This is more of an intuitive question than anything else: the curl of a vector field [itex] \mathbf{F} [/itex], [itex] \nabla \times \mathbf{F} [/itex] is defined by(adsbygoogle = window.adsbygoogle || []).push({});

[tex] (\nabla \times \mathbf{F})\cdot \mathbf{\hat{n}} = \lim_{a \to 0} \frac{\int_{C} \mathbf{F}\cdot d\mathbf{s}}{a} [/tex]

Where the integral is taken around a closed curve [itex] C [/itex], [itex] \mathbf{\hat{n}} [/itex] is the normal unit vector to that curve, and [itex] a [/itex] is the area of the curve.

Now, my question stems from the following: roughly speaking (if we have, for example, a constant vector field and a "flat" curve), the area of the curve decreases as the square of the perimeter, as we make the curve smaller. On the other hand, the line integral of the vector field along the curve decreases proportionally to the perimeter. So how can the ratio of the line integral to the area converge, if the area decreases more rapidly than the integral?

(A similar question could be asked of the divergence, of course.)

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# I Why does curl converge?

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