Dictionary:Divergence theorem

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The flux ${\displaystyle \phi }$ through a surface (or the integral of the vector flux density g over a closed surface) equals the divergence of the flux density integrated over the volume contained by the surface:

${\displaystyle \phi =\iint {\textbf {g}}\cdot d{\textbf {s}}=\iiint \nabla \cdot {\textbf {g}}\;dxdydz}$.

Another way of writing this is

${\displaystyle \phi =\iint _{S}{\textbf {g}}\cdot \mathbf {\hat {n}} \;dS=\iiint _{V}\nabla \cdot {\textbf {g}}\;dV}$,

where ${\displaystyle dS}$ is the area element on the bounding surface ${\displaystyle S}$ of volume ${\displaystyle V}$, with volume element ${\displaystyle dV}$.

Commonly called Gauss's theorem or the Gauss-Ostrogradski theorem .

Generalization to arbitrary dimensions

The divergence theorem is a general mathematical result that may be applied in arbitrary dimensions.

If we have an integral over the volume ${\displaystyle D}$ in ${\displaystyle R^{m}}$ bounded by ${\displaystyle \partial D}$ in ${\displaystyle R^{m-1}}$ then the following relation holds

${\displaystyle \int _{D}\mathbf {\nabla } \cdot \mathbf {Q} \;dV=\int _{\partial D}\mathbf {\hat {n}} \cdot \mathbf {Q} \;dS.}$

Here, ${\displaystyle Q}$ is a vector field and the integrand of the (hyper-)volume integral is an exact divergence of that vector field. The quantity ${\displaystyle \mathbf {\hat {n}} =({\bar {n}}_{1},{\bar {n}}_{2},...{\bar {n}}_{m}),}$ is the outward-pointing normal vector to the (hyper-) surface ${\displaystyle \partial D.}$ Each ${\displaystyle {\bar {n}}_{k}}$ is the respective direction cosine obtained by forming the inner product of the normal to the boundary ${\displaystyle \partial D}$ with the respective ${\displaystyle {\hat {x}}_{k}}$ coordinate axis unit vector.

Because we are in arbitrary dimensions ${\displaystyle \mathbf {x} =(x_{1},x_{2},...,x_{m})}$ and ${\displaystyle \nabla \equiv \left({\frac {\partial }{\partial x_{1}}},{\frac {\partial }{\partial x_{2}}},...,{\frac {\partial }{\partial x_{m}}}\right)}$.

Proof[1]

Consider the case when ${\displaystyle m=3}$ and ${\displaystyle D}$ is a three-dimensional volume bounded by the two-dimensional boundary ${\displaystyle \partial D}$. The volume integral may be written in three terms, one for each coordinate direction

${\displaystyle \int _{D}\mathbf {\nabla } \cdot \mathbf {Q} \;dV=\int _{D}{\frac {\partial Q_{1}}{\partial x_{1}}}\;dV+\int _{D}{\frac {\partial Q_{2}}{\partial x_{2}}}\;dV+\int _{D}{\frac {\partial Q_{3}}{\partial x_{3}}}\;dV.}$

If we concentrate on the ${\displaystyle x_{1}}$ volume integral and apply the fundamental theorem of calculus we obtain

${\displaystyle \int _{D}{\frac {\partial Q_{1}}{\partial x_{1}}}dV=\int _{X_{2}^{-}(x_{1},x_{3})}^{X_{2}^{+}(x_{1},x_{3})}\int _{X_{3}^{-}(x_{1},x_{2})}^{X_{3}^{+}(x_{1},x_{2})}\int _{X_{1}^{-}(x_{2},x_{3})}^{X_{1}^{+}(x_{2},x_{3})}{\frac {\partial Q_{1}}{\partial x_{1}}}\;dx_{1}dx_{2}dx_{3}}$

${\displaystyle =\int _{X_{2}^{-}(X_{1}^{-},x_{3})}^{x_{2}^{+}(X_{1}^{+},x_{3})}\int _{X_{3}^{-}(X_{1}^{-},x_{2})}^{X_{3}^{+}(X_{1}^{+},x_{2})}\left[Q_{1}(X_{1}^{+},x_{2},x_{3})-Q_{1}(X_{1}^{-},x_{2},x_{3})\right]\;dx_{2}dx_{3}.}$

Here, the functions ${\displaystyle X_{k}^{+}}$ and ${\displaystyle X_{k}^{-}}$ define points on ${\displaystyle \partial D}$ of greater ${\displaystyle x_{k}}$ and lesser ${\displaystyle x_{k},}$ respectively.

Now, if we consider the surface integral

${\displaystyle \int _{\partial D}\mathbf {\hat {n}} \cdot \mathbf {Q} \;dS.=\int _{\partial D}\left[{\bar {n}}_{1}Q_{1}+{\bar {n}}_{2}Q_{2}+{\bar {n}}_{3}Q_{3}\right]\;dS}$

and concentrate on the ${\displaystyle x_{1}}$ term

${\displaystyle \int _{\partial D}{\bar {n}}_{1}Q_{1}\;dS=\int _{X_{2}^{-}(X_{1}^{-},x_{3})}^{X_{2}^{+}(X_{1}^{+},x_{3})}\int _{X_{3}^{-}(X_{1}^{-},x_{2})}^{X_{3}^{+}(X_{1}^{+},x_{2})}{\bar {n}}_{1}Q_{1}(x_{1},x_{2},x_{3})\;dS.}$

On the part of ${\displaystyle \partial D}$ that is of greater ${\displaystyle x_{1}}$ the values of ${\displaystyle Q_{1}=Q_{1}(X_{1}^{+},x_{2},x_{3})}$ and ${\displaystyle {\bar {n}}_{1}dS=dx_{2}dx_{3}}$. On the part of the surface that of lesser values of ${\displaystyle x_{1}}$ the values of ${\displaystyle Q_{1}=Q_{1}(X_{1}^{-},x_{2},x_{3})}$ and ${\displaystyle {\bar {n}}_{1}dS=-dx_{2}dx_{3}}$. Here ${\displaystyle {\bar {n}}_{1}\equiv \pm {\hat {x}}_{1}\cdot \mathbf {\hat {n}} }$ on the respective portion of the surface.

Thus, the surface integral for the ${\displaystyle Q_{1}}$ is the same as the volume integral of the ${\displaystyle \partial Q_{1}/\partial x_{1}}$ term:

${\displaystyle \int _{\partial D}{\bar {n_{1}}}Q_{1}\;dS=\int _{X_{2}^{-}(X_{1}^{-},x_{3})}^{X_{2}^{+}(X_{1}^{+},x_{3})}\int _{X_{3}^{-}(X_{1}^{-},x_{2})}^{X_{3}^{+}(X_{1}^{+},x_{2})}\left[Q_{1}(X_{1}^{+},x_{2},x_{3})-Q_{1}(X_{1}^{-},x_{2},x_{3})\right]\;dx_{2}dx_{3}.}$

Here ${\displaystyle X_{1}^{\pm }=X_{1}^{\pm }(x_{2},x_{3}).}$

Similar results may be found for the ${\displaystyle Q_{2}}$ and ${\displaystyle Q_{3}}$, proving the divergence theorem for 3 dimensions.

This same method generalizes for the case of ${\displaystyle m>3}$ dimensions, proving the theorem for the arbitrary case.

References

1. Greenspan, H. P., Benney, D. J., & Turner, J. E. (1986). Calculus: an introduction to applied mathematics. McGraw-Hill Ryerson Ltd.
2. Whaley, J., 2017, Oil in the Heart of South America, https://www.geoexpro.com/articles/2017/10/oil-in-the-heart-of-south-america], accessed November 15, 2021.
3. Wiens, F., 1995, Phanerozoic Tectonics and Sedimentation of The Chaco Basin, Paraguay. Its Hydrocarbon Potential: Geoconsultores, 2-27, accessed November 15, 2021; https://www.researchgate.net/publication/281348744_Phanerozoic_tectonics_and_sedimentation_in_the_Chaco_Basin_of_Paraguay_with_comments_on_hydrocarbon_potential
4. Alfredo, Carlos, and Clebsch Kuhn. “The Geological Evolution of the Paraguayan Chaco.” TTU DSpace Home. Texas Tech University, August 1, 1991. https://ttu-ir.tdl.org/handle/2346/9214?show=full.