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:

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 with respect to itself is expressed in terms of SymbolicIdentityArray[{n}]:

Applying TensorExpand gives the expected answer, namely the dimension:

The divergence of scaled by its squared norm:

The divergence of a constant affine transformation of equals the trace of the linear part:

The power rule for a divergence is the sum of the power rule for the individual components:

Write out this result in three dimensions:

The divergence of the coordinate vector in n dimensions:

Activate the sum to get the simple result:

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:

Div reduces the rank of array by one:

Div[{f₁,f₂,…,f_n},{x₁,x₂,…,x_n}] 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,…,x_n},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,…,x_n},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:

View expressions for the divergence of a vector function in different coordinate systems: