Editing Vector calculus (section)

=== 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.