Suppose F⃗ (x, y) = 4yi⃗ + 2xyj⃗. Use Green's Theorem to calculate the circulation of F⃗ around the perimeter of a circle C of radius 3 centered at the origin and oriented counter-clockwise.

From Green's theorem, the circulation of a function F (x, y) around a circle is given as

∫ (F (x, y). dA = Area of the circle

π (3^2) - π (0^2) = 9π

Since the result is oriented counter-clockwise, the result will take negative value.

The circulation of F (x, y) is - 9π

Step-by-step explanation:

∫c (4y dx + 2xy dy)

= ∫∫ [ (∂/∂x) (2xy) - (∂/∂y) (4y) ] dA, by Green's Theorem

By integrating the function F (x, y) = 4yi + 2xyj, around the circle, the result is πr2[3, 0], from origin 0, to radius of 3