Assignment 2 (Written): Investigating Curvature (Due 3/22)

The written portion of Assignment 2 can be found below. It takes a look at the curvature of smooth and discrete surfaces, which we have been talking about in lecture.

Warning: We renumbered the Exercises in the course notes to make more sense, so you make sure you refer to the updated notes when doing these exercises.

1 thought on “Assignment 2 (Written): Investigating Curvature (Due 3/22)”

  1. Concept question: on page 85 of the course notes, it is mentioned

    > Noting that $dx \wedge dy = d(x \wedge dy) = -d(y \wedge dx)$ …

    Throughout most of our discussions on exterior algebra and differential forms, we were told to regard $dx^i$ as an opaque “basis $k$-form”. So it is surprising to me that we are “allowed” to “separate” the $d$ from the $x$ (I understand that it uses the product rule in the above identity). I think this gets at a more fundamental question of, what exactly *is* the object $x^i$? What is its “type”? It seems like it should be a differential 0-form, because $dx$ for example is a differential 1-form (this is why we can write ex $\int_P dx \wedge dy$ where $P$ is a region in the plane, as $dx \wedge dy$ is a 2-form). But then, what exactly does it yield? What even is a coordinate (I think I’m going crazy)?

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