MMA210, Högre differentialkalkyl, Spring 18

Latest news

Welcome to the course! The schedule for the course can be found in TimeEdit.

Monday May 21:

Here is an updated version of the summary of important definitions, theorems, concepts, examples etc. to focus on while preparing for the exam:

Study guide for the exam

Monday April 30:

Here is a summary of important definitions, theorems, concepts, examples etc. to focus on while preparing for the exam:

Study guide for the exam

The list will be updated with more stuff as we go along.

Teachers

Course coordinator: Daniel Persson (daniel.persson (at) chalmers.se)

Course literature

From Calculus to Cohomology - De Rham Cohomology and Characteristic Classes
by Ib Madsen and Jorgen Tornehave

We will cover the first 10 chapters of the book (4-6 more superficially). The examination will be in the form of a final written exam on Monday 28/5.

There are also nice lectures notes by Pekka Pankka that are based on the book by Madsen and Tornehave. The notes contain more detailed proofs of some of the results in the book. The notes are available from this link:

Introduction to de Rham cohomology

Some other books that provide nice complements are:

Differential forms in algebraic topology by Bott and Tu

Geometry, topology and physics by Nakahara

Calculus on manifolds by Spivak

Introduction to smooth manifolds by Lee

An introduction to manifolds by Tu

See also the following nice notes by Tao which provide a short introduction to differential forms and integration:

Differential forms and integration by Terence Tao

Program

We will cover most of the first 10 chapters of the book, but we will treat 4-6 more superficially. The main point of the course is the careful introduction of the notions of differential forms, De Rham cohomology, manifolds, and vector fields, leading up to Stokes' theorem. Stokes' theorem is a general theorem on integration of differential forms on manifolds with boundaries, containing as special cases Green's formula, the divergence theorem and, yes, 'Stokes' theorem' for surfaces bounded by a curve in three-dimensional space. The difficulty in the subject is not so much the proof of this very central result as the build-up of the abstract notions in its statement.

Rough course outline:

(the numbering is not in one-to-one correspondence with the lectures)

1. Motivation and recap from multivariable calculus (ch. 1)

2. Alternating algebra (ch. 2)

3. Differential forms and de Rham cohomology (ch. 3)

3.1 Differential forms (ch. 3)

3.2 de Rham cohomology (ch. 3, ch. 7)

3.3 Pull-back of differential forms (ch. 3)

3.4 The Poincaré lemma (ch. 3)

4. Manifolds (ch.8)

4.1 Smooth manifolds: definitions (ch.8)

4.2 Examples of manifolds (ch.8)

4.1 Product manifolds, quotient manifolds, submanfolds (ch.8)

5. Calculating de Rham cohomology of some simple manifolds (ch. 4-6)

6. Integration on manifolds (ch.9)

7. Stokes' theorem (ch.10)

Recommended exercises:

Chapter 1: 1.1, 1.2

Chapter 2: 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.9, 2.10

Chapter 3: 3.1, 3.2, 3.3

Chapter 8: 8.2, 8.4, 8.5, 8.6

Computer labs



Reference literature:

Learning MATLAB, Tobin A. Driscoll ISBN: 978-0-898716-83-2 (The book is published by SIAM).

Course requirements

The learning goals of the course can be found in the course plan.

Assignments


Examination

Written exam on Monday 28/5.

Examination procedures

In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.

Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal, where you also can read about what rules apply to examination at University of Gothenburg.

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.

At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination:
Exams are reviewed and retrieved at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

Old exams