Entry requirements

120 credits including 60 credits in mathematics. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

The course aims at strengthening and generalizing results that the student has already learnt from previous calculus courses.

On completion of the course, the student should be able to:

• be able to give an account of various consequences of the mean value theorem;
• master the concept of convergence for numerical sequences and series and for sequences and series of functions, and in particular understand the distinction between pointwise and uniform convergence;
• be able to give an account of the concept of equicontinuity;
• be able to give an account of the contraction principle and to derive some of its consequences in several variable calculus;
• be able to formulate central theorems in the area of the course and to describe the main features of their proofs;
• be able to use the theory, methods and techniques of the course to solve mathematical problems.

Content

Numerical sequences and series: limit points, upper and lower limits, Cauchy sequences, rearrangements. Continuity: discontinuities, monotonic functions. Differentiation of single-variable functions: the mean value theorem, continuity properties of derivatives, l'Hospital's rule and Taylor series. Sequences and series of functions: Uniform convergence, equicontinuous families of functions, the Arzelà-Ascoli theorem, the Stone-Weierstrass theorem. Functions of several variables: the contraction principle, inverse and implicit function theorems, the rank theorem.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination at the end of the course.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.