11. Let F be Cl and exact on R2 \ {(O,O)} (see Exercise 8b).

(a) Suppose that C1 and C2 are disjoint smooth simple curves, oriented in the counterclockwise direction, and E is a two-dimensional region whose topological boundary aE is the union of C1 and C2 . (Note: This means that E has a hole with one of the Cj's as the outer boundary and the other as the inner boundary.) If (0,0) tf- E, prove that 1 F· T ds = 1 F· T ds.

c1 c2

(b) Suppose that E is a two-dimensional region that satisfies (0,0) E EO. If aE is a smooth simple curve oriented in the counterclockwise direction, and

( -y x) F(x,y)= X2+y2'x2+y2 '

compute feE F . T ds.

(c) State and prove an analogue of part

(a) for functions F : R3 \ {(O, 0, O)}, three-dimensional regions, and smooth surfaces.