Chapter 14.7, Problem 44E

Average circulation Let S be a small circular disk of radius R centered at the point P with a unit normal vector n. Let C be the boundary of S.

a.    Express the average circulation of the vector field F on S as a surface integral of ▿ × F.

b.    Argue that for small R, the average circulation approaches (▿ × F)|_p·n (the component of ▿ × F in the direction of n evaluated at P) with the approximation improving as R→0.