Math 73

Syllabus

Lectures	Sections in Text	Brief Description
Day 1:  1/6	Section 1.3	Metric spaces, subspaces, examples. Comparison of sup and norm topology on R^n. Open and closed sets. Continuous mappings.
Day 2:  1/8	Finish Section 1.3 start Section 1.4	Compositions and restrictions of continuous mappings. Limits. Compact sets in R^n
Day 3:  1/10	Section 1.4	Compact sets, product of compact sets, continuous images of compact sets. Uniform continuity. Epsilon-neighborhood of a compact set. Connected sets.
1/11		No class on 1/11 instead we have an x-hour on 1/14
Day 4:  1/13	Section 2.5	Derivatives and directional derivatives, examples of non-differentiable functions with all the directional derivatives being well-defined.
Day 5:  1/14 x-hour instead of the special class on 1/11	Sections 2.5 and 2.6	Differentiability of functions with continuous partial derivatives D_i F.
Day 6:  1/15	Sections 2.6 and 2.7	Iterated partial derivatives. D_i D_j F=D_j D_i F provided that they are continuous.
Day 7:  1/17	Section 2.7	Chain rule. Derivative of the inverse function.
1/20		Martin Luther King Jr. Day. No class. The class is shifted to the x-hour on 1/21
Day 8: 1/21 x-hour instead of the class on 1/20	Section 2.8	Mean value theorem for multivariable functions. Inverse function theorem.
Day 9:  1/22	Section 2.8	Inverse function theorem.
Day 10:  1/24	Section 2.9	Implicit function theorem.
Day 11:  1/27	Section 3.10, 3.11	Riemannian integral over a rectangle, measure zero sets.
Day 12:  1/29	Section 3.11	A bounded function on a rectangle is integrable if and only if the set where it is discontinuous has measure zero. An integral of a nonnegative function is zero if and only if it is zero almost everywhere.
Day 13:  1/31	Section 3.12	Fubini’s Theorem
Day 14:  2/3	Section 3.13,	Integral over a bounded set. Basic properties of integrals. Integral over a set and over its interior.
Day 15:  2/5	Section 3.14	Rectifiable sets. Volume of a rectifiable set. Fubini’s Theorem for simple regions.
2/7		Carnival Holiday. No class. The class is shifted to the x-hour on 2/11
Day 16:  2/10	Sections 3.15,	Extended and improper integrals.
Day 17:  2/11 x-hour instead of the class on 2/7	Section 4.16	Partitions of Unity.
Day 18:  2/12	No class
Day 19:  2/14	Sections 4.16 and 4.18	Integration and partition of unity. Images of the measure zero sets under smooth mappings. Decomposition of a diffeomorphism into a composition of primitive diffeomorphisms.
Day 20:  2/17	Sections 4.17, 4.19	Change of Variables Theorem in Integration
2/18	Homework problems.
Day 21:  2/19	Section 4.20, 5.21, 5.22	Applications of the change of variable. Volume of the parallelepiped. Parametrized manifolds. Integral over a parametrized manifold.
Day 22:  2/21	Sections 5.23, 5.24, 5.25	Manifolds in R^n, examples, boundary of a manifold, transition functions. Integral of a function over a manifold.
Day 23:  2/24	Finish section 5.23, Sections 6.26, 6.27	A ball as a manifold with boundary. Tensors and Alternating tensors. Start wedge product if we have time.
2/25 x-hour x-hour instead of the class on 2/12	Finish section 6.27 and cover section 6.28	Wedge product and its properties.
Day 24:  2/26	Sections 6.29 and 6.30	Tangent vectors and tangent spaces of manifolds. Differential operator on the space of forms on a manifold. Definition of cohomology groups.
Day 25:  2/28	Sections 6.32, 7.33	Pull-back of forms. Commutation between pull back and differential. Integrating forms over parametrized manifolds.
Day 26:  3/3	Sections 7.34, 7.35	Orientable manifolds. Orientation of a manifold. Integrals of forms over oriented manifolds.
Day 27:  3/5	Sections 7.37	Stokes Theorem.
Day 28:  3/7		Final day of classes