# MAT-23306 Multivariate Mathematics Applied

## Course

Credits 6.00

 Teaching method Contact hours Practical intensively supervised 12 Tutorial 48 Self-study
 Course coordinator(s) dr. M de Gee dr. JHJ van Opheusden Lecturer(s) dr. M de Gee dr. JHJ van Opheusden Examiner(s) dr. M de Gee dr. JHJ van Opheusden

English

### Assumed knowledge on:

MAT-14903 and MAT-15003

### Contents:

- linear algebra: vectors, matrices, eigenvalues and eigenvectors;
- complex numbers;
- ordinary differential equations: direction field and equilibria; separation of variables and variation of constants; systems of linear differential equations; systems of non-linear differential equations and classification of equilibria;
- numerical methods for ordinary differential equations: difference quotients and the Euler method; systems of differential equations; trapezium and Runge-Kutta; discretization errors; error propagation, stability and stiffness;
- integration in two or three dimensions: limits of integration; coordinate systems and the Jacobian;
- introduction to partial differential equations: flow models, diffusion and convection; boundary and initial conditions; stationary solutions;
- vector fields: flow fields and force fields; the gradient and the laws of Fick, Fourier and Darcy; the potential function; divergence and the Laplace operator;
- Fourier series for partial differential equations: separation of variables and the Sturm-Liouville problem; boundary value problems and Fourier series;
- use of computer software.

### Learning outcomes:

After this course the student is expected to:
- have working knowledge of concepts, methods and techniques from linear algebra, calculus, vector calculus and numerical mathematics;
- be able to apply mathematical knowledge, insights and methods to solve problems in the technological sciences using a systematic approach;
- be able to critically reflect upon the results;
- be able to interpret the results in terms of the (physical, chemical, biological) problem that was modelled mathematically;
- be able to use mathematical software in elaborating mathematical models.

### Activities:

- preparing for lectures and tutorials by self-study and doing exercises;
- active participation in lectures and tutorials;
- doing the exercises;
- computer practical (compulsory).

### Examination:

Written exam, with open questions and multiple-choice questions.

### Literature:

1. Mathematics at Work, vol. 3: functions of several variables applied.
2. Mathematics at Work, vol. 3: supplements.

Programme Phase Specialization Period Compulsory for: BSW Soil, Water, Atmosphere BSc 1AF BES Environmental Sciences BSc C: Environmental Technology 1AF MHW Hydrology and Water Quality MSc A: Soil and Ecosystem Hydrology 2AF MHW Hydrology and Water Quality MSc B: Hydrology and Quantitative Water Management 2AF MMA Meteorology and Air Quality MSc 1AF MES Environmental Sciences MSc 2AF