Difference between revisions of "Dictionary:Divergence theorem"

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The flux <math>\phi </math> through a surface (or the integral of the vector flux density <b>g</b> over a closed surface) equals the divergence of the flux density integrated over the volume contained by the surface:  
 
The flux <math>\phi </math> through a surface (or the integral of the vector flux density <b>g</b> over a closed surface) equals the divergence of the flux density integrated over the volume contained by the surface:  
 
<center><math>\phi = \iint \textbf{g} \cdot d\textbf{s}=\iiint \nabla \cdot \textbf{g}\; dx dy dz </math>. </center>
 
<center><math>\phi = \iint \textbf{g} \cdot d\textbf{s}=\iiint \nabla \cdot \textbf{g}\; dx dy dz </math>. </center>
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Another way of writing this is
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<center><math>\phi = \iint_S \textbf{g} \cdot \mathbf{\hat{n}} \; dS =\iiint_{V} \nabla \cdot \textbf{g}\; dV </math>, </center>
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where <math> dS </math> is the area element on the bounding surface <math> S </math> of volume <math> V </math>, with volume element <math> dV </math>.
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<!--T:2-->
 
<!--T:2-->
Commonly called <b>Gauss's theorem</b>.
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Commonly called <b>Gauss's theorem</b> or the <b> Gauss-Ostrogradski theorem </b>.
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== Generalization to arbitrary dimensions ==
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The divergence theorem is a general mathematical result that may be applied in arbitrary dimensions.
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If we have an integral over the volume <math> D </math> in <math> R^m </math> bounded by <math> \partial D </math> in <math> R^{m-1} </math> then the following relation holds
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<center> <math> \int_D \mathbf{\nabla} \cdot \mathbf{Q} \; dV = \int_{\partial D} \mathbf{\hat{n}} \cdot \mathbf{Q} \; dS.  </math> </center>
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Here, <math> Q</math> is a vector field and the integrand of the (hyper-)volume integral is an exact divergence of that vector field. The quantity <math>  \mathbf{\hat{n}}=({\bar{n}}_1, {\bar{n}}_2, ... \bar{{n}}_m),</math> is the outward-pointing unit
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normal vector to the (hyper-) surface <math> \partial D. </math> Each <math> {\bar{n}}_k </math> is the respective direction cosine
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obtained by forming the inner product of the unit normal to the boundary <math> \partial D </math> with the respective <math> \hat{x}_k </math> coordinate axis unit vector.
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Because we are in arbitrary dimensions <math> \mathbf{x} = (x_1, x_2, ..., x_m ) </math> and <math> \nabla \equiv \left( \frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, ..., \frac{\partial}{\partial x_m}\right) </math>.
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=== Proof<ref>Greenspan, H. P., Benney, D. J., & Turner, J. E. (1986). Calculus: an introduction to applied mathematics. McGraw-Hill Ryerson Ltd.</ref> ===
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Consider the case when <math> m=3 </math> and <math> D </math> is a three-dimensional volume bounded by the two-dimensional
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boundary <math> \partial D </math>. The volume integral may be written in three terms, one for each coordinate direction
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<center> <math> \int_D \mathbf{\nabla} \cdot \mathbf{Q} \; dV = \int_D \frac{\partial Q_1}{\partial x_1 } \; dV +
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\int_D \frac{\partial Q_2}{\partial x_2 } \; dV + \int_D \frac{\partial Q_3}{\partial x_3 }\; dV .</math> </center>
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If we concentrate on the <math> x_1 </math> volume integral and apply the fundamental theorem of calculus we obtain
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<math> \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 </math>
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<center> <math>
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=  \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 .
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</math> </center>
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Here, the functions <math> X_k^+ </math> and <math> X_k^- </math> define points on <math> \partial D </math> of greater
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<math> x_k </math> and lesser <math> x_k, </math> respectively.
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Now, if we consider the surface integral
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<center> <math> \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  </math> </center>
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and concentrate on the <math> x_1 </math> term
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<center> <math> \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)}
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\bar{n}_1 Q_1(x_1,x_2,x_3) \; dS .</math> </center>
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On the part of <math> \partial D </math> that is of greater <math> x_1 </math> the values of <math> Q_1 =  Q_1(X_1^+,x_2,x_3) </math>
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and <math>\bar{n}_1 dS = dx_2 dx_3</math>.  On the part of the surface that of lesser values of <math> x_1 </math> the values of <math> Q_1 =  Q_1(X_1^-,x_2,x_3) </math> and <math>\bar{n}_1 dS = - dx_2 dx_3</math>.  Here <math> \bar{n}_1 \equiv \pm \hat{x}_1 \cdot \mathbf{\hat{n}} </math> on the respective portion of the surface.
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Thus, the surface integral for the <math> Q_1 </math> is the same as the  volume integral of the <math> \partial Q_1/\partial x_1</math>
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term:
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<center> <math>
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\int_{\partial D} \bar{n_1}Q_1 \; dS =
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\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.
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</math> </center>
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Here <math> X_1^\pm = X_1^\pm (x_2,x_3).</math>
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Similar results may be found for the <math> Q_2 </math> and <math> Q_3 </math>, proving the divergence theorem
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for 3 dimensions.
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This same method generalizes for the case of <math> m > 3 </math> dimensions, proving the theorem for the arbitrary case.
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== References ==
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{{reflist}}
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</translate>
 
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Latest revision as of 16:07, 17 August 2021

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The flux 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:

.

Another way of writing this is

,

where is the area element on the bounding surface of volume , with volume element .



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 in bounded by in then the following relation holds

Here, is a vector field and the integrand of the (hyper-)volume integral is an exact divergence of that vector field. The quantity is the outward-pointing unit normal vector to the (hyper-) surface Each is the respective direction cosine obtained by forming the inner product of the unit normal to the boundary with the respective coordinate axis unit vector.

Because we are in arbitrary dimensions and .

Proof[1]

Consider the case when and is a three-dimensional volume bounded by the two-dimensional boundary . The volume integral may be written in three terms, one for each coordinate direction

If we concentrate on the volume integral and apply the fundamental theorem of calculus we obtain




Here, the functions and define points on of greater and lesser respectively.

Now, if we consider the surface integral

and concentrate on the term

On the part of that is of greater the values of and . On the part of the surface that of lesser values of the values of and . Here on the respective portion of the surface.

Thus, the surface integral for the is the same as the volume integral of the term:

Here

Similar results may be found for the and , proving the divergence theorem for 3 dimensions.

This same method generalizes for the case of dimensions, proving the theorem for the arbitrary case.

References

[2]
[3]
[4]

  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.