(Created 2008-07-17.)

CALCULUS IN SEVERAL VARIABLES | FMA435 |

**Aim**

The course aims at giving a basic treatment of calculus in several variables, including three-dimensional vector analysis. Particular emphasis is on the role calculus in several variables 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 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 to independently choose methods to compute double and triple integrals, and be able to carry out the solution essentially correct.

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

be familiar with the important theorems of vector analysis, and have an understanding of their physical content.

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.

*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 be aware of its possibilities and limitations.

**Contents***Part 1. Calculus in several variables*

*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. Implicit functions.

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

*Double and triple integrals*: Iterated integration. Change of variables. Integration using level curves. Improper integrals. Applications: volume, moment of inertia, centre of gravity.

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

*Part 2. Threedimensional vector analysis*

Surface integrals. Flow integrals. Divergence and rotation. Gauss' and Stokes' theorems. Potential and exact differential. The continuity equation.

Visualisation and formula manipulation using Maple.

**Literature**

Persson A, Böiers L-C: Analys i flera variabler, Chap 1-10. Studentlitteratur 1988, 3rd edition 2005. ISBN 91-44-03869-0.

**Code: **0108.
**Name: **Calculus in Several Variables.

**Higher education credits: ** 6.
**Grading scale: **TH.
**Assessment:** Written test comprising theory and problem solving. The test is identical to the one given for the course FMA430.
**Contents:** See part 1 above. (Coinciding with the contents of the course FMA430.).

**Code: **0208.
**Name: **Three-dimensional Vector Analysis.

**Higher education credits: ** 1,5.
**Grading scale: **UG.
**Assessment:** Written test in the middle of the study period. Retakes in appropriate examination periods. Computer work.
**Contents:** See part 2 above.