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Suppose $\vec{F}(x,y,z)$ is a vector field on $\mathbb{R}^3$, and $f(x,y,z)$ is a scalar function on $\mathbb{R}^3$, determine which of the follow expressions are meaningful:

$$ \begin{align*} I: \quad &\text{div }\vec{F} \\\ \\\ II: \quad &\text{curl } f \\\ \\\ III: \quad &\text{grad }(\text{div }\vec{F}) \\\ \\\ IV: \quad &\text{div }\big(\text{curl }(\text{grad }f)\big) \end{align*}$$

A

Only $I, III$

B

Only $I, IV$

C

Only $I, II, III$

D

Only $I, III, IV$

E

$I, II, III, IV$

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