Course syllabus - Calculus of Several Variables

Objectives

The course aims at broaden the concept of functions to include real-valued functions of several real variables and applications thereof, to introduce the concept of vector fields, to generalize the concept of integrals to include sums on space curves, surfaces and solids in three-dimensional space, as well as providing a basis for further studies in mathematics and its applications in science, engineering and economics.
 

Learning outcomes

At the end of the course the student is expected to be able to …
1. analyse real-valued functions of several real variables based on the concepts domain, range, graph, composition and inverse, and to sketch reasonably simple graphs and contour lines.
 
2. explain the fundamental topological concepts in Rn, as well as from standard limits and arithmetic rules being able to determine limits for functions. Especially, the student should be able to evaluate a function's continuity.
 
3. determine partial derivatives, to evaluate differentiability, to determine differentials, as well as being able to apply the chain rule for derivatives of first and second orders if changing variables or at implicit differentiation.
 
4. determine and geometrically interpret gradients and directional derivatives, and from that when appropriate, to determine tangent lines or tangent planes.
 
5. apply Taylor's formula to classify stationary points, especially in the case of functions of two variables.
 
6. to find the maximum and minimum of a continuous function on a compact set, and to be able to formulate and solve optimization problems including constraints.
 
7. analyse vector-valued functions of one real variable and the corresponding space curves, partly from the kinematic concepts position, velocity and acceleration, and partly from the geometrical concepts tangent line, normal line, arc length and curvature. Especially, the student should be able to parameterize space curves.
 
8. determine double and triple integrals by suitable iterations and change of variables.
 
9. determine and interpret line and surface integrals. Especially, the student should be able to determine the normal vector field for curves in the plane and for surfaces in the 3-space. In cases with conservative fields, the potential functions should be determined, and used for e.g. changes of paths for line integrals.
 
10. apply Green’s, Stokes’ and Gauss’ theorems.

Course content

Fundamental topological concepts in Rn: open set, boundary of a set, closed set, bounded set, compact set. Real-valued functions of several real variables: domain, range, limit, continuity, partially differentiable, differentiable, differentials, chain rule, gradient, direction derivative. Optimization: stationary point, local extreme value, extreme value, optimization with constraints, method of Lagrange multipliers. Vector-valued functions: Motion in space, parametrization of curves, arc length, curvature, radius of curvature, centre of curvature. Multiple integrals: double integral, triple integral, iteration, change of variables (especially to cylindrical and spherical coordinates), generalized multiple integral. Applications of multiple integrals: area of surface, volume of solid. Vector calculus: gradient, divergence, rotation, line integral, conservative field, Greens’ theorem, surface integral, flux integral, Stokes’ theorem, Gauss’ theorem.
 

Tuition

Teaching is given in the form of lectures and classes.
 

Specific requirements

Single Variable Calculus, 7.5 credits and either Vector Algebra 7.5 credits of which 3.5 credits must be completed at the beginning of the course or Basic Vector Algebra of which 5 credits must be completed at the beginning of the course, or equivalent.

Examination

Assigned problems (INL1), 1,5 credits, marks Fail (U), Pass (G)
Written and/or oral examination (TEN1), 2,5 credits, marks Fail (U), 3, 4 or 5
Written and/or oral examination (TEN2), 3,5 credits, marks Fail (U), 3, 4 or 5
 

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Grading scale: 5, 4, 3