# Applied mathematics II

Course Code : | 1004WETWIS |

Study domain: | Mathematics |

Academic year: | 2017-2018 |

Semester: | 2nd semester |

Contact hours: | 75 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | Dutch |

Exam period: | exam in the 2nd semester |

Lecturer(s) | Werner Peeters |

### 1. Prerequisites *

- competences corresponding the final attainment level of secondary school

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. For the most topics in mathematics no prior knowledge is needed, because the course starts from scratch. An acquaintedness with the notions of mathematics as given in secondary school with 8 or 6 hours of mathematics, but given an extra effort also 4 hours, is assumed. The students are held to possess a certain skill in the - mainly algebraic - methods learnt in secondary school.

More specifically, the students already need prior knowledge on:

- General logic, methods of proof

- Working with real numbers and polynomials in a finite number of variables

- Special products and factorisation

- Solving equations and inequalities

- Equations of first- and second order functions

- Basic equations of ellipses, parabola and hyperbola

- Sign determination of products and quotients of terms of degree 1 and 2

- Binomium of Newton

- Solving sets of equations

- Vector geometry and analytic geometry in two dimensions

- Goniometry and trigonometry

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.

Furthermore, the course Applied Mathematics I should have been attended. It is not necessary though to have already passed its exam.

### 2. Learning outcomes *

- You can correctly read and write mathematical formulae.
- You know the terminology used in differential equations, series, matrices and differentiability in several variables.
- You can write down the proofs of the most important theorems.
- You can solve ordinary differential equations of several types.
- You can define sequences and series, control their convergence and you can apply this to Taylor and fourier series
- You can calculate with matrices and determinants, and you can apply this in sets of linear equations and spatial geometry.
- You can calculate the derivative of functions of several variables, and you know a handful of applications.
- You can verify your results numerically.
- You have an elementary knowledge of Maple, and you van verify your exercises with it.
- You can present your results in an orderly fashion in LaTeX, in linguistically sound Dutch.

### 3. Course contents *

The course Applied Mathematics II will most prominently be aimed at the introduction of some new calculating techniques which have not been incorporated in secondary school., in such a way that the students get acquainted for the first time with new mathematical techniques. 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, as well as algebraic structues in several variables. The course comprehends the study of differential equations of one variable, sequences and series, matrices and determinants, geometry in three dimensions and differentiability in multiple variables.

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. Also, some theoretical concepts such as differential equations will thoroughly be worked out so as to guarantee a fluent handling of solution techniques, 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.

VI. Differential equations: generalities, singular solutions, parameter sets solutions, separation of variables, homogenous differential equiations, linear coefficients, exact differential equations, integrating factors, linear differential equations of first order, Bernoulli and Ricatti equations, homgenous differential equations of higher order, Euler equations, method of indetermined coefficients and variation of parameters, order reduction

VII. Sequences and series: affine and geometric sequences and their summations, convergence, limsup and liminf, series in R, convergence criteria, Taylor- and Maclaurin series, pointwise and uniform convergence, fourier series

VIII. Matrices: vector spaces, ring operations, determinants, matrix inversion, solution of sets of linear equations with the methodes of Gauss-Jordan and Cramer

IX. Geometry: points and vectors in R2 and R3, positioning of lines and planes, orthogonality, distance, scalar en vector product, linear transformations, eigenvectors, eigenvalues and eigen spaces, diagonalization of a matrix.

X. Differentiability in several variables: continuity, limits, partial derivation, derivation, differentiation, chain rule, higher derivation, Taylor and Newton-Raphson, implicit function theorem, extremal analysis in multiple variables, Lagrange multipliers

### 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 II

**7.2 Optional reading**

The following study material can be studied voluntarily :

As prior knowledge the following book is being used:

• S. Verwulgen en E. Soetens. Wiskunde voor de beginnende bachelor. Academia Press

Especially the books with a star are recommended.

• A.R. Angel. Elementary algebra for college students, 4th edition. Prentice Hall, New Jersey, 1996

• H. Anton. Multivariable calculus, 4th edition. John Wiley and Sons Inc., 1992

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

• F. Ayres en E. Mendelson. Theory and problems of differential and integral calculus, 3rd edition. Schaum's Outline Series. McGraw Hill Book Company, 1992

• D.D. Benice. Calculus and its applications. Houghton Mifflin Company, 1993

• D.D. Berkey. Applied calculus, 2nd edition. SaundersCollege Publishing, 1991

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

*• C.E. Edwards and D.H. Penney. Calculus, International edition, 6th edition. Prentice Hall, 2002

*• R. Ellis and D. Gulick. Calculus, One and Several Variables. SaundersCollege Publishing, 1991

• H.P. Greenspan, D.J. Benney and J.E. Turner. Calculus, an introduction to applied mathematics, 2nd edition. McGraw Hill Ryerson Limited, 1973

• J.C. Hegarty. Applied calculus. John Wiley and Sons Inc., 1990

• F.E. Hohn. Elementary Matrix Algebra, 3rd edition. Macmillan, New York, 1973

*• M. Nachtegael en J. Buysse. Wiskundig Vademecum -- Een synthese van de leerstof wiskunde. 6e druk. Uitgeverij Pelckmans, 1999

• M.R. Spiegel. Theory and problems of advanced calculus. Schaum's Outline Series. McGraw Hill Book Company, 1974

• J. Stewart. Calculus, 3rd edition. Brooks/Cole Publishing Company

### 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