Question: 4. Let OD be a piecewise CI, simple, closed curve, oriented counterclockwise, enclosing a simply-connected region D in the plane. Let n be the outward

4. Let OD be a piecewise CI, simple, closed curve, oriented counterclockwise, enclosing a simply-connected region D in the plane. Let n be the outward unit normal vector to D. Given two functions, f(x,y) and g(x, y), both C- on an open set containing the domain D, establish the following. (a) $ (109) . di = - $ (9vs) . di. aD aD (b) ( sv 2 9 ) dady - f ( 1vg ) . ads - (vs . V9 ) dady . D aD D (c ) ( v 2 9 - 9021 ) dady = $(1vg-9vs)-ids. D aD In this question, you may need to use the product rules V (fg) = fVg + gVf and V.(fF) = F. VS+ SV.F. where V is del operator, f and g are scalar fields, and F is vector field, all defined in R2. Also, recall that V2g = V . Vg = grz + guy, etc.]

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