Warning: 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.
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 DDG19A6 in the subject line.
Warning: You cannot use late days on this assignment since it’s the last one.
For Exercise 8.3, shouldn’t the vector space to be divided into three parts be V instead of U?
Yes, that’s a typo in the exercise. Good catch!
Will the lecture slide from the last lecture be posted?
Thanks!
For exercise 56 in the notes, how is the inner product defined? If it’s $$\langle\langle d\alpha, \beta\rangle\rangle = \int \star d\alpha \wedge \beta$$ as in exercise 37, I can’t use the inner product to get to Stoke’s theorem.
Yes, that’s the correct definition of the inner product. This is the same sort of integration by parts computation that we’ve been doing in the past few assignments. You might find it easier to start with $\langle\langle \beta, d\alpha\rangle\rangle$ instead. Since the inner product is symmetric, you can start with the arguments in either order.
I thought this definition of the inner product was Hermitian rather than symmetric.
Good point! We wanted a Hermitian inner product for working with complex forms, which meant we needed to add a bar inside of the integral. Now that we’re back to working with real forms, conjugation doesn’t do anything so it becomes a symmetric inner product.
For exercise 58, I end up with $\delta \beta = \star d \star \star \phi = -\star d\phi = -\star (\nabla \phi)^\flat$, which corresponds to a 90 clockwise rotation of the gradient. How do we get a 90 counterclockwise rotation?
$\star\star = +1$ for 0-forms.
There’s another typo in Exercise 8.3. The subspace $Z$ should be defined as $Z:=\text{ker}(B) \cap \text{im}(A)^\perp$ (or you can interpret $A \setminus B$ as “all vectors in $A$ which are orthogonal to $B$).
Just curious, but in general, are you guaranteed to find the exact number of generator loops no matter what $T$ and $T^{*}$ are?