Course Description: The course develops fundamental geometric tools of mathematical analysis, in particular integration theory, and is a preparation for further geometry/topology courses. The central statement is the famous Stokes theorem, a classical version of which appeared for the first time as an examination problem in Cambridge in 1854. Various manifestations of the general Stokes theorem are associated with the names of Newton, Leibniz, Ostrogradski, Gauss, Green. This theorem, both in its infinitesimal and global forms, relates integral over a boundary of a surface or of a solid domain ("circulation" or "flux") with a natural differential operator, known in particular cases as "divergence" or "curl". The prototype and the simplest case of the Stokes theorem is the Newton-Leibniz formula linking the difference of the values of f on endpoints of a segment with the integral of df . The standard modern language for these topics is differential forms and the exterior derivative. Differential forms are used everywhere from pure mathematics to engineering. We give an introduction to the theory of forms, as well as a simplifying treatment for the traditional technique of operations with vector fields in the Euclidean three-space.
1. Differential forms on Rn, n ≤ 3.
Definition, examples. Algebra of forms. Substitutions (pullbacks). Exterior differential: definition, examples, basic properties.
2. Digression: differential calculus on Rn.
Differential of a function as a covector. Geometrical meaning of 1-forms. Differential of a map. Chain rule. Velocity vector. Curvilinear coordinates and associated bases ("moving frames"). Notation for Rn: upper and lower indices, Einstein summation rule.
3. Theory of forms for Rn.
Generalization of notation; algebra of forms and the exterior differential (axiomatic definition, invariance). Pullbacks. Geometric interpretation of k-forms.
4. Integration of forms.
Line integrals. Dependence on paths. Interpretation of d. Changes of parameter. Orientation. Chains. Integration of k-forms; k-dimensional paths and k-chains.
5. The Stokes theorem.
Orientation induced on the boundary. Statement of the theorem and examples. Proof for chains.
6. Forms and vector fields on Euclidean space.
Recollection: areas and volumes. Integrals of the first kind. Forms corresponding to a vector field: "work" (or "circulation") and "flux". Curl and divergence. Hamilton's vector-operator nabla. Classical integral theorems: Ostrogradski--Gauss, Stokes and Green. Geometric meaning of divergence. The Laplace operator.
Theodore Voronov 24 December 2005 (6 January 2006)