0708-1300/Class notes for Tuesday, November 20 - Revision history

Trefor: /* Second Hour */

2008-02-09T20:11:37Z

Second Hour

← Older revision		Revision as of 16:11, 9 February 2008
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	<math>\phi^*\omega = (f\circ\phi) d(x^1\circ\phi)\wedge\ldots\wedge d(x^n\circ\phi) = (f\circ\phi) d\phi^1\wedge\ldots\wedge d\phi^n</math>		<math>\phi^*\omega = (f\circ\phi) d(x^1\circ\phi)\wedge\ldots\wedge d(x^n\circ\phi) = (f\circ\phi) d\phi^1\wedge\ldots\wedge d\phi^n</math>

	<math>= (f\circ\phi)\left(\sum_{i_1} \frac{\partial\phi^1}{\partial y^{i_1}}\right)\wedge\ldots\wedge\left(\sum_{i_n}\frac{\partial\phi^n}{\partial y^{i_n}}\right) = (f\circ\phi)\sum_{i_1,\ldots,i_n =1}^n \frac{\partial\phi^1}{\partial y^{i_1}}\ldots\frac{\partial\phi^n}{\partial y^{i_n}}dy^{i_1}\wedge\ldots\wedge dy^{i_n}</math>		<math>= (f\circ\phi)\left(\sum_{i_1} \frac{\partial\phi^1}{\partial y^{i_1}}dy^{i_1}\right)\wedge\ldots\wedge\left(\sum_{i_n}\frac{\partial\phi^n}{\partial y^{i_n}}dy^{i_n}\right) = (f\circ\phi)\sum_{i_1,\ldots,i_n =1}^n \frac{\partial\phi^1}{\partial y^{i_1}}\ldots\frac{\partial\phi^n}{\partial y^{i_n}}dy^{i_1}\wedge\ldots\wedge dy^{i_n}</math>

	but the wedge product is zero unless <math>(i_1,\ldots,i_n)=\sigma\in S^n</math> and in that case yields <math>(-1)^{\sigma} dy^1\wedge\ldots\wedge dy^n</math>		but the wedge product is zero unless <math>(i_1,\ldots,i_n)=\sigma\in S^n</math> and in that case yields <math>(-1)^{\sigma} dy^1\wedge\ldots\wedge dy^n</math>

Trefor: /* First Hour */

2008-02-09T19:47:30Z

First Hour

← Older revision		Revision as of 15:47, 9 February 2008
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	<math>= (v_1 w)(v_2,v_3) - (v_2 w)(v_1,v_3) + (v_3 w)(v_1,v_2)</math>		<math>= (v_1 w)(v_2,v_3) - (v_2 w)(v_1,v_3) + (v_3 w)(v_1,v_2)</math>

	Now, <math>v_1 f = lim\frac{(p+\epsilon v_1) - f(p)}{\epsilon}</math>, or, loosely <math>f(p+v_1) - f(p)</math>		Now, <math>v_1 f = lim\frac{f(p+\epsilon v_1) - f(p)}{\epsilon}</math>, or, loosely <math>f(p+v_1) - f(p)</math>

	So this corresponds to the difference between <math>\omega</math> calculated on each of the two faces parallel to the <math>v_3,v_2</math> plane.		So this corresponds to the difference between <math>\omega</math> calculated on each of the two faces parallel to the <math>v_3,v_2</math> plane.

Aroesler: /* Second Hour */

2008-02-08T17:34:18Z

Second Hour

← Older revision		Revision as of 13:34, 8 February 2008
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	'''Definition'''		'''Definition'''

	let <math>M^n</math> be an oriented manifold and let <math>\omega\in\Omega^n_c(M)</math>. Let <math>\phi_{\alpha}:U_{\alpha}\mapsto\mathbb{R}^n</math> be a collection of positive charts that cover M. Let <math>\~~lambda^~~{\alpha}</math> be a partition of unity subordinate to this cover then		let <math>M^n</math> be an oriented manifold and let <math>\omega\in\Omega^n_c(M)</math>. Let <math>\phi_{\alpha}:U_{\alpha}\mapsto\mathbb{R}^n</math> be a collection of positive charts that cover M. Let <math>\lambda_{\alpha}</math> be a partition of unity subordinate to this cover then

	<math>\int_M\omega = \int_M 1\omega = \int_M \sum_{\alpha}\lambda_{\alpha}\omega = \sum_{\alpha}\int_{U_{\alpha}}\lambda_{\alpha}\omega = \sum_{\alpha}\int_{\mathbb{R}^n} (\phi^{-1}_{\alpha})^*(\lambda_{\alpha}\omega)</math>		<math>\int_M\omega = \int_M 1\omega = \int_M \sum_{\alpha}\lambda_{\alpha}\omega = \sum_{\alpha}\int_{U_{\alpha}}\lambda_{\alpha}\omega = \sum_{\alpha}\int_{\mathbb{R}^n} (\phi^{-1}_{\alpha})^*(\lambda_{\alpha}\omega)</math>

Aroesler: /* Second Hour */

2008-02-08T17:23:36Z

Second Hour

← Older revision		Revision as of 13:23, 8 February 2008
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	'''Definition'''		'''Definition'''

	An ''orientation of a vector space V'' is an equivalence class of ordered bases <math>v_{\alpha}</math> of V ~~under~~ <math>v_{\alpha}\sim w_{\beta}</math> if the determinant of the transition matrix between these two bases is positive.		An ''orientation of a vector space V'' is an equivalence class of ordered bases <math>v_{\alpha}</math> of V, where <math>v_{\alpha}\sim w_{\beta}</math> if the determinant of the transition matrix between these two bases is positive.

Aroesler: /* Second Hour */

2008-02-08T17:22:34Z

Second Hour

← Older revision		Revision as of 13:22, 8 February 2008
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	'''Definition'''		'''Definition'''

	An ''orientation of a vector space V'' is an equivalence class of ~~order~~ bases <math>v_{\alpha}</math> of V under <math>v_{\alpha}\sim w_{\beta}</math> if the determinant of the transition matrix between these two bases is positive.		An ''orientation of a vector space V'' is an equivalence class of ordered bases <math>v_{\alpha}</math> of V under <math>v_{\alpha}\sim w_{\beta}</math> if the determinant of the transition matrix between these two bases is positive.

Aroesler: /* Second Hour */

2008-02-08T17:05:35Z

Second Hour

← Older revision		Revision as of 13:05, 8 February 2008
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	<math>\phi^*\omega = (f\circ\phi) d(x^1\circ\phi)\wedge\ldots\wedge d(x^n\circ\phi) = (f\circ\phi) d\phi^1\wedge\ldots\wedge d\phi^n</math>		<math>\phi^*\omega = (f\circ\phi) d(x^1\circ\phi)\wedge\ldots\wedge d(x^n\circ\phi) = (f\circ\phi) d\phi^1\wedge\ldots\wedge d\phi^n</math>

	<math>= (f\circ\phi)\left(\frac{\partial\phi^1}{\partial y^{i_1}}\right)\wedge\ldots\wedge\left(\frac{\partial\phi^n}{\partial y^{i_n}}\right) = (f\circ\phi)\sum_{i_1,\ldots,i_n =1}^n \frac{\partial\phi^1}{\partial y^{i_1}}\ldots\frac{\partial\phi^n}{\partial y^{i_n}}dy^{~~1_1~~}\wedge\ldots\wedge dy^{i_n}</math>		<math>= (f\circ\phi)\left(\sum_{i_1} \frac{\partial\phi^1}{\partial y^{i_1}}\right)\wedge\ldots\wedge\left(\sum_{i_n}\frac{\partial\phi^n}{\partial y^{i_n}}\right) = (f\circ\phi)\sum_{i_1,\ldots,i_n =1}^n \frac{\partial\phi^1}{\partial y^{i_1}}\ldots\frac{\partial\phi^n}{\partial y^{i_n}}dy^{i_1}\wedge\ldots\wedge dy^{i_n}</math>

	but the wedge product is zero unless <math>(i_1,\ldots,i_n)=\sigma\in S^n</math> and in that case yields <math>(-1)^{\sigma} dy^1\wedge\ldots\wedge dy^n</math>		but the wedge product is zero unless <math>(i_1,\ldots,i_n)=\sigma\in S^n</math> and in that case yields <math>(-1)^{\sigma} dy^1\wedge\ldots\wedge dy^n</math>

Aroesler: /* First Hour */

2008-02-08T16:02:07Z

First Hour

← Older revision		Revision as of 12:02, 8 February 2008
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	'''Definition'''		'''Definition'''

	~~We define <math>\int_{\mathbb{R}^n}: \Omega^n_c(\mathbb{R}^n)\rightarrow\mathbb{R}</math> by~~ <math>\omega\in\Omega^n_c(\mathbb{R}^n)</math> ~~and thus~~ <math>\omega = fdx^1\wedge\cdots\wedge dx^n</math> ~~then~~		For <math>\omega\in\Omega^n_c(\mathbb{R}^n)</math>, <math>\omega = fdx^1\wedge\cdots\wedge dx^n</math>
			we define <math>\int_{\mathbb{R}^n}: \Omega^n_c(\mathbb{R}^n)\rightarrow\mathbb{R}</math> by

	<math>\int_{\mathbb{R}^n}\omega := \int_{\mathbb{R}^n}f</math>		<math>\int_{\mathbb{R}^n}\omega := \int_{\mathbb{R}^n}f</math>

Aroesler: /* First Hour */

2008-02-08T15:52:40Z

First Hour

← Older revision		Revision as of 11:52, 8 February 2008
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	Then ''<math>\omega</math> has compact support'' if the supp <math>\omega</math> is compact.		Then ''<math>\omega</math> has compact support'' if the supp <math>\omega</math> is compact.

	Define <math>\Omega^_c(M)</math> := the compactly ~~support~~ <math>w\in\Omega^(M)</math>		Define <math>\Omega^_c(M)</math> := the compactly supported <math>w\in\Omega^(M)</math>

Aroesler: /* First Hour */

2008-02-08T15:39:47Z

First Hour

← Older revision		Revision as of 11:39, 8 February 2008
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	3) let <math>\omega=f_I dx^I</math> and <math>\lambda = g_J dx^J</math> then <math>\omega\wedge\lambda = f_I g_J dx^I\wedge dx^J</math>		3) let <math>\omega=f_I dx^I</math> and <math>\lambda = g_J dx^J</math> then <math>\omega\wedge\lambda = f_I g_J dx^I\wedge dx^J</math>

	so <math>d(f_I g_J dx^I\wedge dx^J) = \frac{\partial f_I g_I}{\partial x^j} dx^j\wedge dx^I\wedge dx^J = (\frac{\partial f_I}{\partial x^j}g_J + f_I\frac{\partial g_I}{\partial x^j})dx^j\wedge dx^I\wedge dx^J = d\omega\wedge\lambda \pm\omega\wedge d\lambda</math>		so <math>d(f_I g_J dx^I\wedge dx^J) =\sum_j \frac{\partial f_I g_I}{\partial x^j} dx^j\wedge dx^I\wedge dx^J = \sum_j (\frac{\partial f_I}{\partial x^j}g_J + f_I\frac{\partial g_I}{\partial x^j})dx^j\wedge dx^I\wedge dx^J = d\omega\wedge\lambda \pm\omega\wedge d\lambda</math>

Aroesler: /* First Hour */

2008-02-08T15:36:41Z

First Hour

← Older revision		Revision as of 11:36, 8 February 2008
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	''Proof''		''Proof''

	We need to check that this satisfies the properties:		We need to check that this satisfies the properties 1) - 3) from last lecture:

	1) <math>df(\partial_i) = \sum_j \frac{\partial f}{\partial x_j} dx_j(\partial_i) = \sum_j \frac{\partial f}{\partial x_j} \delta_{ji} = \partial_i f</math> and so satisfies		1) <math>df(\partial_i) = \sum_j \frac{\partial f}{\partial x_j} dx_j(\partial_i) = \sum_j \frac{\partial f}{\partial x_j} \delta_{ji} = \partial_i f</math> and so satisfies property 1)