TITLE: Curl as determinant: technically appropriate?
QUESTION [5 upvotes]: I have seen (not on the most prestigious sources) the curl of a function defined thusly:

If $\vec{F}$ is a trivariate vector-valued function of $x$, of $y$ and of $z$ that takes the form $\vec F = a\hat x +b\hat y + c\hat z$ then the curl of the function $\nabla\times\vec{F}$ equals
  $$\left|
\begin{array} \\
\hat x & \hat y & \hat z \\
\partial\over\partial x & \partial\over\partial y & \partial\over\partial z \\
a & b & c \\
\end{array}
\right|$$

I wonder if this is technically correct, because the definition of a determinant hinges on multiplication. For example, $\displaystyle \left| \begin{array} \\ A & B \\ C & D \\ \end{array} \right|$ is defined to equal $AD-BC$, which is of course equivalent to $DA-CB$. By contrast, a differential operator is not a multiplicand/multiplier, and $\displaystyle{{\partial\over\partial x}b} = \displaystyle{\partial b\over\partial x}$ does not carry the same meaning as $\displaystyle{b{\partial\over\partial x}}$.
So, since a (partial) differential operation is not the same as multiplication, is this determinant definition of the curl of a function technically correct?

REPLY [6 votes]: It's best to view the determinant as a mnemonic for the cross product formula rather than a definition, i.e., as some kind of literal determinant involving vectors, differential operators, and component functions.
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