Editing Divergence theorem (section)

==Mathematical statement==
[[File:Divergence theorem.svg|thumb|right|250px|A region {{mvar|V}} bounded by the surface {{mvar|S}} = ∂{{mvar|V}} with the surface normal {{mvar|n}}]]

Suppose {{mvar|V}} is a [[subset]] of <math>\mathbb{R}^n</math> (in the case of {{math|''n'' {{=}} 3, ''V''}} represents a volume in [[three-dimensional space]]) which is [[compact space|compact]] and has a [[piecewise]] [[smooth surface|smooth boundary]] {{mvar|S}} (also indicated with <math>\partial V = S</math>). If {{math|'''F'''}} is a continuously differentiable vector field defined on a [[Neighbourhood (mathematics)|neighborhood]] of {{mvar|V}}, then:<ref name="Wiley">{{cite book
 | last1  = Wiley
 | first1 = C. Ray Jr.
 | title  = Advanced Engineering Mathematics, 3rd Ed.
 | publisher = McGraw-Hill
 | pages  = 372–373
 }}</ref><ref name="Kreyszig">{{cite book
 | last1  = Kreyszig
 | first1 = Erwin
 | last2  = Kreyszig
 | first2 = Herbert
 | last3  = Norminton
 | first3 = Edward J.
 | title  = Advanced Engineering Mathematics
 | publisher = John Wiley and Sons
 | edition = 10
 | date   = 2011
 | pages  = 453–456
 | url    = https://archive.org/details/AdvancedEngineeringMathematicsKreyszigE.10thEd/page/n477/mode/2up
 | isbn   = 978-0-470-45836-5
 }}</ref>
:{{oiint
| preintegral = <math>\iiint_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)\,\mathrm{d}V=</math>
| intsubscpt = <math>\scriptstyle S</math> 
| integrand = <math>(\mathbf{F}\cdot\mathbf{\hat{n}})\,\mathrm{d}S .</math>
}}
The left side is a [[volume integral]] over the volume {{mvar|V}}, and the right side is the [[surface integral]] over the boundary of the volume {{mvar|V}}. The closed, measurable set <math>\partial V</math> is oriented by outward-pointing [[normal (geometry)|normals]], and <math>\mathbf{\hat{n}}</math> is the outward pointing unit normal at almost each point on the boundary <math>\partial V</math>. (<math>\mathrm{d} \mathbf{S}</math> may be used as a shorthand for <math>\mathbf{n} \mathrm{d} S</math>.)  In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume {{mvar|V}}, and the right-hand side represents the total flow across the boundary {{mvar|S}}.
{{clear}}