**Note:** This assignment is **NOT** required! However, you can complete it for up to 15% extra credit. Or, you can do this one instead of some of the other assignments. Overall, you just need to be sure you completed A0 and A1, as well as 3 of the assignments A2–A6 (your choice which ones). Also, you *cannot* use late days on this assignment since it’s the last one.

In this assignment, you will investigate tools for working with tangent vector fields on surfaces. Tangent vector fields are central to classical differential geometry, and have many interesting applications. For this homework, we’ll look at one algorithm for designing vector fields, and along the way we’ll cover a lot of deep facts about surfaces.

There’s no PDF this week since the exercises are all from the notes. *The notes will also give you all the background you’ll need to complete this assignment.* It builds on a lot of the stuff we’ve already done in the class, especially discrete exterior calculus and the Laplacian.

Do any **12** of Exercise 8.1 – Exercise 8.21 in the notes, except for Exercise 8.13.

**Submission Instructions.** Please submit your solutions to the exercises (whether handwritten, LaTeX, etc.) as a single PDF file by email to Geometry.Collective@gmail.com. *This email must also contain the .zip file for your coding solution*. Scanned images/photographs can be converted to a PDF using applications like Preview (on Mac) or a variety of free websites (e.g., http://imagetopdf.com). Your submission email must include the string **DDG20A6** in the subject line.

**Warning:** You *cannot* use late days on this assignment since it’s the last one.

In Exercise 8.3, does it mean to say that the vector space “V” can be decomposed into three orthogonal subspaces? It is written U, I think it is a typo. Because, V is the “middle one” in the chain complex.

Yes—that’s a typo. Good catch, thanks!

Does “can be decomposed into..” mean direct sum? If so, shouldn’ t Z be the orthogonal complement of the other two spaces?

This is what you are being asked to show.

Yeah… but I also have just noticed that the last year’s website said there is a typo in Z(same page, last comment).

For Exercise 8.3, is $A^*\circ B^*=0$?

This is true, although should not be needed.