1617-257/Homework Assignment 18 - Revision history

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2017-04-20T14:41:57Z

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	</ol>		</ol>

	<u>'''Problem C'''.</u> Let <math>\omega=ydx\in\Omega^1({\mathbb R}^2_{x,y})</math>.		<u>'''Problem C'''.</u> Let <math>\omega=-ydx\in\Omega^1({\mathbb R}^2_{x,y})</math>.
	<ol>		<ol>
	<li> Let <math>\Gamma</math> be the graph in <math>{\mathbb R}^2_{x,y}</math> of some smooth function <math>f\colon[a,b]\to{\mathbb R}</math>. Using the inclusion of <math>\Gamma</math> to <math>{\mathbb R}^2_{x,y}</math>, consider <math>\omega</math> also as a 1-form on <math>\Gamma</math>. What is <math>\int_\Gamma\omega</math>?		<li> Let <math>\Gamma</math> be the graph in <math>{\mathbb R}^2_{x,y}</math> of some smooth function <math>f\colon[a,b]\to{\mathbb R}</math>. Using the inclusion of <math>\Gamma</math> to <math>{\mathbb R}^2_{x,y}</math>, consider <math>\omega</math> also as a 1-form on <math>\Gamma</math>. What is <math>\int_\Gamma\omega</math>?

Drorbn: /* Doing */

2017-03-28T16:32:36Z

Doing

← Older revision		Revision as of 12:32, 28 March 2017
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	Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:		Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:

	<u>'''Problem A.'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\omega=\iota^\ast\left(\sum_i(-1)^~~ix_idx_1~~\wedge\dots\wedge\widehat{dx_i}\wedge\dots\wedge dx_n\right)\in\Omega^{\text{top}}(S^{n-1})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, where the latter integral is integration relative to the volume, as defined a long time ago.		<u>'''Problem A.'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\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})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, 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 <math>\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})</math>. I'd like to thank the students who emailed me the correction.		'''Note.''' Earlier I made a sign mistake in the definition of and wrote <math>\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})</math>. I'd like to thank the students who emailed me the correction.

Drorbn: /* Doing */

2017-03-28T13:12:26Z

Doing

← Older revision		Revision as of 09:12, 28 March 2017
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	Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:		Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:

	<u>'''Problem A.'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\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})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, where the latter integral is integration relative to the volume, as defined a long time ago.		<u>'''Problem A.'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\omega=\iota^\ast\left(\sum_i(-1)^ix_idx_1\wedge\dots\wedge\widehat{dx_i}\wedge\dots\wedge dx_n\right)\in\Omega^{\text{top}}(S^{n-1})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, 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 <math>\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})</math>. I'd like to thank the students who emailed me the correction.

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

Drorbn at 19:24, 22 March 2017

2017-03-22T19:24:25Z

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	{{1617-257/Navigation}}		{{1617-257/Navigation}}
	{{In Preparation}}
	==Reading==		==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.		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.

Drorbn: /* Doing */

2017-03-22T19:24:02Z

Doing

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	<u>'''Problem A.'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\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})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, where the latter integral is integration relative to the volume, as defined a long time ago.		<u>'''Problem A.'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\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})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, where the latter integral is integration relative to the volume, as defined a long time ago.

	<u>'''Problem B'''</u> (an alternative definition for "orientation"). Define a norientation ("new orientation") of a vector space <math>V</math> to be a function <math>\nu\colon\{\text{ordered bases of }V\}\to\{\pm 1\}</math> which satisfies <math>\nu(v)=\operatorname{sign}(\det(C^u_v))\nu(u)</math>, whenever <math>u</math> and <math>v</math> are ordered bases of <math>V</math> and <math>C^u_v</math> is the change-of-basis matrix between them.		<u>'''Problem B'''</u> (an alternative definition for "orientation"). Define a "norientation" ("new orientation") of a vector space <math>V</math> to be a function <math>\nu\colon\{\text{ordered bases of }V\}\to\{\pm 1\}</math> which satisfies <math>\nu(v)=\operatorname{sign}(\det(C^u_v))\nu(u)</math>, whenever <math>u</math> and <math>v</math> are ordered bases of <math>V</math> and <math>C^u_v</math> is the change-of-basis matrix between them.
	<ol>		<ol>
	<li> Explain how if <math>\dim(V)>1</math>, a norientation is equivalent to an orientation.		<li> Explain how if <math>\dim(V)>1</math>, a norientation is equivalent to an orientation.
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	<li> Explain how a norientation of <math>M</math> induces a norientation of <math>\partial M</math>.		<li> Explain how a norientation of <math>M</math> induces a norientation of <math>\partial M</math>.
	<li> What is a <math>0</math>-dimensional manifold? What is a norientation of a <math>0</math>-dimensional manifold?		<li> What is a <math>0</math>-dimensional manifold? What is a norientation of a <math>0</math>-dimensional manifold?
			<li> What is the integral of a <math>0</math>-form on a <math>0</math>-dimensional noriented manifold?
	<li> What is <math>\partial[0,1]</math> as a noriented <math>0</math>-manifold? (Assume that <math>[0,1]</math> is endowed with its "positive" or "standard" orientation/norientation).		<li> What is <math>\partial[0,1]</math> as a noriented <math>0</math>-manifold? (Assume that <math>[0,1]</math> is endowed with its "positive" or "standard" orientation/norientation).
	</ol>		</ol>

Drorbn: /* Doing */

2017-03-20T20:38:27Z

Doing

← Older revision		Revision as of 16:38, 20 March 2017
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	Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:		Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:

	<u>'''Problem 1.'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\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})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, where the latter integral is integration relative to the volume, as defined a long time ago.		<u>'''Problem A.'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\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})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, where the latter integral is integration relative to the volume, as defined a long time ago.

	<u>'''Problem 2'''</u> (an alternative definition for "orientation"). Define a norientation ("new orientation") of a vector space <math>V</math> to be a function <math>\nu\colon\{\text{ordered bases of }V\}\to\{\pm 1\}</math> which satisfies <math>\nu(v)=\operatorname{sign}(\det(C^u_v))\nu(u)</math>, whenever <math>u</math> and <math>v</math> are ordered bases of <math>V</math> and <math>C^u_v</math> is the change-of-basis matrix between them.		<u>'''Problem B'''</u> (an alternative definition for "orientation"). Define a norientation ("new orientation") of a vector space <math>V</math> to be a function <math>\nu\colon\{\text{ordered bases of }V\}\to\{\pm 1\}</math> which satisfies <math>\nu(v)=\operatorname{sign}(\det(C^u_v))\nu(u)</math>, whenever <math>u</math> and <math>v</math> are ordered bases of <math>V</math> and <math>C^u_v</math> is the change-of-basis matrix between them.
	<ol>		<ol>
	<li> Explain how if <math>\dim(V)>1</math>, a norientation is equivalent to an orientation.		<li> Explain how if <math>\dim(V)>1</math>, a norientation is equivalent to an orientation.
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	<li> What is a <math>0</math>-dimensional manifold? What is a norientation of a <math>0</math>-dimensional manifold?		<li> What is a <math>0</math>-dimensional manifold? What is a norientation of a <math>0</math>-dimensional manifold?
	<li> What is <math>\partial[0,1]</math> as a noriented <math>0</math>-manifold? (Assume that <math>[0,1]</math> is endowed with its "positive" or "standard" orientation/norientation).		<li> What is <math>\partial[0,1]</math> as a noriented <math>0</math>-manifold? (Assume that <math>[0,1]</math> is endowed with its "positive" or "standard" orientation/norientation).
			</ol>

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

Drorbn: /* Doing */

2017-03-20T20:29:28Z

Doing

← Older revision		Revision as of 16:29, 20 March 2017
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	Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:		Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:

	<u>'''Problem 1'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\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})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, where the latter integral is integration relative to the volume, as defined a long time ago.		<u>'''Problem 1.'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\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})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, where the latter integral is integration relative to the volume, as defined a long time ago.

			<u>'''Problem 2'''</u> (an alternative definition for "orientation"). Define a norientation ("new orientation") of a vector space <math>V</math> to be a function <math>\nu\colon\{\text{ordered bases of }V\}\to\{\pm 1\}</math> which satisfies <math>\nu(v)=\operatorname{sign}(\det(C^u_v))\nu(u)</math>, whenever <math>u</math> and <math>v</math> are ordered bases of <math>V</math> and <math>C^u_v</math> is the change-of-basis matrix between them.
			<ol>
			<li> Explain how if <math>\dim(V)>1</math>, a norientation is equivalent to an orientation.
			<li> Come up with a reasonable definition of a norientation of a <math>k</math>-dimensional manifold.
			<li> Explain how a norientation of <math>M</math> induces a norientation of <math>\partial M</math>.
			<li> What is a <math>0</math>-dimensional manifold? What is a norientation of a <math>0</math>-dimensional manifold?
			<li> What is <math>\partial[0,1]</math> as a noriented <math>0</math>-manifold? (Assume that <math>[0,1]</math> is endowed with its "positive" or "standard" orientation/norientation).
			</ol>

	==Submission==		==Submission==

Drorbn at 20:15, 20 March 2017

2017-03-20T20:15:33Z

← Older revision		Revision as of 16:15, 20 March 2017
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	==Doing==		==Doing==
			Ponder the questions in sections 34 and 35, yet solve and submit only the following problems:
	'''Solve''' ''all'' the problems in sections 32 and 33, but submit only your solutions of problem <u>3</u> and <u>5</u> in section 32 and of problems <u>2</u> and <u>3</u> in section 33. In addition, ponder the following

			<u>'''Problem 1'''</u> Consider <math>S^{n-1}</math> at the boundary of <math>D^n\subset{\mathbb R}^n</math>, taken with its standard orientation, and let <math>\iota\colon S^{n-1}\to{\mathbb R}^n</math> be the inclusion map. Let <math>\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})</math>. Prove that if <math>(v_1,\ldots,v_{n-1})</math> is a positively oriented basis of <math>T_xS^{n-1}</math> for some <math>x\in S^{n-1}</math>, then <math>\omega(v_1,\ldots,v_{n-1})</math> is the volume of the <math>(n-1)</math>-dimensional parallelepiped spanned by <math>v_1,\ldots,v_{n-1}</math>, and hence for any smooth function <math>f</math> on <math>S^{n-1}</math>, <math>\int_{S^{n-1}}f\omega = \int_{S^{n-1}}fdV</math>, where the latter integral is integration relative to the volume, as defined a long time ago.
	'''Challenge Question''' (do not submit). What was it that you computed, in problem 3 of section 33? Could you have done it without any actual computation?

	==Submission==		==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.		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 <span style="color: blue;">Wednesday March 22 by 2:10PM</span>.		This assignment is due in class on <span style="color: blue;">Wednesday March 29 by 2:10PM</span>.

	===<span style="color: red;">Important</span>===		===<span style="color: red;">Important</span>===

Drorbn: Created page with "{{1617-257/Navigation}} {{In Preparation}} ==Reading== Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really u..."

2017-03-20T16:17:23Z

Created page with "{{1617-257/Navigation}} {{In Preparation}} ==Reading== Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really u..."

New page

{{1617-257/Navigation}}
{{In Preparation}}
==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==
'''Solve''' ''all'' the problems in sections 32 and 33, but submit only your solutions of problem 3 and 5 in section 32 and of problems 2 and 3 in section 33. In addition, ponder the following

'''Challenge Question''' (do not submit). What was it that you computed, in problem 3 of section 33? Could you have done it without any actual computation?

==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 22 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).