Middle East Technical University Fall 1997 Math 457 Calculus on Manifolds

QUESTIONS FOR THE FINAL

(1) The Inverse and Implicit function theorems; formulations and consequences.

(2) Measure zero and content zero sets in ℝ^{n}. Definitions, related theorems and examples.

(3) Lower and upper sums assotiated to a partition. Definition of the Riemann integral in ℝ^{n}. The lower and upper integral.

(4) Jordan measurable sets, the criterion of integrability of functions.

(5) Definition of a manifold in ℝ^{n}. Charts and parametrizations.

(6) Differentiable functions on a manifold.

(7) Tangent vectors (3 approaches). Coordinates of a vector in a chart.

(8) Tangent space. The basis \frac{∂}{∂x_{i}} and dx_{i} in a chart. The coordinate change formulas.

(9) Differential of a mapping f: M → N between differentiable manifolds. Pull back of covectors.

(10) Vector fields and 1-forms in ℝ^{n} and on a manifold. Evaluation of 1-forms on vector fields and derivation of functions along vector fields.

(11) Definition and properties of tensor products of vector spaces. Polylinear functions as tensors; (p, q)-tensors.

(12) Coordinates of a tensor. Symmetric and alternating tensors. Spaces S^{k}(V) and Λ^{k}(V).

(13) Definition and properties of the operators Sym and Alt. Wedge product of tensors, its properties.

(14) k-forms in ℝ^{n} and on a manifold, the space Ω^{k}(M). Local presentation of k-forms with respect to a chart, Wedge product and the differential of k-forms.

(15) Pull back of k-forms and its properties related to the wedge product and derivation of forms.

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

(17) Orientation of a vector space and of a manifold. Induced orientation of the boundary.

(18) Integration along a manifold. General Stokes’ theorem and its relation to the special cases studied in the advanced calculus.