Editing Divergence theorem (section)

==Corollaries==
By replacing {{math|'''F'''}} in the divergence theorem with specific forms, other useful identities can be derived (cf. [[vector identities]]).<ref name=spiegel>{{cite book |author1=M. R. Spiegel |author2=S. Lipschutz |author3=D. Spellman | title = Vector Analysis | edition = 2nd | series = Schaum's Outlines | publisher = McGraw Hill | location = USA | year = 2009 | isbn = 978-0-07-161545-7 }}</ref>

* With <math>\mathbf{F}\rightarrow \mathbf{F}g</math> for a scalar function {{mvar|g}} and a vector field {{math|'''F'''}},

::{{oiint
| preintegral = <math>\iiint_V\left[\mathbf{F}\cdot \left(\nabla g\right) + g \left(\nabla\cdot \mathbf{F}\right)\right] \mathrm{d}V=</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>g\mathbf{F} \cdot \mathbf{n} \mathrm{d}S.</math>
}}

:A special case of this is <math>\mathbf{F} = \nabla f</math>, in which case the theorem is the basis for [[Green's identities]].

* With <math>\mathbf{F}\rightarrow \mathbf{F}\times \mathbf{G}</math> for two vector fields {{math|'''F'''}} and {{math|'''G'''}}, where <math>\times</math> denotes a cross product,

::{{oiint
| preintegral = <math> \iiint_V \nabla \cdot \left( \mathbf{F} \times \mathbf{G}\right) \mathrm{d}V = \iiint_V \left[\mathbf{G}\cdot\left(\nabla\times\mathbf{F}\right) - \mathbf{F}\cdot \left( \nabla\times\mathbf{G}\right)\right]\, \mathrm{d}V =</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>(\mathbf F\times\mathbf{G}) \cdot \mathbf{n} \mathrm{d}S.</math>
}}

* With <math>\mathbf{F}\rightarrow \mathbf{F}\cdot \mathbf{G}</math> for two vector fields {{math|'''F'''}} and {{math|'''G'''}}, where <math>\cdot </math> denotes a [[dot product]],

::{{oiint
| preintegral = <math>\iiint_V \nabla \left( \mathbf{F} \cdot \mathbf{G}\right) \mathrm{d}V = \iiint_V \left[\left(\nabla \mathbf{G}\right) \cdot \mathbf{F} + \left( \nabla \mathbf{F}\right) \cdot \mathbf{G} \right]\, \mathrm{d}V =</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>(\mathbf{F} \cdot \mathbf{G}) \mathbf{n} \mathrm{d}S.</math>
}}

* With <math>\mathbf{F}\rightarrow f\mathbf{c}</math> for a scalar function {{math|&thinsp;''f''&thinsp;}} and vector field '''c''':<ref name=mathworld>[http://mathworld.wolfram.com/DivergenceTheorem.html MathWorld]</ref>

::{{oiint
| preintegral = <math>\iiint_V \mathbf{c} \cdot \nabla f \, \mathrm{d}V =</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>(\mathbf{c} f) \cdot \mathbf{n} \mathrm{d}S - \iiint_V f (\nabla \cdot \mathbf{c})\, \mathrm{d}V.</math>
}}
:The last term on the right vanishes for constant <math>\mathbf{c}</math> or any divergence free (solenoidal) vector field,  e.g. Incompressible flows without sources or sinks such as phase change or chemical reactions etc. In particular, taking <math>\mathbf{c}</math> to be constant:

::{{oiint
| preintegral = <math>\iiint_V \nabla f \, \mathrm{d}V =</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>f\mathbf{n} \mathrm{d}S.</math>
}}

* With <math>\mathbf{F}\rightarrow \mathbf{c}\times\mathbf{F}</math> for vector field {{math|'''F'''}} and constant vector '''c''':<ref name=mathworld/>

::{{oiint
| preintegral = <math>\iiint_V\mathbf{c} \cdot (\nabla\times\mathbf{F}) \, \mathrm{d}V =</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math> (\mathbf{F} \times \mathbf{c}) \cdot \mathbf{n} \mathrm{d}S.</math>
}}

: By reordering the [[triple product]] on the right hand side and taking out the constant vector of the integral,

::{{oiint
| preintegral = <math> \iiint_V (\nabla \times\mathbf{F}) \, \mathrm{d}V \cdot \mathbf{c} = </math>
| intsubscpt = <math> \scriptstyle S</math>
| integrand = <math> (\mathrm{d}\mathbf{S} \times \mathbf{F}) \cdot \mathbf{c}. </math>
}}

: Hence,

::{{oiint
| preintegral = <math> \iiint_V (\nabla \times\mathbf{F}) \, \mathrm{d}V = </math>
| intsubscpt = <math> \scriptstyle S</math>
| integrand = <math> \mathbf{n} \times \mathbf{F} \mathrm{d}S. </math>
}}