MA3402 Differential Forms on Manifolds - Fall 2025

Schedule Room
Lectures: Monday 10:15-12:00 B3
Friday 10:15-12:00 B3
Lecturer
Rune Haugseng
Office: 1250 Sentralbygg 2
Email: rune [dot] haugseng [at] ntnu [dot] no

Exam

Oral exams will take place 26-28 November.

Exam schedule (now with rooms)

Sample exam questions

I will begin each exam by asking (part of) one of these questions; since you see them in advance, some of them may require a bit more thought than a typical oral exam question.

What this course is about

This course could maybe have been called "vector calculus for grown-ups": we'll develop the modern approach to differentiation and integration in higher dimensions via differential forms. We'll start by doing this on open sets in Euclidean space, and see how our new fancy notion of exterior derivative for differential forms recovers the familiar gradient, curl and divergence from vector calculus in dimension 3. Then we'll introduce smooth manifolds and see how our theory extends to these more general geometric objects. With this in hand, we can look at integration on manifolds, leading to a generalization of Stokes's and Green's theorems from vector calculus to manifolds with boundary of arbitrary dimension. Finally, we'll take a look at de Rham cohomology, which gives a way to extract topological information about a manifold from differential forms.

See the study handbook for more information.

What you need to know before this course

You should definitely have taken the standard courses in multivariable analysis and linear algebra. Ideally you have also already taken the course "Differential topology" (TMA4192) so that you already have some experience with manifolds. If you are motivated it is still possible to follow the course even if you haven't done this, however, though you will probably want to do some supplementary reading on your own (e.g. to see some more concrete examples of manifolds than I'll have time to discuss in the lectures). If you have also not taken "Introduction to topology" (TMA4190) you will also need to read up on topological spaces, but the amount of background material we'll need is fairly limited.

Course material

We will not directly follow any particular textbook, but Tu's book (can be downloaded from the link below if you're on NTNU's network) covers most of the material we'll go through in an accessible style; I have listed some relevant sections from this book in the lecture schedule below, but these do not correspond directly to the contents of the lectures. For the last part of the course we will discuss some material that is instead covered in the book of Bott and Tu.

Hand-written lecture notes:

I. Differential forms in Euclidean space:

  • Part 1 (Introduction, Vector fields and differential 1-forms)
  • Part 2 (Multilinear algebra)
  • Part 3 (Differential forms)

II. Manifolds

  • Part 4 (Smooth manifolds and smooth maps)
  • Part 5 (The tangent bundle and vector fields)
  • Part 6 (Differential forms on manifolds)

III. Integration and Stokes's Theorem

  • Part 7 (Orientations)
  • Part 8 (Integration)
  • Part 9 (Manifolds with boundary and Stokes's theorem)

IV. De Rham Cohomology

  • Part 10 (de Rham cohomology)
  • Part 11 (Mayer-Vietoris sequences and Poincaré duality) (The final section on the Čech-de Rham complex is not examinable.)

Exercises

Reference Group

  • David Persson (david.persson@ntnu.no)
  • Jørgen Sønstabø (jorgerso@ntnu.no)
  • Esteban Vargas (etvargas@ntnu.no)

Lecture Plan

Lecture Date Topic Notes Relevant sections in Tu
1 19/08 Introduction, vector fields and 1-forms in Euclidean space 1 1.1, 4.1
2 22/08 Tangent vectors and vector fields as derivations 1 2.1-2.5
3 26/08 Multilinear maps, tensor products, group actions and (co)invariants 2 3.1-3.10
4 29/08 Symmetric and antisymmetric powers, multiplications 2 3.1-3.10
5 01/09 Exterior algebra, wedge product on alternating functionals, differential forms 2,3 3.7-9, 4.2
6 05/09 Exterior multiplication again, differential forms (wedge product, pullback) 2,3 4.2
7 08/09 Exterior derivative, de Rham cohomology, relation to vector calculus 3 4.4-6
8 12/09 Topological manifolds, smooth manifolds, smooth functions 4 5.1-4, 6.1
9 15/09 Smooth maps, the tangent space at a point 4,5 6.2-4, 8.1-2
10 19/09 Tangent vectors and curves, tangent bundle 5 8.6-7, 12.1-2
11 22/09 Vector bundles 5 12.3-4
12 26/09 The cotangent bundle 6 17.3
13 29/09 Pullback of 1-forms, differential of functions, k-forms 6 17.1, 17.5, 18.1-6
14 03/10 Bump functions, exterior derivative 6 13.1, 19.1-5
15 06/10 Orientations of vector spaces, bundles, manifolds 7 21.1-3, 21.5
16 10/10 Oriented manifolds, partitions of unity 7 21.5, 13.2
17 13/10 Partitions of unity, vector fields and derivations 7 13.3 (and App. C), 2.5
18 17/10 Orientations and volume forms, integration 7,8 21.4, 23.1-4
19 20/10 Integration, manifolds with boundary 8,9 23.4, 22.1-3
20 24/10 Tangent bundle with boundary, outward-pointing vector field 9 22.4-5
21 27/10 Boundary orientation, Stokes's Theorem 9 22.6, 23.5
22 31/10 de Rham cohomology (with and without compact support), cochain homotopies 10 24.1-4, 27.1-4, 29.1-2
23 03/11 Homotopy invariance, Poincaré lemma with compact support 10 29.1-5, Bott-Tu §4
24 07/11 Exact sequences, the Mayer-Vietoris sequence in de Rham cohomology 11 25.1-4, 26.1-2
25 10/11 Mayer-Vietoris with compact support, good covers, Poincaré duality 11 Bott-Tu §2,5
26 14/11 Proof of Poincaré duality, Čech-de Rham complex 11 Bott-Tu §5,8
27 17/11 Čech cohomology 11 Bott-Tu §8

References

Since there are many good books on differential forms and differential geometry, it is worth spending some time to look through a few and find a style that works for you. Most of them will cover very similar material to what we will in lectures, though they will differ in notation to a greater or lesser extent.

Suggestions of books:

  • [BT] R. Bott and L. W. Tu, Differential forms in algebraic topology
  • [Tu] L. W. Tu, An Introduction to Manifolds, second edition, Springer Verlag, 2011.
  • [Lee] J.M. Lee, Introduction to Smooth Manifolds, Springer-Verlag.
  • [GH] Guillemin and Haine, Differential Forms, ISBN13: 9789813272774, World Scientific, 2019, Preprint here.
  • [GP] V. Guillemin and A. Pollack, Differential Topology, Prentice Hall, 1974.
  • [M69] J. Milnor, Topology from the differentiable viewpoint, The University Press of Virginia, 1969.
  • [M63] J. Milnor, Morse Theory, Annals of Mathematic Studies No. 51, Princeton University Press, Princeton, N.J., 1963.
  • [D] B. Dundas, Differential Topology. Version January 2013
  • [S] D. Spivak, Calculus on Manifolds, Addison-Wesley, 1965.
  • [Mu91] J.R. Munkres, Analysis on Manifolds, Addison-Wesley, 1991.
  • [Mu75] J.R. Munkres, Topology: a first course, Prentice-Hall, 1975.
  • [RS] Robbin and Salamon, Introduction to Differential Topology, 2018. Preprint here

If you want to go beyond the material in this course, Differential forms in algebraic topology by Bott and Tu is highly recommended as a more advanced book on de Rham cohomology.

If you're interested in physics, you could try to find "Gauge Fields, Knots And Gravity" by Baez and Muniain, which develops the basic theory of differential geometry in parallel with discussions of its applications in physics.

2025-11-20, Rune Gjøringbø Haugseng