0708-1300/Homework Assignment 5 - Revision history

Drorbn: /* Doing */

2007-12-10T16:17:38Z

Doing

← Older revision		Revision as of 12:17, 10 December 2007
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	'''Problem 1.''' Let <math>M^n</math> be a manifold. Show that the following definitions for the orientability of <math>M</math> are equivalent:		'''Problem 1.''' Let <math>M^n</math> be a manifold. Show that the following definitions for the orientability of <math>M</math> are equivalent:
	# There exists a nowhere vanishing <math>n</math>-form on <math>M</math>.		# There exists a nowhere vanishing <math>n</math>-form on <math>M</math>.
	# There exists an atlas <math>\{(U_\alpha,\phi_\alpha:U_\alpha\to{\mathbb R}^n)\}</math> for <math>M</math>, so that <math>\det(\phi_\alpha\~~phi~~^{-1}~~_\beta~~)>0</math> wherever that makes sense.		# There exists an atlas <math>\{(U_\alpha,\phi_\alpha:U_\alpha\to{\mathbb R}^n)\}</math> for <math>M</math>, so that <math>\det((d\phi_\alpha)(d\phi_\beta)^{-1})>0</math> wherever that makes sense.

	'''Problem 2.''' Show that the tangent space <math>TM</math> of any manifold <math>M</math> is orientable.		'''Problem 2.''' Show that the tangent space <math>TM</math> of any manifold <math>M</math> is orientable.

Drorbn: Reverted edit of Franklin, changed back to last version by Drorbn

2007-11-23T22:05:06Z

Reverted edit of Franklin, changed back to last version by Drorbn

← Older revision		Revision as of 18:05, 23 November 2007
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	'''Problem 8.''' Compute the integral <math>\int_{S^2}zdx\wedge dy</math> twice:		'''Problem 8.''' Compute the integral <math>\int_{S^2}zdx\wedge dy</math> twice:
	# Using Stokes' theorem.		# Using Stokes' theorem.
	# Directly from the definition, by using a ~~two~~- or ~~three~~-chart atlas for <math>S^2</math> (or for <math>S^2</math> minus a single point).		# Directly from the definition, by using a one- or two-chart atlas for <math>S^2</math> (or for <math>S^2</math> minus a single point).
	(Repeat 1 and 2 until they stop giving different answers).		(Repeat 1 and 2 until they stop giving different answers).

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	==Due Date==		==Due Date==
	This assignment is due in class on Thursday December 6, 2007.		This assignment is due in class on Thursday December 6, 2007.


	[http://www.brainyquote.com/quotes/authors/d/douglas_adams.html Douglas Adams] - "I love deadlines. I like the whooshing sound they make as they fly by."

	==Just for Fun==		==Just for Fun==

Franklin: /* Doing */ Correcting a minor typo.

2007-11-23T16:29:40Z

Doing: Correcting a minor typo.

← Older revision		Revision as of 12:29, 23 November 2007
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	'''Problem 8.''' Compute the integral <math>\int_{S^2}zdx\wedge dy</math> twice:		'''Problem 8.''' Compute the integral <math>\int_{S^2}zdx\wedge dy</math> twice:
	# Using Stokes' theorem.		# Using Stokes' theorem.
	# Directly from the definition, by using a ~~one~~- or ~~two~~-chart atlas for <math>S^2</math> (or for <math>S^2</math> minus a single point).		# Directly from the definition, by using a two- or three-chart atlas for <math>S^2</math> (or for <math>S^2</math> minus a single point).
	(Repeat 1 and 2 until they stop giving different answers).		(Repeat 1 and 2 until they stop giving different answers).

Franklin: /* Due Date */

2007-11-22T21:33:25Z

Due Date

← Older revision		Revision as of 17:33, 22 November 2007
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	==Due Date==		==Due Date==
	This assignment is due in class on Thursday December 6, 2007.		This assignment is due in class on Thursday December 6, 2007.


			[http://www.brainyquote.com/quotes/authors/d/douglas_adams.html Douglas Adams] - "I love deadlines. I like the whooshing sound they make as they fly by."

	==Just for Fun==		==Just for Fun==

Drorbn at 14:18, 22 November 2007

2007-11-22T14:18:40Z

← Older revision		Revision as of 10:18, 22 November 2007
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	{{0708-1300/Navigation}}		{{0708-1300/Navigation}}
	{{In Preparation}}

	==Reading==		==Reading==
	'''Read''' sections 1-3 of chapter V of Bredon's book three times:		'''Read''' sections 1-3 of chapter V of Bredon's book three times:
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	==Just for Fun==		==Just for Fun==
	A planimeter is a (mechanical!) measuring instrument used to measure the area of an arbitrary two-dimensional shape by tracing along its boundary; if you have a map of Ontario, for example, you can roll the measuring end of a planimeter counterclockwise around the borders of the province and the gadget will tell you its area. Figure out how a planimeter works, both on the mechanical level and on the mathematical level. Wikipedia has an [http://en.wikipedia.org/wiki/Planimeter explanation] which is is right in the spirit but wrong on the details. Fix it, if you feel inspired.		A planimeter is a (mechanical!) measuring instrument used to measure the area of an arbitrary two-dimensional shape by tracing along its boundary; if you have a map of Ontario, for example, you can roll the measuring end of a planimeter counterclockwise around the borders of the province and the gadget will tell you its area. Figure out how a planimeter works, both on the mechanical level and on the mathematical level. Wikipedia has an [http://en.wikipedia.org/wiki/Planimeter explanation] which is is right in the spirit but wrong on the details. Fix it, if you feel inspired (and if you do right and let me know about it, it'll count as a good deed).

Drorbn: /* Just for Fun */

2007-11-22T01:31:50Z

Just for Fun

← Older revision		Revision as of 21:31, 21 November 2007
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	==Just for Fun==		==Just for Fun==
	A planimeter is a (mechanical!) measuring instrument used to measure the area of an arbitrary two-dimensional shape by tracing its boundary; if you have a map of Ontario, for example, you can roll the measuring end of a planimeter counterclockwise around the borders of the province and the gadget will tell you its area. Figure out how a planimeter works, both on the mechanical level and on the mathematical level. Wikipedia has an [http://en.wikipedia.org/wiki/Planimeter explanation] which is is right in the spirit but wrong on the details. Fix it, if you feel inspired.		A planimeter is a (mechanical!) measuring instrument used to measure the area of an arbitrary two-dimensional shape by tracing along its boundary; if you have a map of Ontario, for example, you can roll the measuring end of a planimeter counterclockwise around the borders of the province and the gadget will tell you its area. Figure out how a planimeter works, both on the mechanical level and on the mathematical level. Wikipedia has an [http://en.wikipedia.org/wiki/Planimeter explanation] which is is right in the spirit but wrong on the details. Fix it, if you feel inspired.

Drorbn at 01:30, 22 November 2007

2007-11-22T01:30:12Z

← Older revision		Revision as of 21:30, 21 November 2007
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	'''Problem 8.''' Compute the integral <math>\int_{S^2}zdx\wedge dy</math> twice:		'''Problem 8.''' Compute the integral <math>\int_{S^2}zdx\wedge dy</math> twice:
	# Using Stokes' theorem.		# Using Stokes' theorem.
	# Directly from the definition, by using a one- or two-chart atlas for <math>S^2</math>.		# Directly from the definition, by using a one- or two-chart atlas for <math>S^2</math> (or for <math>S^2</math> minus a single point).
	(Repeat 1 and 2 until they stop giving different answers).		(Repeat 1 and 2 until they stop giving different answers).

Drorbn at 01:26, 22 November 2007

2007-11-22T01:26:43Z

← Older revision		Revision as of 21:26, 21 November 2007
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	==Doing==		==Doing==
	Solve all of the following problems, but submit only your solutions of problems , and *:		Solve all of the following problems, but submit only your solutions of problems 1, 3, 8, 9 and 10:

	'''Problem 1.''' Let <math>M^n</math> be a manifold. Show that the following definitions for the orientability of <math>M</math> are equivalent:		'''Problem 1.''' Let <math>M^n</math> be a manifold. Show that the following definitions for the orientability of <math>M</math> are equivalent:
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	(Repeat 1 and 2 until they stop giving different answers).		(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 orientation-preserving 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>.		'''Problem 9.''' Show that the form <math>\omega_0=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in\Omega^2({\mathbb R}^3_{x,y,z})</math> is invariant under rigid orientation-preserving 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_0=\omega_0</math>.

			'''Problem 10.''' Let <math>\omega_\alpha</math> be the form <math>(x^2+y^2+z^2)^{\alpha/2}\omega_0</math> defined on <math>{\mathbb R}^3\backslash\{0\}</math>, where <math>\alpha</math> is a real number and <math>\omega_0</math> is the form from the previous question.
			# Find a number <math>\alpha</math> so that <math>\omega_\alpha</math> would be closed.
			# For that value of <math>\alpha</math>, compute <math>\int_S\omega_\alpha</math>, where <math>S</math> is the sphere of radius <math>r</math> around a point <math>p\in{\mathbb R}^3</math>. (If you end up writing complicated formulas, you missed the point).
			# For the same <math>\alpha</math>, is <math>\omega_\alpha</math> exact?

	==Due Date==		==Due Date==
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	==Just for Fun==		==Just for Fun==
			A planimeter is a (mechanical!) measuring instrument used to measure the area of an arbitrary two-dimensional shape by tracing its boundary; if you have a map of Ontario, for example, you can roll the measuring end of a planimeter counterclockwise around the borders of the province and the gadget will tell you its area. Figure out how a planimeter works, both on the mechanical level and on the mathematical level. Wikipedia has an [http://en.wikipedia.org/wiki/Planimeter explanation] which is is right in the spirit but wrong on the details. Fix it, if you feel inspired.

Drorbn at 00:52, 22 November 2007

2007-11-22T00:52:32Z

← Older revision		Revision as of 20:52, 21 November 2007
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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 1-form 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 two-chart 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 orientation-preserving 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==

Drorbn at 00:34, 22 November 2007

2007-11-22T00:34:32Z

@@ Line 7: / Line 7: @@
 * 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.
 ==Doing==
-Solve the following problems from Bredon's book, but submit only the solutions of the problems marked with an "S":
+Solve all of the following problems, but submit only your solutions of problems *,* and *:
 ==Due Date==