Difference between revisions of "07081300/Homework Assignment 5"
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'''Problem 5.''' Recall that a form is called closed if it is in the kernel of <math>d</math> and exact if it is in the image of <math>d</math>. Show that every exact form is closed.  '''Problem 5.''' Recall that a form is called closed if it is in the kernel of <math>d</math> and exact if it is in the image of <math>d</math>. Show that every exact form is closed.  
+  
+  '''Problem 6.''' Let <math>f:{\mathbb R}_t\to S^1\subset{\mathbb C}</math> be given by <math>f(t)=e^{it}</math>.  
+  # Show that there exists a unique <math>\omega\in\Omega^1(S^1)</math> such that <math>f^\star\omega=dt</math>.  
+  # Show that <math>\omega</math> is closed but not exact.  
+  
+  '''Problem 7.''' Show, directly from the definitions, that every closed 1form on <math>{\mathbb R}^2</math> is exact.  
+  
+  '''Problem 8.''' Compute the integral <math>\int_{S^2}zdx\wedge dy</math> twice:  
+  # Using Stokes' theorem.  
+  # Directly from the definition, by using a one or twochart atlas for <math>S^2</math>.  
+  (Repeat 1 and 2 until they stop giving different answers).  
+  
+  '''Problem 9.''' Show that the form <math>\omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in\Omega^2({\mathbb R}^3_{x,y,z})</math> is invariant under rigid orientationpreserving rotations of <math>{\mathbb R}^3</math>. That is, if <math>A</math> is such a rotation matrix (<math>AA^T=I</math> and <math>\det A=1</math>) considered also as a linear transformation <math>A:{\mathbb R}^3\to{\mathbb R}^3</math>, then <math>A^\star\omega=\omega</math>.  
==Due Date==  ==Due Date== 
Revision as of 19:52, 21 November 2007

The information below is preliminary and cannot be trusted! (v)
Contents 
Reading
Read sections 13 of chapter V of Bredon's book three times:
 First time as if you were reading a novel  quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
 Second time like you were studying for an exam on the subject  slowly and not skipping anything, verifying every little detail.
 And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.
Also, do the same with your own class notes  much of what we do for this part of the class is not in the textbook!
Doing
Solve all of the following problems, but submit only your solutions of problems *,* and *:
Problem 1. Let be a manifold. Show that the following definitions for the orientability of are equivalent:
 There exists a nowhere vanishing form on .
 There exists an atlas for , so that wherever that makes sense.
Problem 2. Show that the tangent space of any manifold is orientable.
Problem 3.
 Show that if and are orientable then so is .
 Show that if and are orientable then so is .
Problem 4. Show that is always orientable.
Problem 5. Recall that a form is called closed if it is in the kernel of and exact if it is in the image of . Show that every exact form is closed.
Problem 6. Let be given by .
 Show that there exists a unique such that .
 Show that is closed but not exact.
Problem 7. Show, directly from the definitions, that every closed 1form on is exact.
Problem 8. Compute the integral twice:
 Using Stokes' theorem.
 Directly from the definition, by using a one or twochart atlas for .
(Repeat 1 and 2 until they stop giving different answers).
Problem 9. Show that the form is invariant under rigid orientationpreserving rotations of . That is, if is such a rotation matrix ( and ) considered also as a linear transformation , then .
Due Date
This assignment is due in class on Thursday December 6, 2007.