Why Can Only Directional Derivatives Be Computed for Scalar Fields?

[Jhenrique]

Given a vector field f, I can compute the rotational tendency in the direction n (∇×f·n), the translational tendency in the direction t (∇f·t) and the divergence (∇·f) too. So, given a scalar field f, why I can compute only the directional derivative (translational tendency (∇f·t)) in the wanted direction?

 

[chogg]

Because what would it even mean for a scalar field to have a rotational tendency, or to diverge?