Determine if the following expressions result in a scalar or vector field. If undefined, please explain why. $F(x,y,z)$ is a vector field in space and $f(x,y,z)$ is a scalar field in space.


*

*$\nabla \times (\nabla(\nabla \cdot F))$

*$\nabla \times (\nabla \cdot (\nabla f))$

*$ \nabla (\nabla \cdot (\nabla \times F))$

*$\nabla(\nabla \times (\nabla \cdot F))$

*$\nabla \cdot (\nabla \times (\nabla f))$

*$\nabla \cdot (\nabla (\nabla \times f))$
I'm trying to study for a multivariable final and I am having trouble understanding when and why these expressions become undefined.  

For example, in expression 2:


*

*$f$ is a scalar field

*$\nabla f$ is its gradient: it is a vector field

*$\nabla \cdot \nabla f$ is the divergence of this vector field, it is a scalar field

*$\nabla \times (\nabla \cdot \nabla f)$ is the curl of this scalar field: this is undefined as the curl operator takes a vector field as input.

