Editing Divergence theorem (section)

==Generalizations==

===Multiple dimensions===
One can use the [[Generalized Stokes' theorem|generalised Stokes' theorem]] to equate the {{mvar|n}}-dimensional volume integral of the divergence of a vector field {{math|'''F'''}} over a region {{mvar|U}} to the {{math|(''n'' − 1)}}-dimensional surface integral of {{math|'''F'''}} over the boundary of {{mvar|U}}:

:<math> \underbrace{ \int \cdots \int_U }_n \nabla \cdot \mathbf{F} \, \mathrm{d}V = \underbrace{ \oint_{} \cdots \oint_{\partial U} }_{n-1} \mathbf{F} \cdot \mathbf{n} \, \mathrm{d}S </math>

This equation is also known as the divergence theorem.

When {{math|''n'' {{=}} 2}}, this is equivalent to [[Green's theorem]].

When {{math|''n'' {{=}} 1}}, it reduces to the [[fundamental theorem of calculus]], part 2.

===Tensor fields===
{{main|Tensor field}}

Writing the theorem in [[Einstein notation]]:

:{{oiint
| preintegral = <math>\iiint_V \dfrac{\partial \mathbf{F}_i}{\partial x_i} \mathrm{d}V=</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>\mathbf{F}_i n_i\, \mathrm{d}S </math>
}}

suggestively, replacing the vector field {{math|'''F'''}} with a rank-{{mvar|n}} tensor field {{mvar|T}}, this can be generalized to:<ref>{{cite book|author1=K.F. Riley |author2=M.P. Hobson |author3=S.J. Bence | title=Mathematical methods for physics and engineering|url=https://archive.org/details/mathematicalmeth00rile |url-access=registration | publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}</ref>

:{{oiint
| preintegral = <math>\iiint_V \dfrac{\partial T_{i_1i_2\cdots i_q\cdots i_n}}{\partial x_{i_q}} \mathrm{d}V=</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>T_{i_1i_2\cdots i_q\cdots i_n}n_{i_q}\, \mathrm{d}S .</math>
}}

where on each side, [[tensor contraction]] occurs for at least one index. This form of the theorem is still in 3d, each index takes values 1, 2, and 3. It can be generalized further still to higher (or lower) dimensions (for example to 4d [[spacetime]] in [[general relativity]]<ref>see for example: <br />{{cite book |pages=85–86, §3.5|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne | title=Gravitation| publisher=W.H. Freeman & Co| year=1973 | isbn=978-0-7167-0344-0|title-link=Gravitation (book) }}, and <br />{{cite book |author=R. Penrose| title=The Road to Reality| publisher= Vintage books| year=2007 | isbn=978-0-679-77631-4| title-link=The Road to Reality}}</ref>).