1. For C2 scalar fields g(r, y) in R', define the differential operator V- by ag =: TA and so Vig(z, y) = By ag (a) Prove that Vg and Vg are orthogonal at all points in R2. 1 (b) Compute div Vg , and simplify your answer as much as possible. (You need to specify any properties that you are using to earn full credit.) 2 (c) If C is a simple smooth closed curve (oriented counter-clockwise) in R2 enclosing a region S, show that (Vg) . dr = Js V2gdA . 3 (d) Let H be an entity such that, using suffix notation, we can write the differential operator as (V) = Hmn arn a Explaining what sort of object H is, find its components. 2