Upon completion of the course, the student is able to;
1. General competences: 10%
- Formulate a consistent and complete argumentation.
- Use and formulate appropriate theorems and check their conditions.
2. Integration: 30%
- Apply the definition of Riemann Sum, to use the Fundamental Theorem of Calculus and to calculate integrals using partial integration, substitution or direct calculation. All for scalar functions.
- To provide the definition of integrals of functions of several variables, to calculate integrals of functions of several variables through multiple integration, possibly using coordinate transformations such as, but not restricted to, polar, spherical and cylindrical coordinates.
3. Sequences and Series: 20%
- Determine the convergence or divergence of sequences of real numbers,
- Apply this to infinite series, in particular power series and Taylor and MacLaurin series.
- Higher dimensional versions of Taylor Series.
4. Functions of several variable and curves 20%
- To derive parametrisations of curves in the plane of in space, and to determine the length of such curves.
- To provide the definitions of limits and continuity of functions of several variables and to apply these concepts to calculate limits and to determine the (dis)continuity of functions of several variables.
- To formulate and apply the chain rule in higher dimensional cases.
5. Optimisation: 20%
- To use partial derivatives in finding potential extreme values,
- Apply the method of Lagrange Multipliers to determine extreme values of functions of several variables.
The course Calculus 2 is the second course in the line of Mathematics for Applied Mathematics and Applied Physics. The course is divided into six parts.
- Sequences, series and Taylor series. An introduction to sequences, convergence and infinite series is given along with convergence criteria of infinite series. Furthermore a special class of infinite series, power series is introduced. The results are applied in the theory of Taylor series.
- Integration of functions of one variable. The Riemann Integral. Fundamental Theorem of Calculus. Integration techniques: substitution and partial integration.
- Vector Functions. These are functions in 2D or 3D that depend on one variable. The graphical visualisation of such functions are curves in the plane or in space. A central problem is to find parametrisations of curves in the plane or in space.
- Functions of several variables, mainly two or three. These are functions that take their values in the reals, but depend on two or three independent variables.
- Introduction to the optimisation of functions of two or three variables using the theory of Lagrange multipliers
- Integration of functions of several variables. Change of coordinates such as polar, cylindrical and spherical coordinates.