Main field(s) of study and in-depth level: Mathematics G1F
Grading system: Pass with distinction, Pass with credit, Pass, Fail
Finalised by: The Faculty Board of Science and Technology, 19 March 2007
Responsible department: Department of Mathematics

Entry requirements

Linear Algebra and Geometry I, Single Variable Calculus, 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, gradiant and differentiability for functions of several variables;

be able to parametrise curves and surfaces;

be able to compute partial derivatives of elementary functions;

be able to use partial derivatives to compute local and global extreme values - with and without constrains;

be able to outline the definition of the multiple integral, compute multiple integrals and use multiple integrals to compute volumes, centres of gravity, etc.;

be able to give an account of the concepts of line integral and surface integral and know how to compute such integrals;

know how to use the theorems of Green, Stokes and Gauss;

be familiar with the concepts of uniform convergence and uniform continuity, and be able to decide whether a simple sequence of functions is uniformly convergent;

be able to exemplify and interpret important concepts in specific cases;

be able to formulate important results and theorems covered by the course;

be able to express problems from relevant areas of applications in a mathematical form suitable for further analysis;

be able to use the theory, methods and techniques of the course to solve mathematical problems;

be able to 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. The Jacobian. Taylor's formula. Implicit functions. Optimisation: local and global problems, problems with equality constraints. 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. Sequences of functions, function series, uniform convergence. Uniform continuity.

Instruction

Lectures and problem solving sessions.

Assessment

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