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