Div
Div[{f1,…,fn},{x1,…,xn}]
gives the divergence 
.
Div[{f1,…,fn},{x1,…,xn},chart]
gives the divergence in the coordinates chart.
Details
- Div is also known as the contracted covariant derivative.
- Div[f,x] can be input as ∇x.f. The character ∇ can be typed as
del
or \[Del], and the character . is an ordinary period. The list of variables x is entered as a subscript. - An empty template ∇. can be entered as
del.
, and 
moves the cursor from the subscript to the main body. - All quantities that do not explicitly depend on the variables given are taken to have zero partial derivative.
- In Div[f,x], if f is an array of dimensions {n1,…,nk-1,nk}, then x must have length nk, and the resulting divergence is an array of dimensions {n1,…,nk-1}.
- In Div[f,{x1,…,xn},chart], if f is an array, then it must have dimensions {n,…,n}. The components of f are interpreted as being in the orthonormal basis associated with chart.
- For coordinate charts on Euclidean space, Div[f,{x1,…,xn},chart] can be computed by transforming f to Cartesian coordinates, computing the ordinary divergence, and transforming back to chart. »
- A property of Div is that if chart is defined with metric g, expressed in the orthonormal basis, then Div[g,{x1,…,xn]},chart] gives zero. »
- Coordinate charts in the third argument of Div can be specified as triples {coordsys,metric,dim} in the same way as in the first argument of CoordinateChartData. The short form in which dim is omitted may be used.
- Div[f,VectorSymbol[…]] computes the divergence with respect to the vector symbol. »
- Div works with SparseArray and structured array objects.
Examples
open allclose allBasic Examples (4)
Divergence of a vector field in Cartesian coordinates:
Divergence of a vector field in cylindrical coordinates:
Divergence in two-dimensional polar coordinates:
Use 
del
to enter ∇ and
to enter the list of subscripted variables:
Use 
del.
to enter the template ∇., fill in the variables, press 
, and fill in the function:
Scope (6)
In a curvilinear coordinate system, a vector with constant components may have a nonzero divergence:
Divergence of a rank-2 tensor:
Divergence specifying metric, coordinate system, and parameters:
Div works on curved spaces:
The divergence of ![TemplateBox[{"x", n, TemplateBox[{}, Reals]}, VectorSymbol3]](Files/Div.en/35.png "TemplateBox[{\"x\", n, TemplateBox[{}, Reals]}, VectorSymbol3]"){kind=small}
with respect to itself is expressed in terms of SymbolicIdentityArray[{n}]:
Applying TensorExpand gives the expected answer, namely the dimension:
The divergence of a constant affine transformation of ![TemplateBox[{"x", n, TemplateBox[{}, Reals]}, VectorSymbol3]](Files/Div.en/36.png "TemplateBox[{\"x\", n, TemplateBox[{}, Reals]}, VectorSymbol3]"){kind=small}
equals the trace of the linear part:
Applications (3)
Determine whether a fluid flow is incompressible:
For the function 
, define the associated conjugate vector field 
:
The Cauchy–Riemann equations for 
are equivalent to 
being divergence free and curl free:
The divergence of the rank-2 stress tensor equals the force at each point of a static elastic medium:
Properties & Relations (9)
Div reduces the rank of array by one:
Div[{f1,f2,…,fn},{x1,x2,…,xn}] is the trace of the gradient of f:
Compute Div in a Euclidean coordinate chart c by transforming to and then back from Cartesian coordinates:
The result is the same as directly computing Div[f,{x1,…,xn},c]:
The two-argument form Div[f,vars] is essentially Listable in its first argument:
Div[array,vars,sys] is effectively Grad followed by TensorContract on the last two slots:
However, this operation is not, in general, Listable:
If chart is defined with metric g, expressed in the orthonormal basis, Div[g,{x1,…,xn},chart] is zero:
Div contracts the innermost index of the array, which for matrices means acting on rows:
To contract into a different index, use Grad followed by an explicit TensorContract:
In the matrix case, acting on columns can be achieved by first transposing the matrix square:
The divergence of a curl is zero:
Even for non-vector inputs, the result is zero:
This identity is respected by the Inactive form of Div:
Div preserves the structure of SymmetrizedArray objects:
The divergence maintains symmetries not involving the final slot:
Text
Wolfram Research (2012), Div, Wolfram Language function, https://reference.wolfram.com/language/ref/Div.html (updated 2024).
CMS
Wolfram Language. 2012. "Div." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Div.html.
APA
Wolfram Language. (2012). Div. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Div.html