1617-257/Homework Assignment 18

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Contents

Reading

Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read, reread and rereread sections 33-38 (skip 36) of Munkres' book to the same standard of understanding. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread section 39, just to get a feel for the future.

Doing

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

Problem A. Consider S^{n-1} at the boundary of D^n\subset{\mathbb R}^n, taken with its standard orientation, and let \iota\colon S^{n-1}\to{\mathbb R}^n be the inclusion map. Let \omega=\iota^\ast\left(\sum_i(-1)^{i-1}x_idx_1\wedge\dots\wedge\widehat{dx_i}\wedge\dots\wedge dx_n\right)\in\Omega^{\text{top}}(S^{n-1}). Prove that if (v_1,\ldots,v_{n-1}) is a positively oriented basis of T_xS^{n-1} for some x\in S^{n-1}, then \omega(v_1,\ldots,v_{n-1}) is the volume of the (n-1)-dimensional parallelepiped spanned by v_1,\ldots,v_{n-1}, and hence for any smooth function f on S^{n-1}, \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 \omega=\iota^\ast\left(\sum_ix_idx_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 V to be a function \nu\colon\{\text{ordered bases of }V\}\to\{\pm 1\} which satisfies \nu(v)=\operatorname{sign}(\det(C^u_v))\nu(u), whenever u and v are ordered bases of V and C^u_v is the change-of-basis matrix between them.

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

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

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