Suppose we had a vector function $\textbf{F}=\begin{pmatrix}F_x \\ F_y \\ F_z\end{pmatrix}$. We can compute the curl of the vector function by computing $\nabla \times \textbf{F}= \begin{pmatrix}\frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \times \begin{pmatrix}F_x \\ F_y \\ F_z\end{pmatrix} = \begin{pmatrix}\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\\ \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \\ \frac{\partial F_y}{\partial x} - \frac{\partial F_z}{\partial x} \end{pmatrix} $.
My question is what "operation" are we doing when we carry out the cross product? Are we "multiplying" $\frac{\partial}{\partial y}$ by $F_z$ and subtracting $\frac{\partial}{\partial z}$ "multiplied" by $F_y$ to get the $i$ component of the cross product?
If so, what does it even mean to "multiply" a differential operator by a function?
Thanks,
Jack
Technically $\frac{\partial}{\partial y}$ is an operator. That is, a type of function whose domain is itself a set of functions. Because it's a function, we could write it as $\frac{\partial}{\partial y}(F_z)$ and say "the partial of $F_z$" (just like $f(x)$ is said "$f$ of $x$"), but most of the time the phrase we use is that we "apply" $\frac{\partial}{\partial y}$ to $F_z$ and write it without the parentheses, $\frac{\partial}{\partial y}F_z$ or $\frac{\partial F_z}{\partial y}$.
That said, we do write it like multiplication and even sometimes do like to think of it as a type of multiplication. It shouldn't hurt if you think of it that way as long as you remember that this type of "multiplication" is not commutative.