# 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 |

### Language of instruction:

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 | |

Restricted Optional for: | 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 |