11. (a) Consider the rectangular region R in the plane with boundary curve C as shown below: y C 2 R 4 Let F(x, y) = (3 + 5)yi + xj be a vector field in the plane, R2. Use Green's theorem in the plane to calculate F . dr. (b) Consider the region D in the plane with boundary curve OD as shown below: y D aD Suppose F(x, y) = M(x, y) i+ N(x, y) j is a vector field in the plane, R2, which satisfies the condition that F = Vf for some smooth function f defined on R2. Suppose also that F satisfies the conditions of Green's Theorem in the plane. (i) Show that ON OM ax ay (ii) Explain why M dx + N dy = 0