Editing Vector calculus (section)

=== Integral theorems ===
The three basic vector operators have corresponding theorems which generalize the [[fundamental theorem of calculus]] to higher dimensions:

{| class="wikitable" style="text-align:center"
|+Integral theorems of vector calculus
|-
!scope="col"| Theorem
!scope="col"| Statement
!scope="col"| Description
|-
!scope="row"| [[Gradient theorem]]
| <math> \int_{L \subset \mathbb R^n}\!\!\! \nabla\varphi\cdot d\mathbf{r} \ =\ \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right)\ \  \text{ for }\ \ L = L[p\to q]   </math>
| The [[line integral]] of the gradient of a scalar field over a [[curve]] {{math|''L''}} is equal to the change in the scalar field between the endpoints {{math|''p''}} and {{math|''q''}} of the curve.
|-
!scope="row"| [[Divergence theorem]]
| <math> \underbrace{ \int \!\cdots\! \int_{V \subset \mathbb R^n} }_n (\nabla \cdot \mathbf{F}) \, dV 
\  = \ \underbrace{ \oint \!\cdots\! \oint_{\partial V} }_{n-1} \mathbf{F} \cdot d \mathbf{S} </math>
| The integral of the divergence of a vector field over an {{mvar|n}}-dimensional solid {{math|''V''}} is equal to the [[flux]] of the vector field through the {{math|(''n''−1)}}-dimensional closed boundary surface of the solid.
|-
!scope="row"| [[Kelvin–Stokes theorem|Curl (Kelvin–Stokes) theorem]]
| <math> \iint_{\Sigma\subset\mathbb R^3} (\nabla \times \mathbf{F}) \cdot d\mathbf{\Sigma} \ =\ \oint_{\partial \Sigma} \mathbf{F} \cdot d \mathbf{r} </math>
| The integral of the curl of a vector field over a [[Surface (topology)|surface]] {{math|Σ}} in <math>\mathbb R^3</math> is equal to the circulation of the vector field around the closed curve bounding the surface.
|-
!scope="row" colspan=5|<math>\varphi</math> denotes a scalar field and {{mvar|F}} denotes a vector field
|}

In two dimensions, the divergence and curl theorems reduce to the Green's theorem:

{| class="wikitable" style="text-align:center"
|+Green's theorem of vector calculus
|-
! scope="col"| Theorem
! scope="col"| Statement
! scope="col"| Description
|-
!scope="row"| [[Green's theorem]]
| <math> \iint_{A\,\subset\mathbb R^2} \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) dA \ =\ \oint_{\partial A} \left ( L\, dx + M\, dy \right ) </math>|| The integral of the divergence (or curl) of a vector field over some region {{math|''A''}} in <math>\mathbb R^2</math> equals the flux (or circulation) of the vector field over the closed curve bounding the region.
|-
!scope="row" colspan=5|For divergence, {{math|1=''F'' = (''M'', −''L'')}}. For curl, {{math|1=''F'' = (''L'', ''M'', 0)}}. {{mvar|L}} and {{mvar|M}} are functions of {{math|(''x'', ''y'')}}.
|}