Difference between revisions of "0708-1300/Class notes for Tuesday, November 27"

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(Today's Agenda)
(Second Hour)
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===Second Hour===
 
===Second Hour===
  
Coming soon to a wikipedia near you.
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'''Example 2'''
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<math>\Omega^k(\mathbb{R}^3)</math>
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 +
 
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Recall previous we had consider the spaces <math>\Omega^k(\mathbb{R}^3)</math> and showed that <math>\Omega^0</math> and <math>\Omega 3</math> corresponded with functions and that <math>\Omega^2</math> and <math>\Omega^1</math> corresponded with triples of functions (i.e. vector fields). We also showed that the d function between these spaces was the gradient, curl and divergence functions from vector calculus.
 +
 
 +
We are now interested in integrating, using Stokes Theorem, forms in these spaces.
 +
 
 +
 
 +
First, note that to a 0 manifold, assigning an orientation to the manifold is just assigning a plus or minus sign to the manifold as a result of it having a trivial basis.
 +
 
 +
This is consistent with 0 manifolds being the boundary of 1 manifolds. Indeed,
 +
 
 +
<math>\int_{\pm p_0}\omega_0 = \sum\pm f(p_i)</math>
 +
 
 +
 
 +
Now consider a path <math>\gamma:[0,1]\rightarrow\mathbb{R}^3</math>
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 +
<math>\int_{\gamma}\omega_1 = \int_{[0,1]}\gamma^*\omega_1 = \int_{[0,1]}\sum f_i d\gamma^*(x_i) = \int_{[0,1]}\sum f_i d\gamma_i</math>
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 +
<math>= \int_{[0,1]}\sum f_i\dot{\gamma}_i dt = \int_{\gamma}\vec F\cdot \vec T_{\gamma}</math>
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 +
 
 +
Now lets compute <math>\omega_2(v,w)</math>
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 +
First, <math>dx_2\wedge dx_3 (v,w) = v_2 w_3 - v_3 w_2</math>
 +
 
 +
Likewise for each component of <math>\omega_2</math> we thus get
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<math>\omega_2(v,w) = \vec G(p)\cdot (v\times w)</math> where <math>\vec G(p)</math> is the vector of coefficients of <math>\omega_2</math>
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 +
Now we know that <math>v\times w</math> is a vector perpendicular to v and w with magnitude equal to the area of the defined parallelogram. So,
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<math>\int_{\Sigma}\omega_2 = \int_{\Sigma} \vec G\cdot \vec n d\sigma</math>
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where <math>\vec n</math> denotes the normal vector and <math>d\sigma</math> is the area form and <math>\Sigma</math> is a surface
 +
 
 +
 
 +
Now for <math>\omega_3</math>,
 +
 
 +
 
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<math>\int_D \omega_3 = \int_D g</math>
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 +
 
 +
now, <math>f(\gamma(1)) - f(\gamma(0)) = \int_{\gamma} (grad\ f)\cdot\vec T</math>
 +
 
 +
and <math>\int_D div\ G = \int_{\partial D} G\cdot\vec n d\sigma</math>
 +
 
 +
 
 +
This is Gauss' Divergence Theorem.
 +
 
 +
We can think about this as saying that the flow from each point in a domain, when summed up, will be just the flow out of the boundary of the domain.
 +
 
 +
 
 +
We also get Stokes' Theorem:
 +
 
 +
 
 +
<math>\int_{\partial\Sigma} F\cdot\vec T = \int_{\Sigma} curl\ F\cdot\vec n d\sigma</math>
 +
 
 +
 
 +
''End of Example''
 +
 
 +
 
 +
We recall that since <math>d^2 = 0</math>, if <math>\omega = d\lambda</math> then <math>d\omega = 0</math>. But is the converse true? The following Lemma says 'yes', if the domain is <math>\mathbb{R}^n</math>
 +
 
 +
 
 +
'''Poincare's Lemma'''
 +
 
 +
On <math>\mathbb{R}^n, d\omega = 0</math> iff <math>\exists\lambda</math> such that <math>\omega = d\lambda</math>
 +
 
 +
 
 +
This is NOT true for general M, as our homework assignment showed since we had a form <math>d\theta</math> that had <math>d(d\theta) = 0</math> but was not d of a form.
 +
 
 +
 
 +
Likewise, on <math>\mathbb{R}^n-\{0\}</math> we have
 +
<math>
 +
\omega = \frac{1}{||x||^{\alpha}}\sum_{i=1}^{n}x_i dx_1\wedge\cdots\wedge\hat{dx_i}\wedge\cdots\wedge dx_n \in\Omega^{n-1}(\mathbb{R}^n-\{0\})</math>
 +
 
 +
 
 +
Claim:
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 +
For appropriate <math>\alpha,\ d\omega = 0</math> but <math>\exists</math> no <math>\lambda</math> such that <math>d\lambda = \omega</math>
 +
 
 +
This is in our next homework assignment.
 +
 
 +
 
 +
Now, if there was such a <math>\lambda</math>, <math>\int_{\Sigma}\omega = \int_{\sigma}d\lambda = \int_{\partial\Sigma}\lambda = 0</math>
 +
 
 +
If <math>\partial\Sigma = \empty</math> (such as any sphere)
 +
 
 +
But, <math>\int_{S^2}\omega = 4\pi</math>
 +
 
 +
 
 +
 
 +
'''Definition'''
 +
 
 +
<math>Z^k(M) := ker d|_{\Omega^k(M)}</math>
 +
 
 +
<math>
 +
B^k(M) := im d|_{\Omega^{k-1}(M)}</math>
 +
 
 +
 
 +
Clearly <math>B^k\subset Z^k</math> so the following definition makes sense:
 +
 
 +
 
 +
'''Definition''' (de-Rham Cohomology)
 +
 
 +
<math>H^k(M):= Z^k(M)/B^k(M)</math>
 +
 
 +
 
 +
'''Claim'''
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 +
<math>H^k(\mathbb{R}^n) = 0</math> yet <math>H^1(S^1)\neq 0</math>.  
 +
 
 +
Also, <math>H^{n-1}(\mathbb{R}^n - \{x\})\neq 0</math>

Revision as of 15:19, 29 November 2007

Announcements go here

Contents

Today's Agenda

  • The planimeter with a picture from http://whistleralley.com/planimeter/planimeter.htm but our very own plane geometry and Stokes' theorem.
  • Completion of the proof of Stokes' theorem.
  • Completion of the discussion of the two- and three-dimensional cases of Stokes' theorem.
  • With luck, a discussion of de-Rham cohomology, homotopy invariance and Poincaré's lemma.


Class Notes

The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.

First Hour

Planimeter

A planimeter consists of two rods connected with a join where the end of one rod is fixed (but free to rotate) and the opposing end of the second rod traces out the boundary of some surface on the plane. I.e., the planimeter is kind of like a 1 legged roach. At the join of the two rods is a wheel which rotates (and measures the rotation) when the rod tracing the boundary moves in the normal direction and simply slides back an forth when moved in a tangential direction.

Now we recall from plane geometry that we can locate points in the polar form (r,\theta) and have the equations x = rcos\theta and y = rsin\theta

Hence,

dx = cos\theta dr - rsin\theta d\theta

dy = sin\theta dr + rcos\theta d\theta

Hence dx\wedge dy = r(cos^2\theta + sin^2\theta)dr\wedge d\theta = rdr\wedge d\theta



Now, the planimeter is essentially a 1 form corresponding to the speed of the wheel. We consider a diagram where the angle from the horizontal at the fixed end of the planimeter to the measuring end is \theta and the angle from the horizontal to the first rod (the one connected to the fixed point) is \theta + \phi. Hence r = 2cos\phi and dr = -2sin\phi d\phi

With a little plane geometry we can see that \omega = cos2\phi d(\theta + \phi)

Computing,

d\omega = -2 sin2\phi d\phi\wedge(d\theta + d\phi) = -4 sin\phi cos\phi d\phi\wedge d\theta = 2cos\phi dr\wedge d\theta = rdr\wedge d\theta = dx\wedge dy


Now applying stokes theorem, the the planimeter integrates \omega over the boundary of our surface and hence this is just the integral of d\omega over the surface. But this is just the integral of the area form.

Hence the planimeter measure the area of a surface.


Back to Stokes Theorem


Firstly recall that \partial M is oriented so that if you prepend the outward normal to its orientation you get the orientation of M

Alternatively we recall that neighborhoods of points on the boundary look like the half space. Hence we can choose to restrict our attention to atlas's where all charts look like H= \{x\in\mathbb{R}^n:x_1\leq 0\}

We can see that these orientations are the same, i.e., just prepend the outward normal to the half space.


Proof of Stokes


We have now defined all the terms. WLOG \omega is supported in one chart (by linearity)

For a compactly supported n-1 form on H need to show that \int_{\partial H}\omega = \int_H d\omega


We let \omega = \sum f_i dx_1\wedge\cdots\wedge \hat{dx_i}\wedge\cdots\wedge dx_n (where the hat means it is omitted)

d\omega = \sum (-1)^{i-1} \frac{\partial f_i}{\partial x_i} dx_1\wedge\cdots\wedge dx_n


So, \int_H d\omega = \sum \int_{[x_1\leq 0]}(-1)^{i-1}\frac{\partial f_i}{\partial x_i}dx_1\wedge\cdots\wedge dx_n = \sum (-1)^{i-1}\int_{[x_1\leq 0]}\frac{\partial f_i}{\partial x_i}

via fundamental theorem of calculus and that the f_i's are compactly supported we get

= \int_{[x_1\leq 0]} \frac{\partial f_1}{\partial x_1} = \int_{[x_1=0]} f_1


Hence with the standard inclusion of \partial H = \mathbb{R}^{n-1}_{x_2\cdots x_n} we get


\int_{\partial H}\omega = \int_{\mathbb{R}^{n-1}_{x_2\cdots x_n}}\iota^*(\sum f_i dx_1\wedge\cdots\wedge \hat{dx_i}\wedge\cdots\wedge dx_n) = \int_{[x_1=0]}f_1


Thus these are the same and the theorem is proved Q.E.D.


Real Plane

Consider \Omega^0(\mathbb{R}^{n-1}_{x_2\cdots x_n})\rightarrow^d\Omega^1(\mathbb{R}^{n-1}_{x_2\cdots x_n})\rightarrow^d\Omega^2(\mathbb{R}^{n-1}_{x_2\cdots x_n})


Forms in \Omega^1(\mathbb{R}^{n-1}_{x_2\cdots x_n}) look like Fdx +Gdy and map under d to (G_x - F_y)dx\wedge dy

Hence applying Stokes' Theorem:

\int_{\partial D}Fdx + Gdy = \int_D (G_x-F_y)dxdy

This is known as Greens Theorem


In complex analysis we also have a similar result Cauchy's Theorem where the integral of an analytic function around a closed path is zero. This is because analytic functions obey the Cauchy-Riemann equations and hence G_x-F_y is identically zero.


Second Hour

Example 2

\Omega^k(\mathbb{R}^3)


Recall previous we had consider the spaces \Omega^k(\mathbb{R}^3) and showed that \Omega^0 and \Omega 3 corresponded with functions and that \Omega^2 and \Omega^1 corresponded with triples of functions (i.e. vector fields). We also showed that the d function between these spaces was the gradient, curl and divergence functions from vector calculus.

We are now interested in integrating, using Stokes Theorem, forms in these spaces.


First, note that to a 0 manifold, assigning an orientation to the manifold is just assigning a plus or minus sign to the manifold as a result of it having a trivial basis.

This is consistent with 0 manifolds being the boundary of 1 manifolds. Indeed,

\int_{\pm p_0}\omega_0 = \sum\pm f(p_i)


Now consider a path \gamma:[0,1]\rightarrow\mathbb{R}^3

\int_{\gamma}\omega_1 = \int_{[0,1]}\gamma^*\omega_1 = \int_{[0,1]}\sum f_i d\gamma^*(x_i) = \int_{[0,1]}\sum f_i d\gamma_i

= \int_{[0,1]}\sum f_i\dot{\gamma}_i dt = \int_{\gamma}\vec F\cdot \vec T_{\gamma}


Now lets compute \omega_2(v,w)

First, dx_2\wedge dx_3 (v,w) = v_2 w_3 - v_3 w_2

Likewise for each component of \omega_2 we thus get

\omega_2(v,w) = \vec G(p)\cdot (v\times w) where \vec G(p) is the vector of coefficients of \omega_2

Now we know that v\times w is a vector perpendicular to v and w with magnitude equal to the area of the defined parallelogram. So,

\int_{\Sigma}\omega_2 = \int_{\Sigma} \vec G\cdot \vec n d\sigma

where \vec n denotes the normal vector and d\sigma is the area form and \Sigma is a surface


Now for \omega_3,


\int_D \omega_3 = \int_D g


now, f(\gamma(1)) - f(\gamma(0)) = \int_{\gamma} (grad\ f)\cdot\vec T

and \int_D div\ G = \int_{\partial D} G\cdot\vec n d\sigma


This is Gauss' Divergence Theorem.

We can think about this as saying that the flow from each point in a domain, when summed up, will be just the flow out of the boundary of the domain.


We also get Stokes' Theorem:


\int_{\partial\Sigma} F\cdot\vec T = \int_{\Sigma} curl\ F\cdot\vec n d\sigma


End of Example


We recall that since d^2 = 0, if \omega = d\lambda then d\omega = 0. But is the converse true? The following Lemma says 'yes', if the domain is \mathbb{R}^n


Poincare's Lemma

On \mathbb{R}^n, d\omega = 0 iff \exists\lambda such that \omega = d\lambda


This is NOT true for general M, as our homework assignment showed since we had a form d\theta that had d(d\theta) = 0 but was not d of a form.


Likewise, on \mathbb{R}^n-\{0\} we have 
\omega = \frac{1}{||x||^{\alpha}}\sum_{i=1}^{n}x_i dx_1\wedge\cdots\wedge\hat{dx_i}\wedge\cdots\wedge dx_n \in\Omega^{n-1}(\mathbb{R}^n-\{0\})


Claim:

For appropriate \alpha,\ d\omega = 0 but \exists no \lambda such that d\lambda = \omega

This is in our next homework assignment.


Now, if there was such a \lambda, \int_{\Sigma}\omega = \int_{\sigma}d\lambda = \int_{\partial\Sigma}\lambda = 0

If \partial\Sigma = \empty (such as any sphere)

But, \int_{S^2}\omega = 4\pi


Definition

Z^k(M) := ker d|_{\Omega^k(M)}


B^k(M) := im d|_{\Omega^{k-1}(M)}


Clearly B^k\subset Z^k so the following definition makes sense:


Definition (de-Rham Cohomology)

H^k(M):= Z^k(M)/B^k(M)


Claim

H^k(\mathbb{R}^n) = 0 yet H^1(S^1)\neq 0.

Also, H^{n-1}(\mathbb{R}^n - \{x\})\neq 0