Aim
The course aims at giving a basic treatment of calculus in several variables. Particular emphasis is on the role this plays in applications in different subjects of technology, in particular image processing, in order to give the future engineer a good foundation for further studies in mathematics as well as other subjects. The aim is furthermore to develop the students' ability in problem solving and to assimilate mathematical text.

Knowledge and understanding
For a passing grade the student must

be familiar with and be able to compute with different representations of curves, surfaces and volumes in two and three dimensions.

be able to carry out (specified) changes of variables in partial differential equations, and by this means to solve such an equation.

be familiar with the theory of optimisation, local as well as global, and be able to compute the solution in simple cases.

be able to show capability independently to choose methods to compute double and triple integrals, and be able to carry out the solution essentially correct.

be able to show capability independently to choose method to compute a curve integral, and be able to carry out the solution essentially correct.

be able to demonstrate a good algebraic computing ability.

be able to give a general account of and to illustrate the meaning of such mathematical concepts in calculus in several variables that are used to construct and study mathematical models in the applications.

be able to account for the contents of some central definitions, theorems and proofs.

be able to explain how pictures are represented, analysed and manipulated using concepts and methods from calculus in several variables.

Skills and abilities
For a passing grade the student must

in the context of problem solving be able to integrate concepts from different parts of the course.

be able to show capability to construct and analyse some simple mathematical models in calculus in several variables.

be able to show capability to explain mathematical reasoning in a structured and logically clear way.

have a basic ability to use Maple for visualisation and formula manipulation, and to be aware of its possibilities and limitations.

be able to choose a method and on a computer execute filterings of pictures, in order e.g. to smooth out or to bring out edges.

Contents
Calculus
Generalities on functions of several variables: function surfaces, level surfaces, surfaces in parameter form, curvilinear coordinates.

Partial derivatives. Differentiability, tangent planes, error propagation. The chain rule. Applications in partial differential equations. Gradient, directional derivative, level curves. Study of stationary points. Curves, tangent, arc length. Surfaces, normal direction, tangent plane. Functional (Jacobi) matrix and determinant. Implicit functions.

Optimisation on compact and non-compact domains. Optimisation with constraints.

Double and triple integrals. Iterated integration. Change of variables. Improper integrals. Applications: volume, moment of inertia, centre of gravity.

Curve integrals. Green's formula with applications. Potential and exact differential.

Image Processing
Computer exercises treating geometrical visualisation and symbolic computation with Matlab and Maple. Grey level transformations with interpretation in images. Edge detection. Mean and Laplace filtering of images.

Literature
Persson A, Böiers L-C: Analys i flera variabler, Chap 1-9. Studentlitteratur 1988, 3rd edition 2005. ISBN 91-44-03869-0.
Berthilsson, R.&Sparr, G. Kompendium i bildbehandling. Mat. inst.

Code: 0207. Name: Calculus in Several Variables.
Higher education credits: 6. Grading scale: TH. Assessment: Written test comprising theory and problem solving.

Code: 0307. Name: Image Processing.
Higher education credits: 1,5. Grading scale: UG. Assessment: Computer work. Approved project. Contents: See Image processing above.