Why isn't the implicit (explicit ?) definition of curl something along these lines:
$$curl \ \mathbf{F} \overset{\underset{\mathrm{def}}{}}{=} \lim_{V \to 0}\left( \frac{1}{V}\int_{\partial V} \mathbf{\hat{n}} \times (\mathbf{F} - (\mathbf{F} \cdot \mathbf{\hat{n}})\mathbf{\hat{n}}) \ dS\right)$$
vs. the actual definition:
$$(curl \ \mathbf{F}) \cdot \mathbf{\hat{n}} \ \overset{\underset{\mathrm{def}}{}}{=} \lim_{A \to 0}\left( \frac{1}{|A|}\oint_{C} \mathbf{F} \cdot d\mathbf{s}\right)$$
In words, take the integral over a sphere (without loss of generality) of the projection of the vector field onto the tangent plane of the sphere. Finally, divide by the volume of the sphere and take the limit as the sphere shrinks to zero volume. The cross product converts the tangent to an axis tangent.
Pardon me if the equation is not entirely correct - it's the idea I'm trying emphasize.
Intuitively, this is how I would define curl. What's wrong with it?
