MMA211 Advanced Differential Calculus Spring 24

This page contains the program of the course. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.

Program

The schedule of the course is in TimeEdit.

We will cover most of the first 10 chapters of the book (see course-PM for info on the literature), except for chapter 6 which is omitted. 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.

Study guide for the exam

Rough course outline:

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

0. Motivation

1. Recap from multivariable calculus 

2. Alternating algebra 

3. Differential forms 

4. de Rham cohomology 

5. Smooth manifolds

6. Differential forms on manifolds

7. de Rham cohomology of manifolds

8. Examples of de Rham cohomology

9. Integration on manifolds

Lecture notes available here

 

 

Recommended exercises and proofs (a complement to the study guide):

Week 1

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

Week 2

Chapter 3: 3.1, 3.2, 3.3

Chapter 8: 8.2, 8.4, 8.5, 8.6

Week 3

• Prove that the tangent space to R^n at a point p is isomorphic (as a vector space) to the space of derivations of R^n at p.
• Let (R, Id) be a standard chart on R and let X be a copy of R with the non -standard chart (X, \psi), with \psi(x) = x^3. Show that these are non-compatible smooth structures on R. Also show that, nevertheless, R and X are diffeomorphic. 
• Prove Lemma 9.6 in Madsen.
• Construct an atlas on S^2 using the covering U_{+i} and U_{-i}, i =1,2,3, discussed during the lecture today (Tuesday). Each chart has homeomorphisms given by projecting out the i:the coordinate. Verify that this is a smooth atlas.

Week 4

• Let F : R^2 ---> R^3 be the map defined by F(x,y)=(x,y,xy)=(u,v,w). Let p=(x,y)\in R^2. Compute the pushforward F_*(\partial /\partial x) as a linear combination of \partial/\partial u, \partial/\partial v, \partial/\partial w at the point F(p).
• Let U be the open set R^2 - {(x,0)|x\geq 0}. On U the polar coordinates are uniquely defined by x=r \cos\theta, y=r\sin\theta with r>0 and 0<\theta <2\pi. Find \partial/\partial r, and \partial/\partial \theta in terms of \partial /\partial x and \partial /\partial y.

 

 

 

 

 

 

 

Video lectures

As a complement to the on site lectures there are also video lectures and notes available here:

Video lectures

 

 

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