APMAE4990 - Section 002
Calculus of Variations
& Applications

Spring 2022

Time: MW 1:10-2:25

Place: 327 Mudd Building

Zoom Room: https://columbiauniversity.zoom.us/j/98513604877

Canvas Site

Syllabus

Instructor: Dr. James Scott (he/him)

Affiliation: Applied Physics and Applied Mathematics, SEAS

Email: See "Home" on this website

Office Hours: T 2pm-3pm, W 11am-12pm, or by appointment

Office Location: 287D Engineering Terrace (accessible by going through the APAM offices)

Prerequisites: Multivariable Calculus (APMA E2001 or equivalent) is required. Introduction to Real Variables / Real Analysis (MATH UN2000 or equivalent) is highly recommended but not required. PDEs (APMA E4200 or equivalent) is also helpful but not required.

Course Content: A modern introduction to the Calculus of Variations, with both theory and applications. Topics included are existence of solutions, variational formulations, relaxation, and Gamma-convergence. Settings will range from one-dimensional convex problems to multi-dimensional nonconvex problems. Applications will include minimal surfaces, the isoperimetric inequality, solid mechanics and elasticity, composite materials, vibrations and wave propagation.

Textbooks: No book is required for the course, but the course content will follow the presentation from the following books:

Introduction to the Calculus of Variations. Bernard Dacorogna. 2008 Imperial College Press.

Variational Methods with Applications in Science and Engineering. Kevin W. Cassell. 2013 Cambridge University Press, ISBN: 9781107022584

Direct Methods in the Calculus of Variations. Bernard Dacorogna. 2008 Springer-Verlag, 2nd edition.

An Introduction to the Calculus of Variations. Charles Fox. Dover Publications, ISBN: 9780486654997

Gamma-Convergence for Beginners. Andrea Braides. 2002 Oxford University Press. Available electronically.

Grades: Homework (40%), Exam 1 (20%), Exam 2 (20%), Final (20%).

Homework will be assigned every two weeks and submitted via Gradescope (on Canvas) by 11:59 pm on the due date. The lowest homework score will be dropped. If you have questions about the homework problems, please come to office hours or email me. You are welcome to work together on homework. However, each student must turn in their own assignment.

There will be two non-cumulative take-home exams, and one non-cumulative take-home final. The online link for each exam will be available only for the duration of the exam. You will have 48 hours for the two exams, and one week for the final. The exams will be submitted via Gradescope.

Uploading Files to Gradescope: See this PDF and this video.

Additional Resources

Paul's Online Math Notes for Multivariable Calculus

Paul's Online Math Notes for Differential Equations

Mathematical Foundations of Elasticity Theory - Lecture notes by John Ball

Homework