|Programme:||The course can be included in the bachelor’s programmes in Computer systems engineering, Electrical engineering and Mechatronical engineering, the Master's Programme in Information Technology, Master's Programme in Computer Systems Engineering, Master's Programme in Computational Science, and Master's Programme in Microelectronics and Photonics.|
|Course responsible:||Per-Sverre Svendsen|
|Other members of faculty:|
|Solutions for Exam 2011-10-28. (pdf, 61.7 kB)|
Multivariable calculus assumes a working knowledge of the basic concepts and methods in linear algebra and single-variable real analysis. In particular, the student should be familiar with matrices, common transcendental functions (sin, cos, ln etc.), limits, derivatives, integrals, and ordinary differential equations (ODE).
A number of these single-variable concepts (derivative, limit) will be re-defined and extended for functions of more than one variable. This enables us to calculate, e.g., the instantaneous rate of change of a function in a specified direction (’the directional derivative’). Partial differential equations (PDEs) as well as their physical applications are also briefly discussed.
We are often interested in finding the extreme (smallest/largest) values of a function of more than one variable. This typically turns out to be a more difficult problem than in the single-variable case. A completely general method for finding extrema by analytical means does in fact not exist. In certain simple cases, which we will study, it is still possible to calculate global (absolute) and/or local extrema using analytic methods (critical points, Lagrange multipliers/constrained optimization).
Single-variable integrals are useful for calculating areas or the mass/charge of a one-dimensional object. Similarly, double- and triple integrals can be used to calculate the total volume, surface or mass of an extended object. Multiple integrals also have wide-ranging use in other areas like statistics or signal theory (probability distributions and Fourier transforms).
Finally, a brief introduction to vector calculus will be given. Fields, line- and surface integrals as well as the basic theorems of vector calculus are discussed in the context of physical/computational applications.
The emphasis of this course is not so much on memorizing ’a bunch of formulas’. Although it’s probably a good idea to be able to remember, say, the expression for a volume element in spherical coordinates, this particular factoid can be easily looked up in any reference table. Rather, what’s important here is to acquire a certain mathematical ’craftsmanship’ - both in formulating a problem and solving it. Therefore, both lectures and exercises will be geared towards problem-solving.
For the exam a table of formulas will be provided, see Reference formulas and equations. (pdf, 57.7 kB)
No other written or electronic aids are allowed.