Math 457 Calculus On Manifolds
Schedule:
Tuesdays 13:40-15:30 M-102
Thursdays 12:40-13:30 M-102
Office Hours: Tuesdays 10:40-12:30
Course Content
Review of differentiation, inverse and implicit function theorems, integration on subsets of Euclidean
space, tensors, differential forms, integration on chains, integration on manifolds. Stokes` theorem.
Syllabus:
Week 1-2-3 Review of differentiability and derivatives of maps between Euclidean spaces. Review of Inverse and Implicit Function theorems. Review of integration.
Week 4-5-6 Tensors, tensor product, alternating tensors.Wedge
product, orientation on vector spaces, volume element Tangent space of
a Euclidean space at a point, vector fields and differential forms on
Euclidean spaces.
Week 7-8-9 Pull-back
and differential of differential forms, statement of Poincaré Lemma.
Brief review of integration on Euclidean spaces, singular cubes and
chains in Euclidean spaces.
Week 10-11-12 Integration
of forms on chains, Stokes' Theorem on chains. Manifolds and manifolds
with boundary.Tangent space of a manifold at a point, vector fields and
forms on manifolds Differential and pull-backs of forms on manifolds, orientation of manifolds.
Week 13-14 Integration on manifolds. Stokes' Theorem on manifolds.
Course Objective
By
the end of the course the student will learn vector field and
differential form concepts on Euclidean spaces, the meaning of
integration on chains in Euclidean spaces
the manifold concept, the
manifold with boundary concept, vector field and differential form
concepts on manifolds, the meaning of integration on manifolds, and
Stokes' Theorem on manifolds.
Prerequisite Course: Math 252, Math262
Textbook: "Calculus on Manifolds" by Michael Spivak, W.A. Benjamin Inc., 1965.
Supplementary Resources: "A Geometric Approach to Differential Forms" by David Bachman
"Differential Topology" by Victor Guillemin & Allan
Pollack (Chapters 1 and 4).
Exams: There will be two Midterm Exams and a Final Exam.
Midterm 1 October 31, 2019, Thursday, at 17:40, Exam Place is M-07
Midterm 2 December 05, 2019, Thursday, at 17:40, Exam Place is M-07
Final January 07, Tuesday, at 09:30, Exam Place is M-07
Suggested Exercises:
Chapter 1, Problems: 1,2,3,4,5,6,7,8,9,10,13,14,16,18,20,21,22,23,24,25,26,29
Chapter 2, Problems: 1,2,4,5,6,7, 10(a,e,i),11(a,c),12,16,28(a,c,d),29,30,31,32,34,35,36,37,38,41
Chapter 3, Problems: 1,2,5,6,7,9,10,21,22, 32,36,39,41
Chapter 4, Problems: 1,2,3,4,5,6,9,11,13,14,15,16,17,18,19,20,21,25,26,31
Chapter 5, Problems: 1,2,4,5,6,8,19,20,22,23,26,27,28,29