What is this course about?

The answer, in a nutshell, is of course given by its name: the calculus of several variables. To be somewhat more precise, the course is about differentiation, integration and finding extrema of functions on subsets of real n-space. Such a subset may be open or may be something `smooth’ like a curve, a surface or a higher dimensional analogue. Since we always have real n-space in the a background  as an  ambient space,  the course interpolates between a standard course on calculus and a course on smooth manifolds. This is reflected by the fact that it has both computational and conceptual content. These aspects cannot be simply separated, because the care with which the notions we need are introduced often pays off when we  do computations.

We learn to differentiate functions and maps in several variables and derive a chain rule, but find that integration in this setting turns out to have a somewhat unexpected twist and only acquires a sense in the desired generality after  we have derived a transformation formula for integrals. The course ends with a version  of Stokes’s formula.

Although Topology is not a prerequisite,  we often use its  language (which indeed applies to a much wider setting) because that is the most efficient way to express many of the results. As a bonus this also helps to motivate some of the notions of that field.

Prerequisites

Analysis in a single variable and basic linear algebra.

Literature

The course will be based on the 2-volume book J.J. Duistermaat, J.A.C. Kolk: Multidimensional Real Analysis, I, II, Cambridge University Press, Cambridge, ISBN 0-521-55114-5, ISBN 0-521-82925-9. The student society A-Eskwadraat was (and perhaps still is) able to give you a special deal for the set. Additional information (such as corrections and some exams given in the past) can be found on one of the author’s webpages. A larger collection of recent exams is available from this page  of A-eskwadraat.

Place and time

A a rule, the course is given on Tuesdays 13:15-15:00, starting Feb. 7, whereas the exercise sessions are given on Thursdays 9:00-10:45, starting Feb. 9. Both take place in the BBL building, room  169.

Grading

The final grade will have three constituents: the grade M for your midterm exam M, the grade F for your final exam and the grade H for your homework.  You have two options: with or without (H). 

Without H: your final  grade is the average of M and F, but for both you must have a grade 6 or higher.

With H: your final grade is the weighted average 0.3H +0.3M+0.4F, but requires F to be at least 6.

Here is how H is determined:  Exercises with an asterisk will be graded (with grade ‘sufficient’, ‘in between’ or ‘insufficient’) and must for that purpose be handed in within a week after the mentioned date (by mail or hard copy). Then H is the average of all the weekly grades except for the two worst ones, using the rule: sufficient=10, in between=7, insufficient=5 and with failure to hand in a starred (*) homework exercise resulting in a zero grade.

The exercise sessions will be run by Joost Broens broens.joost@gmail.com, Boris Osorno Torres

bosornot@gmail.com and Jaap Eldering J.Eldering@uu.nl. We aim to have your grades available to you via the Blackboard system  http://uu.blackboard.com.

Exam: date, time and place

The midterm exam is on April 17 13:30-16:30 in room Beta of the Educatorium and the final exam is on June 26 13:30-16:30 in room Alfa of the Educatorium.

Material covered and exercises

  To be updated weekly, the starred exercises will be graded and will count towards your H-grade.

All items of any given exercise must be made unless mentioned otherwise.

   Feb. 7  Standard notation for Rn  inner product, norms,  Hilbert-Schmidt norm;

metric space, neighborhood, open set, continuity of a map between metric spaces;

Differentiability of a map at a point, directional derivative.

  Exercises: (2.8), (2.10)* and: Prove that two norms on a real vector space give rise to the same collection of open sets if and only if either norm is continuous with respect to the metric defined by the other. Prove that this is the case for ||x|| and the norms on Rn given  by

||x||′ =max{ |x1| ,.., |xn| } and  ||x||″= |x1| +...+| xn|.

   Feb. 14 Jacobian matrix, chain rule, Ck-map,  Lipschitz property of a C1-map on a convex compact set, Ck-diffeomorphism, Lemma 3.2.1.

  Exercises: Prove that a nonempty  open subset of Rn cannot be C1-diffeomorphic to an open subset of Rp unless n=p, 2.22, 2.27*.

  Feb. 21 Mention of a strengthened Lemma 3.2.1, Contraction Lemma 1.7.2 for complete metric spaces, proof of the Inverse Function Theorem, discussion of the Implicit Function Theorem.

  Exercises: 3.2, 3.31*, 3.36

  Feb. 28 Diffeomorphism interpreted as a new coordinate system, examples (such as polar coordinates), Implicit Function Theorem as providing a normal form, example: a simple root of a polynomial depends differentiably on the coefficients, notion of a  Submanifold of Rn .

  Exercises: 3.45*, 4.3.

  March 6 Immersion, Embedding, the image of an embedding is a submanifold, Submersion, fiber of a submersion is a submanifold.

  Exercises: 4.8, 4.12. 4.29*.

  March 13 No class.

  March 20 Normal forms for submersions  and  immersions, Tangent Space of a submanifold, derivative as a linear maps between tangent spaces.

  Exercises: Do this extra exercise first: Let f  be a submersion defined on a neighborhood of a point a in Rn and denote by V the fiber of f through a. Prove that the tangent space of V  at a is the kernel of Df(a). The other exercises are 5.2, 5.10, 5.18*.

  March 27 Example of a cuspical curve. Critical point and value of a real valued function on a subvariety, Theorem of the Lagrange multipliers. Example.

  Exercises: 5.35, 5.47* (NB: this differs from what was given in class.)

  April 3 More examples. Review of Riemann integrability of compactly supported functions on Rn.

  Exercises: 5.40, 5.41* parts i) and ii) only, 5.42.

  April 10 Jordan-measurability, negligible subsets until the end of subsection 6.3.

  Exercises: 6.6, 6.7*, 6.8 (NB 6.8 comes instead of 6.9 given in class).

  April 17 Midterm exam.

  April 24 No class (but there will be an exercise session on April 26; Exercises: 6.11, 6.14).

  May 1 Integrabilty of a continuous function on a Jordan measurable set, admissibility of changing the order of integration, formulation of the transformation formula for integrals, partitions of 1.

  Exercises: 6.15, 6.16*, 6.20.

  May 8 (Dr Stienstra will substitute for me) Partitions of 1.

  Exercises: 6.26, 6.32*, 6.33.

  May 15 Local integrability, absolute integrability, Haar measure on vector space, density on a submanifold. Integrability properties of a density on an open subset of  Rn. Transformation formula  for an integrable density on an open subset of  Rn.

  May 22

  May 29

  June 5

  June 12

  June 19: no class.

  June 22: Final exam.