Editing Vector calculus (section)

== Operators and theorems ==
{{main|Vector calculus identities}}
=== Differential operators ===
{{main|Gradient|Divergence|Curl (mathematics)|Laplacian}}
Vector calculus studies various [[differential operator]]s defined on scalar or vector fields, which are typically expressed in terms of the [[del]] operator (<math>\nabla</math>), also known as "nabla". The three basic [[vector operator]]s are:<ref>{{Cite web|title=Differential Operators|url=http://192.168.1.121/math2/differential-operators/|access-date=2020-09-17|website=Math24|language=en-US}}</ref>
{| class="wikitable" style="text-align:center"
|+Differential operators in vector calculus
|-
!scope="col"|Operation
!scope="col"|Notation
!scope="col"|Description
!scope="col"|[[Notation_for_differentiation#Notation_in_vector_calculus|Notational<br/>analogy]]
!scope="col"|Domain/Range
|-
!scope="row"|[[Gradient]]
|<math>\operatorname{grad}(f)=\nabla f</math>
|Measures the rate and direction of change in a scalar field.
|[[Scalar multiplication]]
|Maps scalar fields to vector fields.
|-
!scope="row"|[[Divergence]]
|<math>\operatorname{div}(\mathbf{F})=\nabla\cdot\mathbf{F}</math>
|Measures the scalar of a source or sink at a given point in a vector field.
|[[Dot product]]
|Maps vector fields to scalar fields.
|-
!scope="row"|[[Curl (mathematics)|Curl]]
|<math>\operatorname{curl}(\mathbf{F})=\nabla\times\mathbf{F}</math>
|Measures the tendency to rotate about a point in a vector field in <math>\mathbb R^3</math>.
|[[Cross product]]
|Maps vector fields to (pseudo)vector fields.
|-
!scope="row" colspan=5|{{mvar|f}} denotes a scalar field and {{mvar|F}} denotes a vector field
|}

Also commonly used are the two Laplace operators:

{| class="wikitable" style="text-align:center"
|+Laplace operators in vector calculus
|-
!scope="col"|Operation
!scope="col"|Notation
!scope="col"|Description
!scope="col"|Domain/Range
|-
!scope="row"|[[Laplace operator|Laplacian]]
|<math>\Delta f=\nabla^2 f=\nabla\cdot \nabla f</math>
|Measures the difference between the value of the scalar field with its average on infinitesimal balls.
|Maps between scalar fields.
|-
!scope="row"|[[Vector Laplacian]]
|<math>\nabla^2\mathbf{F}=\nabla(\nabla\cdot\mathbf{F})-\nabla \times (\nabla \times \mathbf{F})</math>
|Measures the difference between the value of the vector field with its average on infinitesimal balls.
|Maps between vector fields.
|-
!scope="row" colspan=4|{{mvar|f}} denotes a scalar field and {{mvar|F}} denotes a vector field
|}

A quantity called the [[Jacobian matrix and determinant|Jacobian matrix]] is useful for studying functions when both the domain and range of the function are multivariable, such as a [[change of variables]] during integration.

=== 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'')}}.
|}