# Applied Mathematics III

Course Code : | 1005WETWIS |

Study domain: | Mathematics |

Academic year: | 2017-2018 |

Semester: | 1st semester |

Contact hours: | 60 |

Credits: | 5 |

Study load (hours): | 140 |

Contract restrictions: | No contract restriction |

Language of instruction: | Dutch |

Exam period: | exam in the 1st semester |

Lecturer(s) | Werner Peeters |

### 1. Prerequisites *

an active knowledge of

- Dutch

- general knowledge of the use of a PC and the Internet

specific prerequisites for this course

The students need to have aquired a general mathematical attitude and insight in the scientific method, as taught in the courses Applied Mathematics I and II.

Apart from that, the basic notions of working with computers is assumed: installing software on a Windows operating system, some applications like Excel are redeemed necessary. Furthermore, the students are required to have the basic notions to be able to surf on the internet. Also, the students have to be able to work fluently with a scientific calculator. Finally, the students are expected to be able to express themselves in decent Dutch.

### 2. Learning outcomes *

- You can correctly read and write mathematical formulae.
- You know the terminology used in calculus in several variables, general differential equations and difference equations.
- You can write down the proofs of the most important theorems.
- You can solve exercises in calculus in several variables, general differential equations and difference equations.
- You can integrate functions over 2D and 3D regions, as well as general curves and surfaces, potentially by using integraOpl theorems like Green, Stokes and Gauss.
- You can solve differential equations by using power series, and you know a few examples that occur in physics.
- You can solve linear vectorial differential equations of first order.
- You can solve partial differential equations by separation of variables.
- You can calculate the Laplace and inverse Laplace transform of a function, and solve differential equations with it.
- You can solve difference equations, and you know the analogy with differential equations.
- You can verify your results numerically.
- You can present your results in an orderly fashion in LaTeX, in linguistically sound Dutch.

### 3. Course contents *

The course Applied Mathematics III will most prominently be aimed at calculus in several variables. Again, the students will be pushed to acquire more depth and insight, and they are expected to make links between mathematics and other scientific courses, such as chemistry, physics and biology

The aim is to make the students familiar with techniques from calculus in several variables, which may be of use in a broader (applied) scientific context. The course comprehends the study of multiple integrals, integration theorems on surfaces and curves, power series, vectorial differential equations, partial differential equations and difference equations.

Again, much attention is being paid to to the recognition of problems and the application of a correct mathematical instrumentarium to solve these, or at least to formulate the problem in a mathematical sound way, so that a further solution (e.g. by computer) is made possible. In other words, the students must familiarize themselves with the formal mathematical language. Besides, the students also have to be able to produce numerical results and interpret these critically. To this end, an initiation to the mathematical computer environment of Maple will be taught, for which the basic techniques will be introduced in a laboratory session. Less attention will be drawn to the studying of proofs, although a rigid mathematical expression of certain concepts is considered to be of utmost importance. The majority of the course will focus on advanced theory of differential equations, and a lot of focus will be allotted to the application of these in other scientific subdisciplines.

Participation in the lessons implies that the students are expected to participate actively. Among other things, this means that the students will in turn put solutions to exercises on the blackboard. With this, also the end competence is being incorporated that the students must learn to present their results in an organized way before their fellow-students. The students are also expected to present their results in a scientific word processing environment.

XI. Multiple integrals: double and triple integrals, Fubini theorem, coordinat transformations

XII. Integral theorems: line integrals, curves and surfaces, potentials, Green's theorem, surfaces, orientation, flux, curl, divergence, theorems of Stokes and Gauss-Ostrogradski

XIII. Power series: real power series, solution of differential equations with the method of Frobenius-Fuchs, Bessel and Legendre equations

XIV. Vectorial differential equations: linear sets of differential equations, generalized eigenvalue problems, stability

XV. Partial differential equiations: differential operators, separation of variables, problems with border constraints,

XVI. Laplace transforms, the Heaviside function and the Dirac distribution

XVII. Difference equations: linear and homogenous difference equations, first order difference equations, difference calculus, non-homogenous difference equations

### 4 International dimension*

### 5. Teaching method and planned learning activities

Personal work

### 6. Assessment method and criteria

Continuous assessment

### 7. Study material *

#### 7.1 Required reading

** **

The students will be provided with a Dutch course, where the stress will be put on the demonstration of examples with the theory, which can also be used as a reference work for further self study. A major portion is taken up by (unsolved) exercises, some of which are treated in the course, others are the subject of self study.

• W. Peeters. Toegepaste Wiskunde III

**7.2 Optional reading**

The following study material can be studied voluntarily :

Especially the books with a star are recommended.

• F. Ayres. Theory and problems of differential equations. Schaum's Outline Series. McGraw Hill Book Company, 1981

• M. Braun. Differential Equations And Their Applications. 2nd edition, Springer Verlag, New York, 1975

*• P. Dawkins. Differential equations. http://tutorial.math.lamar.edu/

• S. Elaydi. An Introduction to Difference Equations. 3rd edition. Springer, 2000

• T. Myint--u. Partial Differential Equations Of Mathematical Physics. Elsevier, 1973

*• M. Nachtegael en J. Buysse. Wiskundig Vademecum -- Een synthese van de leerstof wiskunde. 6e druk.

• R. Kent Nagle en Edward B. Saff. Fundamentals of Differential Equations and Boundary Value Problems. Addison--Wesley, 1999

• L. Perko. Differential Equations and Dynamical Systems. Springer--Verlag, Berlijn, 1991

### 8. Contact information *

dr Werner Peeters

Dept. Wiskunde en Informatica

Campus Middelheim gebouw G lokaal G3.14

Tel. 03/265.32.93

E-mail: werner.peeters@ua.ac.be