Meetkunde en Natuurkunde - 2006-2007

The RuG page with information on the course is here.

There will be a non-compulsory homework for the course around Christmas. For people that decide to do the homework the final grade will be 0.75*(final exam grade)+0.25*(homework grade). Otherwise the final grade for the course will be just the final exam grade.

Kerstmis huiswerk: Download.

The homework should be returned by January 10, 2007.

Answers to the Kerstmis huiswerk: Download.

Lessons

Lesson 1 (15/11/2006)

Covered the definition of derivatives for maps $f:R^n->R^m$ and the inverse and implicit function theorems. Proved the implicit function theorem.

Lesson 2 (16/11/2006)

Definition of (sub)manifolds in Euclidean space and examples.

Exercise session 1 (17/11/2006)

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Lesson 3 (22/11/2006)

Defined vector fields in R^n and their relation to ODEs and their integral curves. Explained that if we have a map f: R^m -> R^n, the vectors at x in R^m are mapped to vectors at f(x) in R^n using the derivative Df(x). Defined the tangent space of a manifold.

Exercise session 2 (23/11/2006)

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Lesson 4 (24/11/2006)

Further remarks about vector fields. Defined coordinates on a manifold M as maps from M to R and corresponding tangent vectors to M. Introduced the dual space of a vector space.

Lesson 5 (29/11/2006)

Defined the cotangent space of R^m at a point x, smooth 1-form fields, the dual basis dx_1, ..., dx_m, closed and exact forms and proved the Poincare lemma in R^m.

Exercise session 3 (30/11/2006)

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Lesson 6 (1/12/2006)

Definition of sharp, flat. The gradient of a function. Pull-back of forms.

Lesson 7 (6/12/2006)

Pull-back of functions, push-forward of vector fields. Integrals of 1-forms along curves.

Exercise session 4 (7/12/2006)

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Lesson 8 (8/12/2006)

Special 1-forms (area form, angles form, winding number). Definition of 2-forms.

Lesson 9 (13/12/2006)

Definition of k-forms. Exterior (wedge) product, interior product.

Exercise session 5 (14/12/2006)

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Lesson 10 (15/12/2006)

Pull-back and exterior derivative of k-forms. Closed and exact forms. Poincare lemma.

Lesson 11 (20/12/2006)

Integration of k-forms. Singular k-cubes, k-chains. The Stokes theorem.

Exercise session 6 (21/12/2006)

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Lesson 12 (22/12/2006)

Lie derivatives.

Lesson 13 (10/01/2007)

Covariant and contravariant k-tensors.

Exercise session 7 (11/01/2007)

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Lesson 14 (12/01/2007)

Mixed type tensors. Coordinate transformations.

Lesson 15 (17/01/2007)

Flux in dimension 2 and 3. The Gauss, Green and Stokes theorems. Maxwell's equations using differential forms.

Exercise session 8 (18/01/2007)

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Lesson 16 (19/01/2007)

No lesson