The diff(f, v) calling sequence is an extension of the top-level diff that performs coordinate-wise differentiation on Vectors (including vector fields), in the coordinate systems of the Student[VectorCalculus] package.  If f is not a Vector, the arguments are passed to the top-level diff command.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right)\:$

$\mathrm{diff}\left(⟨t\,t^2\,t^3⟩\,t\right)$

$\left(1\right)e_x\+\left(2t\right)e_y\+\left(3t^2\right)e_z$

$\mathrm{diff}\left(⟨\cos \left(st\right)\,\sin \left(st\right)⟩\,s\,t\right)$

$\left(-\sin \left(st\right)-ts\cos \left(st\right)\right)e_x\+\left(\cos \left(st\right)-ts\sin \left(st\right)\right)e_y$

$\mathrm{SetCoordinates}\left(\mathrm{spherical}\left[r,\mathrm{\phi },\mathrm{\theta }\right]\right)$

${\mathrm{spherical}}_{r,\mathrm{\phi },\mathrm{\theta }}$

$F≔\mathrm{VectorField}\left(⟨ar\,0\,0⟩\right)$

$F≔\left(ar\right){\stackrel{\_}{e}}_r\+\left(0\right){\stackrel{\_}{e}}_{\mathrm{\φ}}\+\left(0\right){\stackrel{\_}{e}}_{\mathrm{\θ}}$

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

$\left(ar\cos \left(\mathrm{\phi }\right){\cos \left(\mathrm{\theta }\right)}^2\sin \left(\mathrm{\phi }\right)+ar\cos \left(\mathrm{\phi }\right){\sin \left(\mathrm{\theta }\right)}^2\sin \left(\mathrm{\phi }\right)-ar\sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\phi }\right)\right){\stackrel{\_}{e}}_r\+\left(ar{\cos \left(\mathrm{\phi }\right)}^2{\cos \left(\mathrm{\theta }\right)}^2+ar{\cos \left(\mathrm{\phi }\right)}^2{\sin \left(\mathrm{\theta }\right)}^2+ar{\sin \left(\mathrm{\phi }\right)}^2\right){\stackrel{\_}{e}}_{\mathrm{\φ}}\+\left(0\right){\stackrel{\_}{e}}_{\mathrm{\θ}}$

$\mathrm{diff}\left(F\,a\right)$

$\left(r\right){\stackrel{\_}{e}}_r\+\left(0\right){\stackrel{\_}{e}}_{\mathrm{\φ}}\+\left(0\right){\stackrel{\_}{e}}_{\mathrm{\θ}}$