(Bonus 5 points) Let F
=⟨y,z,−x> and let surface S be the paraboloid z=1−x 2
−y 2
. Assume the surface is outward oriented and z≥0. Evaluate ∫ C
​
F
⋅d r
with and without using Stokes' Theorem, where C is the boundary of S (intersection of the paraboloid and the plane z=0. THEOAEM 6.12 Green's Theorem, Circulation Form Let D be an open, simply connected region with a boundary curve C that is a piecewise smooth, simple closed curve oriented counterclockwise (Figure 6.33). Let F=⟨P, Q⟩ be a vector field with component functions that have continuous partial derivatives on D. Then, ∮ C
​
F⋅dr=∮ C
​
Pdx+Qdy=∬ D
​
(Q x
​
−P y
​
)dA.