If f is a scalar field and f is a vector field , which one of the following expressions does not make sense?

Question

If f is a scalar field and f is a vector field , which one of the following expressions does not make sense?

1)▽×f = (2x,3yz,-xz^2)

2)▽(▽•f)

3)(▽×f)×(▽f)

4)▽•(▽f)

Answer 1

The first is not because if you take the divergence then it is naot zero. Indeed $$ \nabla \cdot (\nabla \times \mathbf f) = \nabla \cdot (2x,3yz,-xz^2)^T = 2+3z-2xz \ne 0 $$ I assume you know that $\nabla \cdot \nabla \times= 0$