(Created 2009-08-11.)

APPLIED MATHEMATICS | FMA062 |

**Aim**

The aim of the course is to treat such mathematical concepts and methods above the basic level that are important for further studies within e.g. mechanics, solid mechanics, fluid mechanics, control theory, kinetics, ecology, electrical engineering and for further professional activities.

*Knowledge and understanding*

For a passing grade the student must

have good knowledge of trigonometric series and their application for solving model problems in partial differential equations.

be familiar with and be able to describe the different properties of linear systems, and how they can be modelled in the time domain and in the frequency domain.

be familiar with the Laplace transform and its significance in connection with input/output relations and solving differential equations, and be well versed in handling simple transform tables.

have good knowledge of such matrix algebra which is the foundation of eigenvalue problems and solving systems of differential equations.

*Skills and abilities*

For a passing grade the student must

show ability to use the concepts in connection with problem solving.

with proper terminology, suitable notation, well structured and with clear logic be able to explain the solution to a problem.

**Contents***Vector analysis:* Scalar and vector fields. Gradient, divergence, rotation. Conservative fields, potential. Curve integrals and surface integrals. Gauss' and Stokes' theorems. The continuity equation.

*Fourier series and partial differential equations:* Trigonometric Fourier series. Half period expansions. The heat conduction and diffusion equation. The wave equation. Method of separation of variables.

*The Laplace transform:* Step and impulse functions. Computational rules for the two-sided Laplace transform. Inverse transforms, in particular of rational functions. Use of transform tables. Convolution.

*Linear systems:* Mathematical models of linear, time invariant systems. Transfer function. Step response and impulse response. The frequency function.

*Matrix algebra:* Eigenvalues and eigenvectors. Diagonalization, in particular of symmetric matrices. Quadratic forms, diagonalization and classification. Systems of differential equations: solution by diagonalization, solution using exponential matrix.

**Literature**

Persson, A. & Böiers, L.C.: Analys i flera variabler, Chapter 10. Studentlitteratur 2004. ISBN 91-44-03869-0.

Sparr, A.: Tillämpad matematik 1. KF-Sigma.

Spanne, S. & Sparr, A.: Föreläsningar i Tillämpad matematik, Lineära system. KF-Sigma.