The diff command of Vectors is the same as the diff command of Physics, but for two additional things:

It takes into account the geometrical relation between cartesian, cylindrical and spherical coordinates, as well as the coordinate dependence of curvilinear unit vectors. Example: $x=\mathrm{\rho }\cos \left(\mathrm{\phi }\right)$, so diff(x, rho) returns $\cos \left(\mathrm{\phi }\right)$ even when has(x, rho) returns false.

It takes into account any functional dependency of the geometrical coordinates and unit vectors found in the derivand. Example: When the derivand has $x\left(t\right)$ the formula used is $x\left(t\right)=\mathrm{\rho }\left(t\right)\cos \left(\mathrm{\phi }\left(t\right)\right)$ so diff(x(t), rho) (also diff(x(t), rho(t))) returns $\cos \left(\mathrm{\phi }\left(t\right)\right)$.

Although taking into account relations between geometrical coordinates, unit vectors and their functional dependency, is natural when working with vectors, you can turn OFF these two differences between Physics[Vectors][diff] and Physics[diff] using the Setup command, entering Setup(geometricdifferentiation = false). In such a case you can still differentiate taking into account the geometrical relation between curvilinear coordinates unit vectors by passing the optional argument geometricdifferentiation.

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

The result of diff is always expressed in the coordinate system of the differentiation variable. When that is ambiguous (e.g. $z$ 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 $\mathrm{\phi }$, that could be cylindrical or spherical.

In the derivand, the cylindrical and spherical coordinates and related unit vectors can have functional dependency, say as in $\mathrm{\rho }\left(t\right)$, or for a unit vector, $\mathrm{\_\ρ}\left(\mathrm{\phi }\left(t\right)\right)$, 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 $\frac{\ⅆ}{\ⅆt}\mathrm{\_\ρ}\left(\mathrm{\phi }\left(t\right)\right)$ taking into account that $\frac{\ⅆ\mathrm{\_\ρ}}{\ⅆ\mathrm{\phi }}=\mathrm{\_\φ}$.

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

If $q$ does not belong to $\left\{\mathrm{\phi }\,r\,\mathrm{\rho }\,\mathrm{\theta }\,x\,y\,z\right\}$ (the geometrical coordinates - see conventions), then send the task to Physics[diff] returning $\mathrm{Physics}:-\mathrm{diff}\left(a\,q\right)$.

Otherwise,  if $A$ is a projected vector then

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

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

the differentiation is performed using the standard Physics[diff];

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

If $A$ 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

$\mathrm{with}\left(\mathrm{Physics}\left[\mathrm{Vectors}\right]\right)$

$\left[\mathrm{\&x}\,\mathrm{`+`}\,\mathrm{`.`}\,\mathrm{Assume}\,\mathrm{ChangeBasis}\,\mathrm{ChangeCoordinates}\,\mathrm{CompactDisplay}\,\mathrm{Component}\,\mathrm{Curl}\,\mathrm{DirectionalDiff}\,\mathrm{Divergence}\,\mathrm{Gradient}\,\mathrm{Identify}\,\mathrm{Laplacian}\,\nabla \,\mathrm{Norm}\,\mathrm{ParametrizeCurve}\,\mathrm{ParametrizeSurface}\,\mathrm{ParametrizeVolume}\,\mathrm{Setup}\,\mathrm{Simplify}\,\mathrm{`^`}\,\mathrm{diff}\,\mathrm{int}\right]$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$R≔\mathrm{\_\ρ}$

$R≔\stackrel{\wedge }{\mathrm{\rho }}$

$\mathrm{diff}\left(R\,x\right)$

$-\frac{y\stackrel{\wedge }{\mathrm{\phi }}}{x^2+y^2}$

$\mathrm{diff}\left(R\,y\right)$

$\frac{x\stackrel{\wedge }{\mathrm{\phi }}}{x^2+y^2}$

$\mathrm{diff}\left(R\,z\right)$

$0$

$\mathrm{diff}\left(x\,x\,\mathrm{\rho }\right)$

$0$

$\mathrm{simplify}\left(\mathrm{diff}\left(x\,\mathrm{\rho }\,x\right)\right)$

$\frac{y^2}{{\left(x^2+y^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

$\mathrm{\_\ρ}\left(\mathrm{\phi }\left(t\right)\right)$

$\stackrel{\wedge }{\mathrm{\rho }}\left(\mathrm{\phi }\left(t\right)\right)$

$\mathrm{diff}\left(\,\mathrm{\phi }\left(t\right)\right)$

$\stackrel{\wedge }{\mathrm{\phi }}\left(t\right)$

$\mathrm{diff}\left(\,t\right)$

$\stackrel{\mathbf{.}}{\phi }\left(t\right)\stackrel{\wedge }{\mathrm{\phi }}\left(t\right)$

$\mathrm{Setup}\left(\mathrm{geom}=\mathrm{false}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{geom}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{geometricdifferentiation}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{geometricdifferentiation}=\mathrm{false}\right]$

$\mathrm{diff}\left(\,t\right)$

$\mathrm{D}\left(\stackrel{\wedge }{\mathrm{\rho }}\right)\left(\mathrm{\phi }\left(t\right)\right)\stackrel{\mathbf{.}}{\phi }\left(t\right)$

$\mathrm{diff}\left(R\,x\right)$

$0$

$\mathrm{diff}\left(R,x,\mathrm{geometricdifferentiation}\right)$

$-\frac{y\stackrel{\wedge }{\mathrm{\phi }}}{x^2+y^2}$