0708-1300/Class notes for Tuesday, October 9 - Revision history

Trefor: /* Typed Notes - Second Hour */

2007-11-08T20:59:32Z

Typed Notes - Second Hour

← Older revision		Revision as of 16:59, 8 November 2007
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	1) <math>N_1\cap N_2</math> is a manifold of dimension <math>n_1 + n_2 - m</math>		1) <math>N_1\cap N_2</math> is a manifold of dimension <math>n_1 + n_2 - m</math>

	2) Locally can find charts so that <math>N_1 = \mathbb{R}^{n_1} \times 0^{m-n_1}</math> and <math>N_2 = 0^{m-n_2} \times \mathbb{R}^{n_2}</math> with <math>N_1\cap N_2 = 0^{m-n_2} \times \mathbb{R}^{n_1+n_2 - 2m} \times 0^{m-n_1}</math>		2) Locally can find charts so that <math>N_1 = \mathbb{R}^{n_1} \times 0^{m-n_1}</math> and <math>N_2 = 0^{m-n_2} \times \mathbb{R}^{n_2}</math> with <math>N_1\cap N_2 = 0^{m-n_2} \times \mathbb{R}^{n_1+n_2 - m} \times 0^{m-n_1}</math>

ABjorndahl: /* Typed Notes - Second Hour */

2007-11-08T15:33:28Z

Typed Notes - Second Hour

← Older revision		Revision as of 11:33, 8 November 2007
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	''Assistance needed'': There is a symbol for "intersects transversally" but I am not sure of the latex command. See the scanned notes for what this looks like, I will just write it.		''Assistance needed'': There is a symbol for "intersects transversally" but I am not sure of the latex command. See the scanned notes for what this looks like, I will just write it.

			''RE: Assistance needed'': Try the command \pitchfork. I don't think you need any special packages, and it's pretty close to the "intersects transversally" symbol.

Trefor: /* Typed Notes - Second Hour */

2007-11-07T19:39:52Z

Typed Notes - Second Hour

← Older revision		Revision as of 15:39, 7 November 2007
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	''Recall a Thm from Linear Algebra''		''Recall a Thm from Linear Algebra''

	If <math>W_1</math> and <math>W_2</math> subspaces of V then <math>dim(W_1 + W_2) + dim(W_1\cap W_2) = dim(W_1) + dim)W_2)</math>		If <math>W_1</math> and <math>W_2</math> subspaces of V then <math>dim(W_1 + W_2) + dim(W_1\cap W_2) = dim(W_1) + dim(W_2)</math>

	In particular if <math>W_1 + W_2 = V</math> then <math>dim(W_1\cap W_2) = -dim(V) + dim(W_1) + dim(W_2)</math>		In particular if <math>W_1 + W_2 = V</math> then <math>dim(W_1\cap W_2) = -dim(V) + dim(W_1) + dim(W_2)</math>

Trefor: /* Typed Notes - First Hour */

2007-11-07T18:55:07Z

Typed Notes - First Hour

← Older revision		Revision as of 14:55, 7 November 2007
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	'''Example 3'''		'''Example 3'''

	Consider a function <math>\gamma</math> from a segment in <math>\mathbb{R}</math> onto a curve in <math>\mathbb{R}^2</math> such that <math>d\gamma</math> is never zero. Thus, rank(<math>d\gamma = 1</math>) and so <math>d\gamma</math> is never onto. Hence, ALL points are critical in the segment. The points on the curve are critical values, as they are images of critical points, and all points in <math>\mathbb{R}^2</math> NOT on the curve are vacuously regular.		Consider a function <math>\gamma</math> from a segment in <math>\mathbb{R}</math> onto a curve in <math>\mathbb{R}^2</math> such that <math>d\gamma</math> is never zero. Thus, rank(<math>d\gamma) = 1</math> and so <math>d\gamma</math> is never onto. Hence, ALL points are critical in the segment. The points on the curve are critical values, as they are images of critical points, and all points in <math>\mathbb{R}^2</math> NOT on the curve are vacuously regular.

Aptikuis: /* Typed Notes - Second Hour */

2007-10-31T00:48:39Z

Typed Notes - Second Hour

← Older revision		Revision as of 20:48, 30 October 2007
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	'''Proof Scheme for Theorem 3'''		'''Proof Scheme for Theorem 3'''

	We can write <math>N_i = \varphi_i^{-1}(0)</math> for some such <math>\varphi_i:U\subset M \rightarrow\mathbb{R}^{m-~~n_1~~ = cod(N_i)}</math>		We can write <math>N_i = \varphi_i^{-1}(0)</math> for some such <math>\varphi_i:U\subset M \rightarrow\mathbb{R}^{m-n_i = cod(N_i)}</math>

	We then write <math>\varphi = \varphi_1\times\varphi_2: M\rightarrow \mathbb{R}^{m-n_1}\times\mathbb{R}^{m-n_2} = \mathbb{R}^{2m - n_1 - n_2}</math>		We then write <math>\varphi = \varphi_1\times\varphi_2: M\rightarrow \mathbb{R}^{m-n_1}\times\mathbb{R}^{m-n_2} = \mathbb{R}^{2m - n_1 - n_2}</math>
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	as we wanted.		as we wanted.

	This shows that 0 is a regular ~~point~~ and hence by our previous theorem <math>N_1\cap N_2</math> is a submanifold.		This shows that 0 is a regular value and hence by our previous theorem <math>N_1\cap N_2</math> is a submanifold.

	Now, we know we can construct the following diagram,		Now, we know we can construct the following diagram,

	<math>\begin{matrix}		<math>\begin{matrix}
	N_1\cap N_2 & \rightarrow^{\varphi} & \mathbb{R}\\		N_1\cap N_2 & \rightarrow^{\varphi} & \mathbb{R}^{2m-n_1-n_2}\\
	\downarrow &&\downarrow^{\iota}\\		\downarrow &&\downarrow^{\iota}\\
	M&\rightarrow^{\lambda} & \mathbb{R}^m\\		M&\rightarrow^{\lambda} & \mathbb{R}^m\\

Aptikuis: /* Typed Notes - First Hour */

2007-10-31T00:33:10Z

Typed Notes - First Hour

← Older revision		Revision as of 20:33, 30 October 2007
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	'''Proof of Theorem 2'''		'''Proof of Theorem 2'''

	Let <math>f:M^m\rightarrow N^n</math> is smooth and y is a regular value. Pick a <math>p\in f^{-1}(y)</math>. p is a regular point and thus <math>df_p</math> is onto. Hence, by the submersion property (Reminder 2) we can find a "good charts" thats maps a neighborhood U of p by projection to a neighborhood V about y. Indeed, on U f looks like <math>\mathbb{R}^n\times\mathbb{R}^{m-1}\rightarrow \mathbb{R}^n</math> by <math>(x,z)\mapsto x</math>.		Let <math>f:M^m\rightarrow N^n</math> is smooth and y is a regular value. Pick a <math>p\in f^{-1}(y)</math>. p is a regular point and thus <math>df_p</math> is onto. Hence, by the submersion property (Reminder 2) we can find a "good charts" thats maps a neighborhood U of p by projection to a neighborhood V about y. Indeed, on U f looks like <math>\mathbb{R}^n\times\mathbb{R}^{m-n}\rightarrow \mathbb{R}^n</math> by <math>(x,z)\mapsto x</math>.

	So<math> f^{-1}(0) = \{(0,z)\} = \mathbb{R}^{m-n}</math>. ''Q.E.D''		So<math> f^{-1}(0) = \{(0,z)\} = \mathbb{R}^{m-n}</math>. ''Q.E.D''
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	Let <math>N_1^{n_1}, N_2^{n_2} \subset M</math> be smooth submanifolds. Let <math>p\in N_1^{n_1} \bigcap N_2^{n_2}</math>		Let <math>N_1^{n_1}, N_2^{n_2} \subset M</math> be smooth submanifolds. Let <math>p\in N_1^{n_1} \bigcap N_2^{n_2}</math>

	We say <math>N_1</math> is ''transverse'' to <math>N_2</math> in M at p if ~~for~~ <math>T_p N_1\subset T_p M</math> and<math> T_p N_2 \subset T_p M</math> ~~satisfies~~ <math>T_p N_1 + T_p N_2 = T_p M</math>		We say <math>N_1</math> is ''transverse'' to <math>N_2</math> in M at p if <math>T_p N_1\subset T_p M</math> and<math> T_p N_2 \subset T_p M</math> satisfy <math>T_p N_1 + T_p N_2 = T_p M</math>

Aptikuis: /* Typed Notes - Second Hour */

2007-10-11T01:21:14Z

Typed Notes - Second Hour

Franklin: /* Example 1 */

2007-10-09T23:13:01Z

Example 1

← Older revision		Revision as of 19:13, 9 October 2007
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	===Example 1===		===Example 1===
		⚫	'''Nota benne:''' Here we are using the name Cantor set for any perfect set with empty interior.

	There exists a function <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> smooth such that its set of critical values is homeomorphic to a Cantor set.		There exists a function <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> smooth such that its set of critical values is homeomorphic to a Cantor set.


	Remember that		Remember that
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	Observe that, since <math>h</math> is non negative, <math>f</math> is increasing (observe it is strictly increasing)(it is continuous too!).		Observe that, since <math>h</math> is non negative, <math>f</math> is increasing (observe it is strictly increasing)(it is continuous too!).
	Since increasing continuous functions have continuous inverses it is a homeomorphism.		Since increasing continuous functions have continuous inverses it is a homeomorphism.

⚫	'''Nota benne:''' Here we are using the name Cantor set for any perfect set with empty interior.

Franklin: /* Example 1 */

2007-10-09T23:10:35Z

Example 1

← Older revision		Revision as of 19:10, 9 October 2007
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	is a smooth function such that <math>g_{a,b}(a)=g_{a,b}(b)=0</math>		is a smooth function such that <math>g_{a,b}(a)=g_{a,b}(b)=0</math>

	We can define the function <math>h</math> in the complement of a Cantor set using the appropriate <math>g_{a,b}</math> in ~~its~~ of the ~~intervals~~.		We can define the function <math>h</math> in the complement of a Cantor set using the appropriate <math>g_{a,b}</math> in the intervals of the complement.

	Notices that <math>f(x)=\int_{0}^{x}h(t)dt</math> holds the conditions of the example.		Notices that <math>f(x)=\int_{0}^{x}h(t)dt</math> holds the conditions of the example.

			We can divide the new <math>g_{a,b}</math> in each step of the construction by <math>1/2^n</math> just to make the integral converge. And of course define h(t)=0 on the Cantor set.

			Observe that, since <math>h</math> is non negative, <math>f</math> is increasing (observe it is strictly increasing)(it is continuous too!).
			Since increasing continuous functions have continuous inverses it is a homeomorphism.

			'''Nota benne:''' Here we are using the name Cantor set for any perfect set with empty interior.

Trefor at 21:36, 9 October 2007

2007-10-09T21:36:30Z

← Older revision		Revision as of 17:36, 9 October 2007
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	=== Scanned Notes ===		=== Scanned Notes ===
			Scanned notes for today's class can be found [[0708-1300/Scanned Class notes for Tuesday, October 9\| here]]
	[[Image:top_10-09p1.jpg\|thumb\|left\|200px]]

			(see discussion)
	[[Image:top_10-09p2.jpg\|thumb\|left\|200px]]

	[[Image:top_10-09p3.jpg\|thumb\|left\|200px]]

	[[Image:top_10-09p4.jpg\|thumb\|left\|200px]]

	[[Image:top_10-09p5.jpg\|thumb\|left\|200px]]


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	1) More about "locally things look like their differential"		1) More about "locally things look like their differential"

	2) The ~~trick~~ Sard's Theorem: "Evil points are rare, good points everywhere"		2) The tricky Sard's Theorem: "Evil points are rare, good points everywhere"


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	Now on to some examples and comments about why Sard's Theorem is expected, but not obvious:		Now on to some examples and comments about why Sard's Theorem is expected, but not obvious:

	'''Example 4'''		'''Example 5'''

	Consider a standard "first year" smooth function <math>f:\mathbb{R}\rightarrow\mathbb{R}</math>. The "critical points" are in the first year calculus sense where the derivative is zero and the critical values are the images of these points. hence, the set of critical values we expect to be "small"		Consider a standard "first year" smooth function <math>f:\mathbb{R}\rightarrow\mathbb{R}</math>. The "critical points" are in the first year calculus sense where the derivative is zero and the critical values are the images of these points. hence, the set of critical values we expect to be "small"

	'''Example 5'''		'''Example 6'''

	Consider the function that folds the plane in half. The critical points are along the fold, as are the critical values and this line has 1 dimension and so of trivial measure in the plane (not that we have given it a measure yet!)		Consider the function that folds the plane in half. The critical points are along the fold, as are the critical values and this line has 1 dimension and so of trivial measure in the plane (not that we have given it a measure yet!)

← Older revision		Revision as of 21:21, 10 October 2007
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	'''Theorem 3'''		'''Theorem 3'''

	If <math>N_1^{n_1}\~~bigcap~~ N_2^{n_2} \subset M</math> transversally then		If <math>N_1^{n_1}\cap N_2^{n_2} \subset M</math> transversally then

	1) <math>N_1\~~bigcap~~ N_2</math> is a manifold of dimension <math>n_1 + n_2 - m</math>		1) <math>N_1\cap N_2</math> is a manifold of dimension <math>n_1 + n_2 - m</math>

	2) Locally can find charts so that <math>N_1 = \{(n_1\ ~~arb.~~\ ~~pts.),(~~ m-n_1~~\ zeros)\~~}</math> and <math>N_2 = \{(m-n_2\ ~~zeros),( m-n_2~~\ ~~arb.\ pts.)~~\}</math> with <math>N_1\~~bigcap~~ N_2 = \{(m-n_2\ ~~arb.~~\ ~~pts.),(~~n_1+n_2 - 2m\ ~~zeros), (~~m-n_1~~\ arb.\ pts.)\~~}</math>		2) Locally can find charts so that <math>N_1 = \mathbb{R}^{n_1} \times 0^{m-n_1}</math> and <math>N_2 = 0^{m-n_2} \times \mathbb{R}^{n_2}</math> with <math>N_1\cap N_2 = 0^{m-n_2} \times \mathbb{R}^{n_1+n_2 - 2m} \times 0^{m-n_1}</math>


	''Recall a Thm from Linear Algebra''		''Recall a Thm from Linear Algebra''

	If <math>W_1</math> and <math>W_2</math> subspaces of V then <math>dim(W_1 + W_2) + dim(W_1\~~bigcap~~ W_2) = dim(W_1) + dim)W_2)</math>		If <math>W_1</math> and <math>W_2</math> subspaces of V then <math>dim(W_1 + W_2) + dim(W_1\cap W_2) = dim(W_1) + dim)W_2)</math>

	In particular if <math>W_1 + W_2 = V</math> then <math>dim(W_1\~~bigcap~~ W_2) = -dim(V) + dim(W_1) + dim(W_2)</math>		In particular if <math>W_1 + W_2 = V</math> then <math>dim(W_1\cap W_2) = -dim(V) + dim(W_1) + dim(W_2)</math>


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	We then write <math>\varphi = \varphi_1\times\varphi_2: M\rightarrow \mathbb{R}^{m-n_1}\times\mathbb{R}^{m-n_2} = \mathbb{R}^{2m - n_1 - n_2}</math>		We then write <math>\varphi = \varphi_1\times\varphi_2: M\rightarrow \mathbb{R}^{m-n_1}\times\mathbb{R}^{m-n_2} = \mathbb{R}^{2m - n_1 - n_2}</math>

	Hence, <math>N_1\~~bigcap~~ N_2 = \varphi^{-1}(0)</math>		Hence, <math>N_1\cap N_2 = \varphi^{-1}(0)</math>

	We want <math>rank(d\varphi_p) = 2m-n_1-n_2</math>. To prove this, we consider the aforementioned theorem from linear algebra with respect to the vector spaces obeying <math>ker(d\varphi) = ker(d\varphi_1)\~~bigcap~~ ker(d\varphi_2)</math>		We want <math>rank(d\varphi_p) = 2m-n_1-n_2</math>. To prove this, we consider the aforementioned theorem from linear algebra with respect to the vector spaces obeying <math>ker(d\varphi) = ker(d\varphi_1)\cap ker(d\varphi_2)</math>

	hence, and by rank nullity, <math>rank(d\varphi_p) = m - dim(ker(d\varphi)) = m - \left( dim(TN_1) + dim(TN_2) - dim(TN_1 + TN_2)\right) = </math>		hence, and by rank nullity, <math>rank(d\varphi_p) = m - dim(ker(d\varphi)) = m - \left( dim(TN_1) + dim(TN_2) - dim(TN_1 + TN_2)\right) = </math>
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	as we wanted.		as we wanted.

	This shows that 0 is a regular point and hence by our previous theorem <math>N_1\~~bigcap~~ N_2</math> is a submanifold.		This shows that 0 is a regular point and hence by our previous theorem <math>N_1\cap N_2</math> is a submanifold.

	Now, we know we can construct the following diagram,		Now, we know we can construct the following diagram,

	<math>\begin{matrix}		<math>\begin{matrix}
	N_1\~~bigcap~~ N_2 & \rightarrow^{\varphi} & \mathbb{R}\\		N_1\cap N_2 & \rightarrow^{\varphi} & \mathbb{R}\\
	\downarrow &&\downarrow^{\iota}\\		\downarrow &&\downarrow^{\iota}\\
	M&\rightarrow^{\lambda} & \mathbb{R}^m\\		M&\rightarrow^{\lambda} & \mathbb{R}^m\\
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	We then set <math>\psi =</math> (the function that takes the first <math>n_1 +n_2-m</math> coordinates only)<math>\circ\lambda</math>		We then set <math>\psi =</math> (the function that takes the first <math>n_1 +n_2-m</math> coordinates only)<math>\circ\lambda</math>

	hence, <math>\psi: M\rightarrow\mathbb{R}^{n_1 + n_2 - m}</math> and <math>\psi \|_{N_1\~~bigcap~~ N_2}</math> is a chart for <math>N_1\~~bigcap~~ N_2</math>		hence, <math>\psi: M\rightarrow\mathbb{R}^{n_1 + n_2 - m}</math> and <math>\psi \|_{N_1\cap N_2}</math> is a chart for <math>N_1\cap N_2</math>

	We now consider <math>\zeta: M\rightarrow\mathbb{R}^m</math> given by <math>(\varphi_2,\psi, \varphi_1)</math>		We now consider <math>\zeta: M\rightarrow\mathbb{R}^m</math> given by <math>(\varphi_2,\psi, \varphi_1)</math>
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	'''Example 6'''		'''Example 6'''

	Consider the function that folds the plane in half. The critical points are along the fold, as are the critical values and this line has 1 dimension and so of trivial measure in the plane (not that we have given it a measure yet!)		Consider the function that folds the plane in half. The critical points are along the fold, as are the critical values and this line has 1 dimension and so of trivial measure in the plane (not that we have not given it a measure yet!)


	'''Claims:'''		'''Claims:'''

	1) <math>\exists f:\mathbb{R}\rightarrow\mathbb{R}</math> whose critical values are homeomorphic to a ~~canter~~ set.		1) <math>\exists f:\mathbb{R}\rightarrow\mathbb{R}</math> whose critical values are homeomorphic to a cantor set.

	2) <math>\exists</math> cantor sets with measure arbitrarily close to 1		2) <math>\exists</math> cantor sets with measure arbitrarily close to 1

	3) <math>\exists g\in C^1:\mathbb{R}^2\rightarrow\mathbb{R}</math> whose critical points are a cantor set cross a cantor set and whose critical values are everything. Hence we will need ~~out~~ functions to be <math>C^{\infty}</math> in the theorem.		3) <math>\exists g\in C^1:\mathbb{R}^2\rightarrow\mathbb{R}</math> whose critical points are a cantor set cross a cantor set and whose critical values are everything. Hence we will need our functions to be <math>C^{\infty}</math> in the theorem.

	==Evil functions==		==Evil functions==