(Created 2011-09-01.)

CALCULUS IN SEVERAL VARIABLES | FMA430 |

**Aim**

The course aims at giving a basic treatment of calculus in several variables. Particular emphasis is given to the role this plays in applications in different subjects of technology, 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 student's ability to solve problems and to assimilate mathematical text.

*Knowledge and understanding*

For a passing grade the student must

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

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

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

be able to demonstrate an ability to independently choose methods to evaluate double and triple integrals, and be able to carry out the solution essentially correctly.

be able to demonstrate an ability to independently choose method to evaluate a curve integral, and be able to carry out the solution essentially correctly.

be able to demonstrate a good ability to carry out algebraic calculations within the context of the course.

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.

*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 demonstrate an ability 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 be aware of its possibilities and limitations.

**Contents***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, linearisation. Implicitly given functions.

*Optimization* on compact and non-compact domains. Optimization 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.

*Computer work:* Visualisation and formula manipulation using Maple.

**Literature**

Persson A, Böiers L-C: Analys i flera variabler, Chap 1-9. Studentlitteratur 1988, 3rd edition 2005. ISBN 91-44-03869-0.