PlanetMath: proof of general Stokes theorem

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proof of general Stokes theorem

(Proof)

We divide the proof in several steps.

Step One.

Suppose $M=(0,1]\times (0,1)^{n-1}$ and

$\displaystyle \omega(x_1,\ldots,x_n)=f(x_1,\ldots,x_n)\, d x_1\wedge\cdots\wedge \widehat{d x_j}\wedge \cdots \wedge d x_n$

(i.e. the term

is missing). Hence we have

$\displaystyle d\omega(x_1,\ldots,x_n)$	$\displaystyle =$	$\displaystyle \left(\frac{\partial f}{\partial x_1} d x_1+\cdots +\frac{\partia... ...ight)\wedge d x_1\wedge \cdots \wedge \widehat{d x_j}\wedge \cdots \wedge d x_n$
	$\displaystyle =$	$\displaystyle (-1)^{j-1} \frac{\partial f}{\partial x_j} d x_1\wedge \cdots \wedge d x_n$

and from the definition of integral on a manifold we get

$\displaystyle \int_M d\omega = \int_0^1\cdots \int_0^1 (-1)^{j-1} \frac{\partial f}{\partial x_j} d x_1 \cdots d x_n.$

From the fundamental theorem of Calculus we get

$\displaystyle \int_M d\omega =(-1)^{j-1} \int_0^1\cdots\widehat{\int_0^1}\cdots... ...dots,x_n)-f(x_1,\ldots,0,\ldots,x_n) d x_1\cdots \widehat{ d x_j}\cdots d x_n.$

Since $\omega$ and hence

have compact support in

we obtain

$\begin{displaymath} \int_M d\omega=\left\{ \begin{array}{lcl} \int_0^1\cdots\int... ...\text{if} & j=1 \\ \ 0 &\text{if} & j>1 . \end{array}\right. \end{displaymath}$

On the other hand we notice that $\int_{\partial M}\omega$ is to be understood as $\int_{\partial M}i^* \omega$ where $i:\partial M\to M$ is the inclusion map. Hence it is trivial to verify that when $j\neq 1$ then $i^*\omega=0$ while if it holds

$\displaystyle i^*\omega(x) = f(1,x_2,\ldots,x_n) d x_2\wedge\ldots\wedge d x_n$

and hence, as wanted

$\displaystyle \int_{\partial M}i^*\omega = \int_0^1\cdots\int_0^1 f(x) d x_2 \cdots d x_n.$

Step Two.

Suppose now that $M=[0,1)\times (0,1)^{n-1}$ and let $\omega$ be any differential form. We can always write

$\displaystyle \omega(x)=\sum_j f_j(x) d x_1 \wedge\cdots \wedge \widehat{ d x_j} \wedge \cdots \wedge d x_n$

and by the additivity of the integral we can reduce ourself to the previous case.

Step Three.

When we could follow the proof as in the first case and end up with $\int_M d \omega = 0$ while, in fact, $\partial M=\emptyset$ .

Step Four.

Consider now the general case.

First of all we consider an oriented atlas $(U_i,\phi_i)$ such that either is the cube $(0,1]\times (0,1)^{n-1}$ or . This is always possible. In fact given any open set in $[0,+\infty)\times \mathbb{R}^{n-1}$ and a point $x\in U$ up to translations and rescaling it is possible to find a “cubic” neighbourhood of contained in .

Then consider a partition of unity $\alpha_i$ for this atlas.

From the properties of the integral on manifolds we have

$\displaystyle \int_{M} d \omega$	$\displaystyle =$	$\displaystyle \sum_i \int_{U_i} \alpha_i \phi^* d \omega = \sum_i \int_{U_i} \alpha_i d (\phi^* \omega)$
	$\displaystyle =$	$\displaystyle \sum_i \int_{U_i} d (\alpha_i\cdot \phi^\omega) - \sum_i \int_{U_i} (d\alpha_i)\wedge(\phi^\omega).$

The second integral in the last equality is zero since $\sum_i d\alpha_i= d \sum_i \alpha_i = 0$ , while applying the previous steps to the first integral we have

$\displaystyle \int_{M} d\omega = \sum_i \int_{\partial U_i}\alpha_i\cdot \phi^*\omega.$

On the other hand, being $(\partial U_i,\phi_{\vert\partial U_i})$ an oriented atlas for $\partial M$ and being ${\alpha_i}_{\vert\partial U_i}$ a partition of unity, we have

$\displaystyle \int_{\partial M} \omega = \sum_i \int_{\partial U_i} \alpha_i \phi^* \omega$

and the theorem is proved.

"proof of general Stokes theorem" is owned by paolini.

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Cross-references: equality, properties, partition of unity, contained, neighbourhood, translations, point, open set, cube, atlas, oriented, additivity, differential form, inclusion map, support, compact, fundamental theorem of calculus, manifold, integral, term, divide

This is version 4 of proof of general Stokes theorem, born on 2003-06-16, modified 2003-06-16.
Object id is 4370, canonical name is ProofOfGeneralStokesTheorem.
Accessed 14214 times total.

Classification:

AMS MSC:

58C35 (Global analysis, analysis on manifolds :: Calculus on manifolds; nonlinear operators :: Integration on manifolds; measures on manifolds)

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