In vectorial Analysis, the theorem of flow-divergence , also called the theorem of Green-Ostrogradski is a Théorème connecting the Divergence of a vector Field to the value of the Intégrale of surface of the flow defined by this field.

It stipulates that the flow of a vector through a closed surface is equal to the integral of the divergence of this vector on the volume delimited by this surface.

The expression of the theorem is the following:

$\backslash \; iiint\_\; \{\backslash \; mathcal\; \{V\}\}\; \backslash \; vec\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; vec\; F\; \backslash \; \{\backslash \; rm\; D\}\; V\; =\; \backslash \; iint\_\; \{\backslash \; Sigma\}\; \backslash \; vec\; F\; \backslash \; cdot\; \{\backslash \; rm\; D\}\; \backslash \; vec\; S$

where:

$\backslash \; mathcal\; \{V\}\; \backslash ,$ represents volume, and $\backslash \; Sigma\; \backslash ,$ the Bord of $\backslash \; mathcal\; \{V\}\; \backslash ,$, which one notes mathematically $\backslash \; Sigma=\; \backslash \; share\; \backslash \; mathcal\; \{V\}\; \backslash ,$.

$\{\backslash \; rm\; D\}\; \backslash \; vec\; S$ is the normal Vecteur on the surface, directed towards outside, and of length equal to the element which it represents.

$\backslash \; vec\; \backslash \; nabla\; \backslash \; cdot\; \backslash \; vec\; F$ is also noted $\backslash \; mathrm\; \{div\}\; \backslash \; vec\; F$

It will be noted that this theorem rises from the Théorème of Stokes, which itself generalizes the fundamental Théorème of differential and integral calculus.

It is an important result in in particular Mathematical physics, in electrostatic and Dynamique of the fluids.

One can use this theorem to deduce certain useful formulas from vector calculus:

$\backslash \; iiint\_\; \backslash \; mathcal\; \{V\}\; \backslash \; vec\; \{F\}\; \backslash \; cdot\; \backslash \; vec\; \{\backslash \; nabla\}\; G\; +\; G\; \backslash \; left\; (\backslash \; vec\; \{\backslash \; nabla\}\; \backslash \; cdot\; \backslash \; vec\; \{F\}\; \backslash \; right)\; \{\backslash \; rm\; D\}\; V=\; \backslash \; iint\_\; \{\backslash \; share\; \backslash \; mathcal\; \{V\}\}\; G\; \backslash \; vec\; \{F\}\; \backslash \; cdot\; \{\backslash \; rm\; D\}\; \backslash \; vec\; \{S\},$

$\backslash \; iiint\_\; \backslash \; mathcal\; \{V\}\; \backslash \; vec\; \{\backslash \; nabla\}\; G\; \backslash ,\; \{\backslash \; rm\; D\}\; V=\; \backslash \; iint\_\; \{\backslash \; share\; \backslash \; mathcal\; \{V\}\}\; G\; \{\backslash \; rm\; D\}\; \backslash \; vec\; \{S\},$

$\backslash \; iiint\_\; \backslash \; mathcal\; \{V\}\; \backslash \; vec\; \{G\}\; \backslash \; cdot\; \backslash \; left\; (\backslash \; vec\; \{\backslash \; nabla\}\; \backslash \; wedge\; \backslash \; vec\; \{F\}\; \backslash \; right)$

- \ vec {F} \ cdot \ left (\ vec {\ nabla} \ wedge \ vec {G} \ right) {\ rm D} V

\ iint_ {\ share \ mathcal {V}} \ left (\ vec {F} \ wedge \ vec {G} \ right) \ cdot {\ rm D} \ vec {S},

$\backslash \; iiint\_\; \backslash \; mathcal\; \{V\}\; \backslash \; vec\; \{\backslash \; nabla\}\; \backslash \; wedge\; \backslash \; vec\; \{F\}\; \{\backslash \; rm\; D\}\; V\; =\; \backslash \; iint\_\; \{\backslash \; share\; \backslash \; mathcal\; \{V\}\}\; \{\backslash \; rm\; D\}\; \backslash \; vec\; \{S\}\; \backslash \; wedge\; \backslash \; vec\; \{F\}.$

This theorem in particular makes it possible to find the integral version of the Théorème of Gauss starting from the equation of Maxwell-Gauss:

See too

• Integral of volume
• Integral of surface
• Theorem of rotational the

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