For your next assignment, due on Friday, March 4th, you will simply complete the exercises you previously skipped over that required use of exterior calculus. These exercises can all be found in the class notes, and include:
- Chapter 5, Exercise 13 (Hint: read very carefully the end of Section 3.2)
- Chapter 5, Exercise 16
- Chapter 6, Exercise 20
- Chapter 6, Exercise 26
You should also carefully read Section 6.3 (which leads up to Exercise 26), which provides an alternative perspective on the Laplacian to the finite element approach you derived in your last homework, and will be very useful to understand for upcoming assignments. For the exercises in Chapter 5, it may also be helpful to review your reading of Chapter 2 (on the geometry of surfaces), re-thinking these concepts in terms of differential forms. (For example, the Weingarten map \(dN\) and differential \(df\) of the immersion can both be viewed as vector-valued 1-forms.) These exercises are a bit more challenging than the “warm up” exercises from your readings, so please post comments if you have any trouble.