VectorDiff computes the partial derivative of the expression , taking into account the geometrical relation existing between cartesian, cylindrical and spherical coordinates, as well as the coordinate dependence of curvilinear unit vectors. Though for people usually working with vectors VectorDiff works "as expected", from the computational point of view its user must take some care:  does not return 0, for instance, even when has() returns false.

The %VectorDiff is the inert form of VectorDiff, that is: it represents the same mathematical operation while holding the operation unperformed. To activate the operation use value.

The result of VectorDiff is always expressed in the coordinate system of the differentiation variable. When that is ambiguous (e.g.  may be cartesian or cylindrical), the ambiguity is resolved looking at the derivand, whether it is a cartesian or cylindrical vector, and when it is neither, then cartesian coordinates are used. The same approach is used when the differentiation variable is , that could be cylindrical or spherical.

In the derivand, the cylindrican and spherical coordinates and related unit vectors can have functional dependency, say as in , or for a unit vector, , and the differentiation variables can be names or functions, as it is the case when using the Physics[diff] command. This permits computing things for instance like  taking into account that .

In the results, all unevaluated , where  is in turn a non-projected vector, are substituted by unevaluated . So the differentiation knowledge of the standard diff is taken into account when evaluating derivatives using VectorDiff. Note however that, though high order derivatives w.r.t coordinates of the same type commute, this is not true w.r.t coordinates of different types. Thus, when evaluating high order derivatives with VectorDiff, the order of the differentiation variables is respected and the evaluation happens from the inside to the outside.

The computation of VectorDiff(A, q) is performed as follows.

If  does not belong to  (the geometrical coordinates - see conventions), then send the task to diff returning .

Otherwise,  if  is a projected vector then

 is reprojected in the cartesian orthonormal basis (using ChangeBasis), where unit vectors are constant;

a change of variables if performed on the components of  (using dchange), in order to express  in the coordinate system to which  belongs;

the differentiation is performed using the standard diff;

the result is reprojected into the original orthonormal basis and returned.

If  is a non projected vector or a scalar function, the task is restricted to steps 2. and 3. above.

For the conventions about the geometrical coordinates and vectors see Identify