I'm not sure if my understanding about curl is correct but I hope someone can show me the truth. According to the definition, curl of a vector field is:
$(\nabla \times \textbf {F} )(p)\cdot \hat {\textbf {n}} \ {\overset {\underset {\textrm {def} }{}}{=}}\lim _{A\to 0}{\frac {1}{|A|}}\oint _{C}\textbf {F} \cdot \textrm {d} \textbf {r} $
so at a point where field direction is tangent to the radius of the area and magnitude is a constant (i.e., 1), like the following:
the curl will be:
$(\nabla \times \textbf {F} )(p)\cdot \hat {\textbf {n}} =\lim _{r\to 0}{\frac {1}{\pi r^2}}2 \pi r \hat {\textbf {n}}= \lim _{r\to 0} \frac{2}{r}\hat {\textbf {n}} = \infty$
but my common sense tell me that is not correct. Did I get it wrong?