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Multivariable Calculus, 7.5 credits

Administrative information
Syllabus (pdf)external link, opens in new window  
Study period:  
Course code: MA4005
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:  
Examiner: Per-Sverre Svendsen
  Solutions for Exam 2011-10-28.PDF (pdf, 0 bytes)

Course purpose:

The general aim of the course is to provide an introduction to real analysis of several variables. Particular emphasis will be given to subjects of relevance to applications in science and engineering.

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.

Instruction and examination:

Instruction is organized into lectures and exercises. The exercises are conducted in smaller groups under the supervision of a teaching assistant. Grades for the course are mainly based on a 4-hour written examination giving a maximum of 30 points. In addition to this, ’bonus’ points will be added from 3 home assignments. Each assignment will be awarded with up to 2 points upon successful completion within a given deadline; typically 10 days after the assignment is handed out. The requirements for the highest and the lowest passed grade are 27 and 15 points, respectively.

For the exam a table of formulas will be provided, see Reference formulas and equations.PDF (pdf, 57.7 kB)
No other written or electronic aids are allowed.

Course literature:

C.H. Edwards, D.E. Penney: Multivariable Calculus, 6th ed., Prentice Hall (2002).

Preliminary schedule.

Home assignments.

Old exams.


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