# 1617-257/Homework Assignment 18

## Doing

Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:

Problem A. Consider ${\displaystyle S^{n-1}}$ at the boundary of ${\displaystyle D^{n}\subset {\mathbb {R} }^{n}}$, taken with its standard orientation, and let ${\displaystyle \iota \colon S^{n-1}\to {\mathbb {R} }^{n}}$ be the inclusion map. Let ${\displaystyle \omega =\iota ^{\ast }\left(\sum _{i}(-1)^{i-1}x_{i}dx_{1}\wedge \dots \wedge {\widehat {dx_{i}}}\wedge \dots \wedge dx_{n}\right)\in \Omega ^{\text{top}}(S^{n-1})}$. Prove that if ${\displaystyle (v_{1},\ldots ,v_{n-1})}$ is a positively oriented basis of ${\displaystyle T_{x}S^{n-1}}$ for some ${\displaystyle x\in S^{n-1}}$, then ${\displaystyle \omega (v_{1},\ldots ,v_{n-1})}$ is the volume of the ${\displaystyle (n-1)}$-dimensional parallelepiped spanned by ${\displaystyle v_{1},\ldots ,v_{n-1}}$, and hence for any smooth function ${\displaystyle f}$ on ${\displaystyle S^{n-1}}$, ${\displaystyle \int _{S^{n-1}}f\omega =\int _{S^{n-1}}fdV}$, where the latter integral is integration relative to the volume, as defined a long time ago.

Note. Earlier I made a sign mistake in the definition of and wrote ${\displaystyle \omega =\iota ^{\ast }\left(\sum _{i}x_{i}dx_{1}\wedge \dots \wedge {\widehat {dx_{i}}}\wedge \dots \wedge dx_{n}\right)\in \Omega ^{\text{top}}(S^{n-1})}$. I'd like to thank the students who emailed me the correction.

Problem B (an alternative definition for "orientation"). Define a "norientation" ("new orientation") of a vector space ${\displaystyle V}$ to be a function ${\displaystyle \nu \colon \{{\text{ordered bases of }}V\}\to \{\pm 1\}}$ which satisfies ${\displaystyle \nu (v)=\operatorname {sign} (\det(C_{v}^{u}))\nu (u)}$, whenever ${\displaystyle u}$ and ${\displaystyle v}$ are ordered bases of ${\displaystyle V}$ and ${\displaystyle C_{v}^{u}}$ is the change-of-basis matrix between them.

1. Explain how if ${\displaystyle \dim(V)>1}$, a norientation is equivalent to an orientation.
2. Come up with a reasonable definition of a norientation of a ${\displaystyle k}$-dimensional manifold.
3. Explain how a norientation of ${\displaystyle M}$ induces a norientation of ${\displaystyle \partial M}$.
4. What is a ${\displaystyle 0}$-dimensional manifold? What is a norientation of a ${\displaystyle 0}$-dimensional manifold?
5. What is the integral of a ${\displaystyle 0}$-form on a ${\displaystyle 0}$-dimensional noriented manifold?
6. What is ${\displaystyle \partial [0,1]}$ as a noriented ${\displaystyle 0}$-manifold? (Assume that ${\displaystyle [0,1]}$ is endowed with its "positive" or "standard" orientation/norientation).

Problem C. Let ${\displaystyle \omega =-ydx\in \Omega ^{1}({\mathbb {R} }_{x,y}^{2})}$.

1. Let ${\displaystyle \Gamma }$ be the graph in ${\displaystyle {\mathbb {R} }_{x,y}^{2}}$ of some smooth function ${\displaystyle f\colon [a,b]\to {\mathbb {R} }}$. Using the inclusion of ${\displaystyle \Gamma }$ to ${\displaystyle {\mathbb {R} }_{x,y}^{2}}$, consider ${\displaystyle \omega }$ also as a 1-form on ${\displaystyle \Gamma }$. What is ${\displaystyle \int _{\Gamma }\omega }$?
2. Prove that if ${\displaystyle E}$ is an ellipse in ${\displaystyle {\mathbb {R} }_{x,y}^{2}}$ (of whatever major and minor axes, placed anywhere and tilted as you please), then ${\displaystyle \int _{\partial E}\omega }$ is the area of ${\displaystyle E}$.
3. Compute also ${\displaystyle d\omega }$ and ${\displaystyle \int _{E}d\omega }$.

## Submission

Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

This assignment is due in class on Wednesday March 29 by 2:10PM.

### Important

Please write on your assignment the day of the tutorial when you'd like to pick it up once it is marked (Wednesday or Thursday).