a. Show that the outward flux of the position vector field F = xi + yj across any closed curve to which Green’s Theorem applies is twice the area of the region enclosed by the curve.

b. Let n be the outward unit normal vector to a closed curve to which Green’s Theorem applies. Show that it is not possible for F = xi + yj to be orthogonal to n at every point of C.

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THEOREM 5-Green's Theorem (Flux-Divergence or Normal Form) Let C be a piecewise smooth, simple closed curve enclosing a region R in the plane. Let F = Mi + Nj be a vector field with M and N having continuous first partial derivatives in an open region containing R. Then the outward flux of F across C equals the double integral of div F over the region R enclosed by C. fr-nds = M dy-1 Outward flux N dx = J R ƏM ON + dx dy ax dy Divergence integral (4)