How does this definition of curl make sense? In our maths textbook, curl is defined as $\text{curl} \textbf{F} = \nabla \times \textbf{F}$.  If the $\nabla$ operator isn't a vector, but an operation on a vector, how can it be an operand of a cross product?
$$\textbf{F} \in \mathbb R^3 \\ (\nabla) \in \mathbb R^3 \rightarrow \mathbb R^3 \\ (\times) \in \mathbb R^3 \rightarrow \mathbb R^3 \rightarrow \mathbb R^3 \\ \nabla \times \textbf{F} \in \mathbb R^3$$
This seems as silly to me as:
$$2 \in \mathbb R \\ \text{sin} \in \mathbb R \rightarrow \mathbb R \\ (+) \in \mathbb R \rightarrow \mathbb R \rightarrow \mathbb R \\ \text{sin} + 2 \in \mathbb R$$
Obviously, you can't treat sine as a number and add it to 2.  How come you can treat functions as numbers in the definition of curl?  Finding this cross product makes less sense, since you have the determinant of a matrix containing 3 vectors, 3 functions, and 3 scalars. 

$
\mathrm{\nabla}
$
can be regard as a vector
$
\left({\frac{}{\mathrm{\partial}{x}}\mathrm{,}\frac{}{\mathrm{\partial}{y}}\mathrm{,}\frac{}{\mathrm{\partial}{z}}}\right)
$
to participate in calculation
