Curl and Divergence This is an online solution to one of the problems I'm working on. The question is to analyze the statement at the beginning of each sentence and determine whether its meaningful and if so then is it a vector field or scalar field. However I'm confused by B and D and E. Are these answers incorrect? What should the right answers be?

 A: (B) is false.  We can only take gradients of scalar functions and not vector functions, so in fact $\text{grad }f$ makes sense (i.e. is not meaningless).
(D) is true: $\text{grad }f$ is a vector field.
(E) is true.
A: $f$ = scalar field, $F$ = vector field
Gradient: The vector derivative of a scalar field $f$ is called the gradient, and it can be represented as:
$$\nabla f = {\partial f \over \partial x} \mathbf{\hat{x}} + {\partial f \over \partial y} \mathbf{\hat{y}} + {\partial f \over \partial z} \mathbf{\hat{z}}$$
Divergence: The divergence of a vector field  $\vec{F}(x, y, z) = F_x \mathbf{\hat{x}}  + F_y \mathbf{\hat{y}} + F_z \mathbf{\hat{z}}$  is a scalar function that can be represented as:
$$\mbox{div}\,\vec F = \nabla \cdot \vec F= {\partial F_x \over \partial x} + {\partial F_y \over \partial y} + {\partial F_z \over \partial z}  $$
Curl: The curl of a vector field $\vec{F}(x, y, z) = F_x\mathbf{\hat{x}}  + F_y\mathbf{\hat{y}} + F_z\mathbf{\hat{z}}$ is a vector function that can be represented as:
$$\mbox{curl}\;\vec F = \nabla \times \vec F = \left( {\partial F_z \over \partial y} - {\partial F_y \over \partial z} \right) \mathbf{\hat{x}} + \left( {\partial F_x \over \partial z} - {\partial F_z \over \partial x} \right) \mathbf{\hat{y}} + \left( {\partial F_y \over \partial x} - {\partial F_x \over \partial y} \right) \mathbf{\hat{z}}$$

Summary:
| Operator   | Input  | Output |
|------------|--------|--------|
| Gradient   | Scalar | Vector |
| Divergence | Vector | Scalar |
| Curl       | Vector | Vector |


Therefore,
(A) True
(B) False
(C) True
(D) True
(E) True
(F) True
A: Largely depends on what f and F are.  If f is a scalar, and F a vector, then the answers are correct.  For a better understanding of what curl and divergence are, check out the book called "Mathematical Mechanic."
