1617-257/Homework Assignment 14 - Revision history

Stefan.divic: /* Student Solutions */

2017-03-31T04:41:20Z

Student Solutions

← Older revision		Revision as of 00:41, 31 March 2017
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	==Student Solutions==		==Student Solutions==
	[[Media:1617-257-~~set14~~.pdf\|Student 1]]		[[Media:1617-257-pset14.pdf\|Student 1]]

Stefan.divic at 04:40, 31 March 2017

2017-03-31T04:40:52Z

← Older revision		Revision as of 00:40, 31 March 2017
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	===<span style="color: red;">Important</span>===		===<span style="color: red;">Important</span>===
	<span style="color: red;">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).</span>		<span style="color: red;">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).</span>

			==Student Solutions==
			[[Media:1617-257-set14.pdf\|Student 1]]

Drorbn at 17:06, 8 February 2017

2017-02-08T17:06:26Z

← Older revision		Revision as of 13:06, 8 February 2017
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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 27-29 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 sections 30-31, 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 27-29 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 sections 30-31, just to get a feel for the future.

Drorbn: /* Doing */

2017-02-06T17:52:06Z

Doing

← Older revision		Revision as of 13:52, 6 February 2017
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	==Doing==		==Doing==
	'''Solve''' ''all'' the problems in sections 27-28, but submit only your solutions of problem 1 in section 27 and problems 1 and 6 in section 28. In addition, solve the following problems three problems, though submit only your solutions of the first two. These problems require more creativity than the usual and the are less precisely specified. You may wish to listen carefully at the tutorials this week!		'''Solve''' ''all'' the problems in sections 27-28, but submit only your solutions of problem <u>1</u> in section 27 and problems <u>1</u> and <u>6</u> in section 28. In addition, solve the following problems three problems, though submit only your solutions of the first two. These problems require more creativity than the usual and the are less precisely specified. You may wish to listen carefully at the tutorials this week!

	'''<u>Problem A.</u>''' Along the lines of our development of a theory of "tensors" and a theory of "alternating tensors", develop a theory of "symmetric tensors" <math>S^k(V)</math> (a symmetric tensor is a tensor whose values are unchanged if its arguments are permuted). Your theory should include definitions for specific tensors <math>\sigma_I</math> for <math>I\in\underline{n}^k_s</math> (what should <math>\underline{n}^k_s</math> be)?), a proof that the <math>\sigma_I</math> exist and are unique and that they form a basis of <math>S^k(V)</math>, and a computation of the dimension of <math>S^k(V)</math>.		'''<u>Problem A.</u>''' Along the lines of our development of a theory of "tensors" and a theory of "alternating tensors", develop a theory of "symmetric tensors" <math>S^k(V)</math> (a symmetric tensor is a tensor whose values are unchanged if its arguments are permuted). Your theory should include definitions for specific tensors <math>\sigma_I</math> for <math>I\in\underline{n}^k_s</math> (what should <math>\underline{n}^k_s</math> be)?), a proof that the <math>\sigma_I</math> exist and are unique and that they form a basis of <math>S^k(V)</math>, and a computation of the dimension of <math>S^k(V)</math>.

Drorbn: /* Doing */

2017-02-06T17:50:27Z

Doing

← Older revision		Revision as of 13:50, 6 February 2017
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	==Doing==		==Doing==
	'''Solve''' ''all'' the problems in sections 27-28, but submit only your solutions of problem 1 in section 27 and problems 1 and 6 in section 28. In addition, solve ~~and~~ submit your solutions of the ~~following~~ ~~problems~~. These problems require more creativity than the usual and the are less precisely specified. You may wish to listen carefully at the tutorials this week!		'''Solve''' ''all'' the problems in sections 27-28, but submit only your solutions of problem 1 in section 27 and problems 1 and 6 in section 28. In addition, solve the following problems three problems, though submit only your solutions of the first two. These problems require more creativity than the usual and the are less precisely specified. You may wish to listen carefully at the tutorials this week!

	'''<u>Problem A.</u>''' Along the lines of our development of a theory of "tensors" and a theory of "alternating tensors", develop a theory of "symmetric tensors" <math>S^k(V)</math> (a symmetric tensor is a tensor whose values are unchanged if its arguments are permuted). Your theory should include definitions for specific tensors <math>\sigma_I</math> for <math>I\in\underline{n}^k_s</math> (what should <math>\underline{n}^k_s</math> be)?), a proof that the <math>\sigma_I</math> exist and are unique and that they form a basis of <math>S^k(V)</math>, and a computation of the dimension of <math>S^k(V)</math>.		'''<u>Problem A.</u>''' Along the lines of our development of a theory of "tensors" and a theory of "alternating tensors", develop a theory of "symmetric tensors" <math>S^k(V)</math> (a symmetric tensor is a tensor whose values are unchanged if its arguments are permuted). Your theory should include definitions for specific tensors <math>\sigma_I</math> for <math>I\in\underline{n}^k_s</math> (what should <math>\underline{n}^k_s</math> be)?), a proof that the <math>\sigma_I</math> exist and are unique and that they form a basis of <math>S^k(V)</math>, and a computation of the dimension of <math>S^k(V)</math>.

	'''<u>Problem B.</u>''' Find a good way of identifying <math>A^1({\mathbb R}^3)</math> and <math>A^2({\mathbb R}^3)</math> with <math>{\mathbb R}^3</math>. Under that identification, <math>\wedge\colon A^1({\mathbb R}^3)\times A^1({\mathbb R}^3)\to A^2({\mathbb R}^3)</math> becomes a map <math>P\colon{\mathbb R}^3\times{\mathbb R}^3\to{\mathbb R}^3</math>. If you chose your identifications right, <math>P</math> is the vector product of two vectors in <math>{\mathbb R}^3</math>. See to it that this is the indeed the case!		'''<u>Problem B.</u>''' Find a good way of identifying <math>A^1({\mathbb R}^3)</math> and <math>A^2({\mathbb R}^3)</math> with <math>{\mathbb R}^3</math>. Under that identification, <math>\wedge\colon A^1({\mathbb R}^3)\times A^1({\mathbb R}^3)\to A^2({\mathbb R}^3)</math> becomes a map <math>P\colon{\mathbb R}^3\times{\mathbb R}^3\to{\mathbb R}^3</math>. If you chose your identifications right, <math>P</math> is the vector product of two vectors in <math>{\mathbb R}^3</math>. See to it that this is the indeed the case!

			'''Problem C.''' The determinant, as a function of a list of column vectors, is alternating. Write it in terms of the elementary alternating functions <math>\psi_I</math>.

	==Submission==		==Submission==

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..."

2017-02-06T17:08:31Z

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 27-29 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 sections 30-31, just to get a feel for the future.

==Doing==
'''Solve''' ''all'' the problems in sections 27-28, but submit only your solutions of problem 1 in section 27 and problems 1 and 6 in section 28. In addition, solve and submit your solutions of the following problems. These problems require more creativity than the usual and the are less precisely specified. You may wish to listen carefully at the tutorials this week!

'''Problem A.''' Along the lines of our development of a theory of "tensors" and a theory of "alternating tensors", develop a theory of "symmetric tensors" <math>S^k(V)</math> (a symmetric tensor is a tensor whose values are unchanged if its arguments are permuted). Your theory should include definitions for specific tensors <math>\sigma_I</math> for <math>I\in\underline{n}^k_s</math> (what should <math>\underline{n}^k_s</math> be)?), a proof that the <math>\sigma_I</math> exist and are unique and that they form a basis of <math>S^k(V)</math>, and a computation of the dimension of <math>S^k(V)</math>.

'''Problem B.''' Find a good way of identifying <math>A^1({\mathbb R}^3)</math> and <math>A^2({\mathbb R}^3)</math> with <math>{\mathbb R}^3</math>. Under that identification, <math>\wedge\colon A^1({\mathbb R}^3)\times A^1({\mathbb R}^3)\to A^2({\mathbb R}^3)</math> becomes a map <math>P\colon{\mathbb R}^3\times{\mathbb R}^3\to{\mathbb R}^3</math>. If you chose your identifications right, <math>P</math> is the vector product of two vectors in <math>{\mathbb R}^3</math>. See to it that this is the indeed the case!

==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 February 15 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).