Theorem of Stokes

In differential Geometry, the theorem of Stokes is a central result on the integration of differential forms, which generalizes many theorems on the vectorial Analyze. After the statement and the demonstration, this article proposes many applications of it: in particular, it provides a form which readily physicists and engineers use.

The theorem is allotted to Sir George Gabriel Stokes, but the first to know this result is actually William Thomson. The mathematician and the physicist maintain an active correspondence during 5 years 1847 to 1853.

The proof requires to have the good definition of integration; it is necessary to go obviously that connect it simplicity of the current demonstration is misleading.

Demonstrations

The idea is to use a Partition of the unit adapted to the problem in the definition of the integral of a differential form; and to bring back itself to an almost obvious case.

That is to say $\backslash \; \{U\_i\; \backslash \}\; \_I$ a covering locally finished of M by fields of local charts $\backslash \; phi\_i:\; U\_i\; \backslash \; rightarrow\; \backslash \; phi\_i\; (U\_i)\; \backslash \; subset\; R^n$, such as:

$\backslash \; phi\_i\; (U\_i\; \backslash \; course\; NR)\; =\; \backslash \; phi\_i\; (U\_i)\; \backslash \; course\; (R\_+\; \backslash \; times\; R^\; \{n-1\})$

Let us introduce $\backslash \; phi\_i$ a partition of the unit subordinate to $\backslash \; \{U\_i\; \backslash \}$. As the support of $\backslash \; omega$ is closed, the differential form $\backslash \; omega$ is written:

$\backslash \; omega=\; \backslash \; sum\; f\_\; \{I\}\; \backslash \; omega$

where the summation is with finished support. Let us pose $\backslash \; beta\_i=\; \backslash \; phi\_i^*\; \backslash \; left$, differential form with compact support of $M\text{'}=R\_+\; \backslash \; times\; R^n$. $\backslash \; phi\_i|\_\; \{\backslash \; partial\; M\}$ is a diffeomorphism on its image preserving the outgoing orientations. One thus has:

$\backslash \; int\_\; \{\backslash \; partial\; M\}\; \backslash !\; \backslash \; left=\; \backslash \; int\_\; \{\backslash \; partial\; Me\}\; \backslash !\; \backslash \; beta\_i$

How $\backslash \; phi\_i^*$ commutates with the operator of differentiation D , one a:

$\backslash \; int\_M\; \backslash !\; \backslash \; mathrm\; D\; \backslash \; left=\; \backslash \; int\_\; \{Me\}\; \backslash !\; \backslash \; mathrm\; \backslash \; mathrm\; D\; \backslash \; alpha\_i$

By summation, the theorem of Stokes is shown once established the particular case $M\text{'}=R\_+\; \backslash \; times\; R^n$.

One (n-1) - form $\backslash \; omega$ on $M=R\_+\; \backslash \; times\; R^\; \{n-1\}$ is written:

$\backslash \; omega=\; \backslash \; sum\_\; \{i=1\}\; ^n\; f\_i\; \backslash \; cdot\; \backslash \; mathrm\; dx\_1\; \backslash \; wedge\; \backslash \; dowries\; \backslash \; wedge\; \backslash \; widehat\; \{\backslash \; mathrm\; dx\_i\}\; \backslash \; wedge\; \backslash \; dowries\; \backslash \; wedge\; \backslash \; mathrm\; dx\_n$

where the hat indicates an omission. The Théorème of Fubini gives:

$\backslash int\_\{R\_+\backslash times\; R^n\}\backslash !\; \backslash \; mathrm\; D\; \backslash \; omega=\; \backslash \; sum\_\; \{i=1\}\; ^n\; \backslash \; int\_\; \{R^\; \{n-1\}\}\; \backslash !\; \backslash \; partial\; left\; f\_i\}\; \{\backslash \; x\_1\}\; \backslash \; right\; \backslash \; mathrm\; dx\_2\; \backslash \; dowries\; \backslash \; mathrm\; dx\_n=\; \backslash \; int\_\; \{R^\; \{n-1\}\}\; f\_1\; (0,\; x\_2,\; \backslash \; dowries,\; x\_n)\; \backslash \; mathrm\; dx\_2\; \backslash \; dowries\; \backslash \; mathrm\; dx\_n=\; \backslash \; int\_\; \{R^\; \{n-1\}\}\; i^*\; \backslash \; omega$

From where the result.

Fundamental theorem of integration

If F is a function $C^\; \{\backslash \; infty\}$ of the real variable, then F is a differential form of degree zero, whose Différentielle is $f^\; \backslash \; premium\; (X)\; \backslash \; mathrm\; dx\; \backslash ,$. The directed edge of $$ is $\backslash \; \{B\; \backslash \}\; -\; \backslash \; \{has\; \backslash \}$ (end with the orientation + and origin with the orientation - ), whatever the relative values of has and B . The formula of Stokes gives in this situation:

$\backslash \; int\_a^bf\text{'}\; (X)\; \backslash \; mathrm\; dx=f\; (B)\; -\; F\; (a)$

In fact, the theorem of Stokes is the generalization of this formula to higher dimensions. The difficulty is well more in the installation of the good framework (forms differential, variety S on board or possibly more general, orientation S) that in the demonstration, which rests on the fundamental theorem of integration and an argument of Partition of the unit.

Formulate of Green-Riemann

See also: Theorem of Green

Either U a compact field smoothes $\backslash \; R^2$ and $\backslash \; alpha=f\; \backslash \; cdot\; \backslash \; mathrm\; dx+g\; \backslash \; cdot\; \backslash \; mathrm\; dy$ a differential 1-form on $\backslash \; R^2$. Then, the formula of Stokes is written:

$\backslash \; int\_\; \{\backslash \; partial\; U\}\; \backslash \; alpha\; =\; \backslash \; int\_\; \{\backslash \; partial\; U\}\; \backslash !\; \backslash \; left\; \backslash \; mathrm\; dx+g\; \backslash \; cdot\; \backslash \; mathrm\; Dy\; \backslash \; right=\; \backslash \; int\_U\; \backslash \; left\; G\}\; \{\backslash \; partial\; X\}\; -\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; there\}\; \backslash \; right\; \backslash ,\; \backslash \; mathrm\; dx\; \backslash ,\; \backslash \; mathrm\; dy$

The formula of Green-Riemann is used in geometry to show the Inégalité of Poincaré.

Formulate of Ostrogradsky

See also: Theorem of Green-Ostrogradsky

That is to say U a compact field on smooth board of $\backslash \; R^3$, and let us pose $\backslash \; eta=\; \backslash \; mathrm\; dx\; \backslash \; wedge\; \backslash \; mathrm\; Dy\; \backslash \; wedge\; \backslash \; mathrm\; dz$. If X is a Champ of vectors in the vicinity of the adherence of U , then its divergence checks:

$\backslash \; mathrm\; D\; \backslash \; left=\; \backslash \; mathrm\; \{div\}\; (X)\; \backslash \; cdot\; \backslash \; omega$

The formula of Stokes gives then:

$\backslash \; int\_\; \{\backslash \; partial\; U\}\; \backslash !\; \backslash \; left\; \backslash \; mathrm\; Dy\; \backslash \; wedge\; \backslash \; mathrm\; dz+g\; \backslash \; cdot\; \backslash \; mathrm\; dz\; \backslash \; wedge\; \backslash \; mathrm\; dx+h\; \backslash \; cdot\; \backslash \; mathrm\; dx\; \backslash \; wedge\; \backslash \; mathrm\; Dy\; \backslash \; right=\; \backslash \; int\_U\; \backslash !\; \backslash \; left\; F\}\; \{\backslash \; partial\; X\}\; +\; \backslash \; frac\; \{\backslash \; partial\; G\}\; \{\backslash \; partial\; there\}\; +\; \backslash \; frac\; \{\backslash \; partial\; H\}\; \{\backslash \; partial\; Z\}\; \backslash \; right\; \backslash ,\; \backslash \; mathrm\; dx\; \backslash ,\; \backslash \; mathrm\; Dy\; \backslash ,\; \backslash \; mathrm\; dz$

Physical direction of the formula of Stokes

Let us note $\backslash \; mathrm\; D\; \backslash \; vec\; S$ the outgoing field of vectors normal of a field U relatively compact on regular board. That is to say X a field of vectors defined in the vicinity of the adherence of D . One defines the surface form on $\backslash \; partial\; U$ by:

$\backslash \; eta=\; \backslash \; iota\; (NR)\; (\backslash \; mathrm\; dx\; \backslash \; wedge\; \backslash \; mathrm\; Dy\; \backslash \; wedge\; \backslash \; mathrm\; dz)\; \backslash \; big|\_\; \{\backslash \; partial\; U\}$

One defines the Flux X by:

$\backslash \; oint\_\; \{\backslash \; partial\; U\}\; \backslash \; vec\; X\; \backslash \; \backslash \; mathrm\; D\; \backslash \; vec\; S=\; \backslash \; int\_\; \{\backslash \; partial\; U\}\; \backslash \; langle\; X\; \backslash \; mid\; NR\; \backslash \; rangle\; \backslash \; cdot\; \backslash \; eta$

The formula of Ostrogradsky is rewritten then:

$\backslash \; oint\_\; \{\backslash \; partial\; U\}\; \backslash \; vec\; X\; \backslash \; \backslash \; mathrm\; D\; \backslash \; vec\; S=\; \backslash \; int\_U\; (\backslash \; mathrm\; \{div\}\; X)\; \backslash ,\; \backslash \; mathrm\; dx\; \backslash ,\; \backslash \; mathrm\; Dy\; \backslash ,\; \backslash \; mathrm\; dz$

That is to say $\backslash \; partial\; S$, a closed curve directed in $\backslash \; R^3$, S a directed surface whose contour is $\backslash \; partial\; S$. The orientation of $\backslash \; partial\; S$ is induced by the orientation of S. If the vector field $\backslash \; vec\; \{V\}$ admits derivative partial continuous, then:

$\backslash \; oint\_\; \{\backslash \; partial\; S\}\; \backslash \; vec\; V\; \backslash \; cdot\; \backslash \; mathrm\; D\; \backslash \; vec\; L\; =\; \backslash \; iint\_\; \{S\}\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; \backslash \; vec\; V\; \backslash \; cdot\; \backslash \; mathrm\; D\; \backslash \; vec\; S$

where $\backslash \; mathrm\; D\; \backslash \; vec\; l$ is the directing vector of the curve in any point, $\backslash \; overrightarrow\; \backslash \; mathrm\; \{belch\}\; \backslash \; \backslash \; vec\; V=\; \backslash \; nabla\; \backslash \; wedge\; \backslash \; vec\; V$ the Rotationnel of $\backslash \; vec\; V$, and $\backslash \; mathrm\; D\; \backslash \; vec\; S$ the normal Vecteur with an infinitesimal element of surface whose standard is equal to the surface of the element.

Its direct application is the Théorème of Amp (one applies it to the magnetic field).

Application to the Homology

The formula of Stokes is used to show the Théorème of duality of Rham.

The formula of Stokes also makes it possible to show the Lemme of Poincaré. This last proves of a great utility to include/understand isotopies in homology. It is also used notably in the proof of the theorem of Darboux in symplectic Géométrie.

References

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