Calculus on Manifolds (FA23)

Course Info:

  • Lectures: Tuesday, Thursday, block 10A (10:10 am - 12:00 pm ET)
  • x-period: Friday, block 10AX (3:30 - 4:20 pm ET); generally used for extra office hours
  • Room: Kemeny 120
  • Instructor: Alena Erchenko
  • Instructor office: Kemeny 331
  • E-mail: alena.erchenko@dartmouth.edu
  • Office hours: Friday (x-period) 3:30-4:20pm ET, Thursday 12:10-1:10pm, Mondays 11:30am-12:30pm ET, or by appt in my office Kemeny 331
  • Prerequisites: Math 11 or 13 and Math 22 or 24.  If you are unsure about your preparation, please talk to the instructor!
  • Recommended Texts:
  • James R. Munkres, Analysis on Manifolds. References in the schedule below are to this book.
  • Michael Spivak, Calculus on Manifolds: A modern approach to classical theorems of advanced calculus.
  • Grading: Grades will be based on weekly homework (70%) and a final project (30%). Your lowest homework score will be dropped.

Homework:

Homework is due on Gradescope on the day indicated below before 11:59 pm ET. Please upload your solutions by following the instructions in this video.  Feel free to submit handwritten or work typeset in LaTeX (Overleaf is a great resource and tool for the latter). If you are skipping a problem, it is easier for the grader if you write "Skip" and upload that as your solution to the problem. You are more than welcome to work with others on the homework problems. However, the solutions should be written up on your own and reflect your understanding.

Due Date Homework Solutions
9/15 Homework 0 N/A
9/22 Homework 1 Solutions
9/29 Homework 2 Solutions
10/6 Homework 3 Solutions
10/13 Homework 4 Solutions
10/20 Homework 5 Solutions
11/10 Homework 6 Solutions

Schedule:

The specific topics covered in each class are tentative and will be updated as the semester progresses. Moreover, the notes posted for each lecture may contain slightly more content than the lecture and may bleed a bit into the next lecture. Let me know if you find any errors.

Date Topic References
9/12 Introduction. Review of linear algebra. Metric spaces. Lecture1.pdf 
Munkres: Chapter 1 Sections 1-3
 9/14 Continuous functions. Topology. Compact sets. Lecture 2-3
Munkres: Chapter 1 Section 4
9/19 Continue about compact sets. Connectedness. Definitions of differentiable function and derivative. Lecture 2-3
Lecture 3-4.pdf
Munkres: Chapter 1 Section 4, Chapter 2 Section 5
9/21 Properties of the derivative. Chain Rule. Jacobian Matrix. Recognizing differentiability.  Lecture 3-4
Lecture 4-5
Munkres: Chapter 2, Sections 6,7
9/26 Operator norm. Mixed derivatives. Differentiation with respect to a parameter an integral. C^k-diffeomorphism.  Lecture 5
Munkres: Chapter 2 Sections 6,7,8
9/28 Banach fixed point theorem. The inverse function theorem. Lecture 6
Continuation of Lecture 6-7
Munkres, Chapter 2 Section 8
10/3 Continue on the inverse function theorem. The implicit function theorem. Lecture 7
Munkres, Chapter 2 Sections 8 and 9
10/5 More on the implicit function theorem. Integral over a rectangle.  Lecture 8
Munkres: Chapter 2 Section 9 and Chapter 3 Section 10
10/10 Integrable function. Measure zero sets. Lecture 9-10
Munkres: Chapter 3 Section 11,12
10/12 Integrability and measure zero. Fubini's theorem. Integrals over bounded sets. Rectifiable sets. Properties of integrals. Lecture 9-10
Lecture 10
Section 11-14
10/17 Smooth transition functions. Partition of unity. Integrals of continuous functions over open sets.  Lecture 11
Munkres: Chapter 4 Sections 16
10/19 Change of variables in an integral. Ideas of the proof.

Lecture 12
Munkres: Chapter 4 Section 17-19
10/24 Continues the proof of the theorem on change of variables. Volume of parallelopiped. Lecture 13
Munkres: Chapter 5 Section 21
10/26 Parametrized manifolds. Manifolds. Lecture 14
Munkres: Chapter 5 Sections 22-23
10/31 Manifolds with boundary. Integration of scalar functions on manifolds. Forms on R^n. Lecture 15
H. Hubbard and Burke Hubbard "Vector Calculus, Linear Algebra, and Differential Forms": Chapter 6, Sections 6.0, 6.1
11/2 Elementary forms form a basis of the space of k-forms on R^n. Lecture 16
Hubbard-Hubbard: Chapter 6, Section 6.2
11/7 Wedge Product. Form fields and their integration over parametrized manifolds. Orientation of manifolds. Lecture 17
Hubbard-Hubbard: Chapter 6, Sections 6.2-6.3
11/9 Induced orientation on boundary. Exterior derivative. Generalized Stokes's Theorem. Lecture 18
Hubbard-Hubbard: Chapter 6, Sections 6.4, 6.6, 6.7, 6.9
11/14 Project Presentations

Course Catalogue Description:

Manifolds provide mathematicians and other scientists with a way of grappling with the concept of “space” (from a global viewpoint). The space occupied by an object. The space that we inhabit. The space of solutions to a system of equations. Or, perhaps, the space of configurations of a mechanical system. While manifolds are central to the study of geometry and topology, they also provide an appropriate framework in which to explore aspects of mathematical physics, dynamics, control theory, medical imaging, and robotics, to name just a few. This course will demonstrate how ideas from calculus can be generalized to manifolds, providing a new perspective and toolkit with which to explore problems where “space” plays a fundamental role.

Learning Outcomes:

By the end of this course, you should be able to:

  1. Understand the multivariable differentiable calculus and the concept of a manifold;
  2. Solve mathematical problems: utilize abstraction and think creatively; 
  3. Write clear mathematical proofs: recognize and construct mathematically rigorous arguments.

Additional Pages:

  1. Academic Honor Principle
  2. Expectations
  3. Student Accessibility Needs
  4. Mental Health
  5. Title IX
  6. Religious Observances
  7. Additional Support for your Learning

Additional Resources: 

  • It is highly recommended to solve as many exercises from the textbook or other resources as you have time for. This will be more useful than rereading your class notes or textbook as it will often force you to go back to look at the material anyway. The only way to truly learn mathematics is through practice. Feel free to ask about your solutions.

  • There are many other textbooks and online materials that you can consult. If you have questions on whether a particular one might be useful, do not hesitate to reach out.