0708-1300/Class notes for Tuesday, November 27: Difference between revisions
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===Second Hour=== |
===Second Hour=== |
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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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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. |
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We are now interested in integrating, using Stokes Theorem, forms in these spaces. |
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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. |
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This is consistent with 0 manifolds being the boundary of 1 manifolds. Indeed, |
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<math>\int_{\pm p_0}\omega_0 = \sum\pm f(p_i)</math> |
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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> |
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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 |
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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> |
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and <math>\int_D div\ G = \int_{\partial D} G\cdot\vec n d\sigma</math> |
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This is Gauss' Divergence Theorem. |
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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. |
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We also get Stokes' Theorem: |
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<math>\int_{\partial\Sigma} F\cdot\vec T = \int_{\Sigma} curl\ F\cdot\vec n d\sigma</math> |
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''End of Example'' |
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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> |
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'''Poincare's Lemma''' |
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On <math>\mathbb{R}^n, d\omega = 0</math> iff <math>\exists\lambda</math> such that <math>\omega = d\lambda</math> |
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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. |
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Likewise, on <math>\mathbb{R}^n-\{0\}</math> we have |
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<math> |
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\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> |
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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> |
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This is in our next homework assignment. |
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Now, if there was such a <math>\lambda</math>, <math>\int_{\Sigma}\omega = \int_{\sigma}d\lambda = \int_{\partial\Sigma}\lambda = 0</math> |
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If <math>\partial\Sigma = \empty</math> (such as any sphere) |
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But, <math>\int_{S^2}\omega = 4\pi</math> |
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'''Definition''' |
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<math>Z^k(M) := ker d|_{\Omega^k(M)}</math> |
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<math> |
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B^k(M) := im d|_{\Omega^{k-1}(M)}</math> |
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Clearly <math>B^k\subset Z^k</math> so the following definition makes sense: |
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'''Definition''' (de-Rham Cohomology) |
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<math>H^k(M):= Z^k(M)/B^k(M)</math> |
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'''Claim''' |
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<math>H^k(\mathbb{R}^n) = 0</math> yet <math>H^1(S^1)\neq 0</math>. |
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Also, <math>H^{n-1}(\mathbb{R}^n - \{x\})\neq 0</math> |
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Revision as of 15:19, 29 November 2007
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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 and have the equations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = rcos\theta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = rsin\theta}
Hence,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx = cos\theta dr - rsin\theta d\theta}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dy = sin\theta dr + rcos\theta d\theta}
Hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta}
and the angle from the horizontal to the first rod (the one connected to the fixed point) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta + \phi}
. Hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = 2cos\phi}
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dr = -2sin\phi d\phi}
With a little plane geometry we can see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = cos2\phi d(\theta + \phi)}
Computing,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega}
over the boundary of our surface and hence this is just the integral of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega}
is supported in one chart (by linearity)
For a compactly supported n-1 form on H need to show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\partial H}\omega = \int_H d\omega}
We let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \sum f_i dx_1\wedge\cdots\wedge \hat{dx_i}\wedge\cdots\wedge dx_n}
(where the hat means it is omitted)
So, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \int_{[x_1\leq 0]} \frac{\partial f_1}{\partial x_1} = \int_{[x_1=0]} f_1}
Hence with the standard inclusion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial H = \mathbb{R}^{n-1}_{x_2\cdots x_n}}
we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^1(\mathbb{R}^{n-1}_{x_2\cdots x_n})}
look like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Fdx +Gdy}
and map under d to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G_x - F_y)dx\wedge dy}
Hence applying Stokes' Theorem:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 is identically zero.
Second Hour
Example 2
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^k(\mathbb{R}^3)}
Recall previous we had consider the spaces and showed that and corresponded with functions and that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^2}
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\pm p_0}\omega_0 = \sum\pm f(p_i)}
Now consider a path Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma:[0,1]\rightarrow\mathbb{R}^3}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \int_{[0,1]}\sum f_i\dot{\gamma}_i dt = \int_{\gamma}\vec F\cdot \vec T_{\gamma}}
Now lets compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_2(v,w)}
First,
Likewise for each component of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_2} we thus get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_2(v,w) = \vec G(p)\cdot (v\times w)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec G(p)} is the vector of coefficients of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_2}
Now we know that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\times w} is a vector perpendicular to v and w with magnitude equal to the area of the defined parallelogram. So,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\Sigma}\omega_2 = \int_{\Sigma} \vec G\cdot \vec n d\sigma}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec n} denotes the normal vector and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\sigma} is the area form and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma} is a surface
Now for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_3}
,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_D \omega_3 = \int_D g}
now, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\gamma(1)) - f(\gamma(0)) = \int_{\gamma} (grad\ f)\cdot\vec T}
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\partial\Sigma} F\cdot\vec T = \int_{\Sigma} curl\ F\cdot\vec n d\sigma}
End of Example
We recall that since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^2 = 0}
, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = d\lambda}
then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\omega = 0}
. But is the converse true? The following Lemma says 'yes', if the domain is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n}
Poincare's Lemma
On Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n, d\omega = 0} iff Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists\lambda} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = d\lambda}
This is NOT true for general M, as our homework assignment showed since we had a form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\theta}
that had Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(d\theta) = 0}
but was not d of a form.
Likewise, on we have
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha,\ d\omega = 0} but Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists} no Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\lambda = \omega}
This is in our next homework assignment.
Now, if there was such a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda}
, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\Sigma}\omega = \int_{\sigma}d\lambda = \int_{\partial\Sigma}\lambda = 0}
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial\Sigma = \empty} (such as any sphere)
But, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{S^2}\omega = 4\pi}
Definition
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z^k(M) := ker d|_{\Omega^k(M)}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B^k(M) := im d|_{\Omega^{k-1}(M)}}
Clearly Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B^k\subset Z^k}
so the following definition makes sense:
Definition (de-Rham Cohomology)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^k(M):= Z^k(M)/B^k(M)}
Claim
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^k(\mathbb{R}^n) = 0} yet Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^1(S^1)\neq 0} .
Also, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^{n-1}(\mathbb{R}^n - \{x\})\neq 0}
