1) grad is an operation , that works on a scalar function
2) Embedding grad in a VectorPlot3D command :
grad[f_] := {D[f, x], D[f, y], D[f, z]}
VectorPlot3D[
Evaluate[grad[x + y^3 Cos[x] - Sin[z]]], {x, 0, 3}, {y, 0, 3}, {z, 0, 3}]
3) that should work with curl as well
4) div is a scalar, that should be plotted in another way
div[a_] := D[a, x] + D[a, y] + D[a, z]
ContourPlot3D[
Evaluate[div[{x^2, Sin[x y], Exp[- x y z]}]], {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]
