Middle East Technical University Department of Mathematics 1997/98

MATH 457 CALCULUS ON MANIFOLDS

THE COURSE SYLLABUS

Week 1. Review of the point set topology in ℝ^{n} (open and closed sets, continuity, compactness). Review of the advanced calculus (differentiation): directional derivatives; Jacobi matrix and the differential; the Inverse and Implicit function theorems (formulations); regular points and regular values; diffeomorphisms (coordinate change).

Week 2. Vectors (located at a point) as derivations C^{∞}(ℝ^{n}) → ℝ; \{\frac{∂}{∂x^{i}} \} as the basis vectors in ℝ^{n}. Covectors, dx^{i} as the basis of (ℝ^{n})^{∗}. Vector fields and 1-forms in ℝ^{n}. Linear space Ω^{1}(ℝ^{n}). Evaluation of 1-forms on vector fields and differentiation of functions along vector fields. The total differential of a function as a 1-form.

Week 3. Identification of (located, tangent) vectors in a local (curvilinear) coordinate system {y^{α}}. Invariant meaning: vectors correspond to a class of tangent curves. Coordinate change formulas. Notation T_{x}(ℝ^{n}) for the space of (tangent) vectors located at a point x ∈ ℝ^{n}. Differential of a mapping f: ℝ^{n} → ℝ^{m} as a linear mapping between the tangent spaces, T_{x}ℝ^{n} → T_{f(x)}ℝ^{n}. Differential of an overlap mapping identifies corresponding vectors of ℝ^{n} = T_{x}(ℝ^{n}). The pull back of covectors.

Week 4. Pull back f^{∗}(ω) of 1-forms in ℝ^{n} along mappings f: U → ℝ^{n}, U ⊂ ℝ^{k}. Properties: linearity, pull back along a product mapping, commuting with the total differential (as the chain rule). Line integral in ℝ^{n} in terms of the pull back of a 1-form. Independence of parametrization as the coordinate change formula.

Week 5. Differentiable manifolds in ℝ^{n}. Charts and local parametrizations. Manifolds with boundary. Examples: curves and surfaces. Manifolds defined by a system of equations and parametrized manifolds (cutting manifold into pieces). Differentiable functions on a manifold.

Week 6. Tangent vectors and covectors on a manifold. Coordinates of a vector in a chart. Tangent space, cotangent space. Vector fields and 1-forms on a manifold. Coordinate representation and transformation after coordinate change.

Week 7. Review of multilinear algebra. Definition and properties of tensor products of vector spaces. Multilinear functions as tensors. Coordinates of tensors with respect to a basis. Coordinate change formulas. Symmetric and alternating tensors. Spaces S^{k}(V) and Λ^{k}(V). Definition and properties of the symmetrization and alternation operators Sym and Alt. Wedge product of tensors, its properties. Orientation of a vector space and the volume element as a vector in Λ^{n}(V), its relation to determinants.

Week 8. k-forms in ℝ^{n} (as Λ^{k}(ℝ^{n})^{∗}-valued functions and as a family of alternating multilinear function ω_{x} on ℝ^{n} ≅ T_{x}(ℝ^{n}), x ∈ ℝ^{n}. The linear space Ω^{k}(ℝ^{n}). Wedge product and differentiation of k-forms, properties. Closed and exact forms, statement of the Poincare Lemma.

Week 9. Pull-back of k-forms along a mapping V → ℝ^{n}. Properties: linearity, commuting with the wedge product and differentiation of forms. Coordinate change formulas for k-forms (as the pull back with respect to the overlap mapping).

Week 10. Review of Riemann integral. Partitions, the upper and lower sums. Measure zero sets. Integration of functions on Jordan measurable regions R ⊂ ℝ^{n}.

Week 11. Integration of k-forms in ℝ^{n} along mappings f: R → ℝ^{n}. Relation to the line and surface integrals as they appear in the advanced calculus. Properties. Examples.

Week 12. Singular chains and the boundary of chains. Integration along chains. Stokes’ theorem for chains. Examples (integration of exact/closed forms, the case of 1-forms).

Week 13. Oriented manifolds. The induced orientation of the boundary. Construction of a fundamental cycle (as a picewise parametrisation). Integration along a manifold. Stokes’ theorem for manifolds, its relation to the special cases studied in the advanced calculus.

Midterm 1 may include the content of Weeks 1–5.

Midterm 2 may include the content of Weeks 6–9.

At least 3 homeworks are desirable before Midterms 1–2 and the Final.

General remarks and suggestions

The above syllabus does not follow too closely the Spivak’s course. The essential point of difference is the order manifolds–differential forms–integration, which differs from Spivak’s. The reason is that manifolds and differential forms are the key notions of the course and must be considered in detail rather then discussed briefly in the end of the course.

The essential content of the course is included into the last two chapters of the Spivak’s book. The main content of the first three chapters, namely,

(1) rigorous proofs of the Inverse and Implicit function theorems

(2) a rigorous definition of the Riemann integral in ℝ^{n} which includes the upper and lower sums, the upper and lower integrals and Jordan measurable sets

(3) proof of the Fubini theorem

(4) proof of the coordinate change formula seems to be a deviation from the main line of the course, which makes it overloaded. Actually, we believe that these topics should rather belong to the syllabus of the advance calculus course.