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