Answer Happy

(d) For each of the following statements, either briefly explain why it is true or give an example in which it is not. [You may assume that all partial derivatives exist and are continuous.] (i) If F = V f for some scalar function f then V * F = 0. (ii) If $ F. dr = 0 then F = V f for some scalar function f. (iii) The Jacobian of a reverse transformation is the inverse of the forward Jacobian: -1 a(s, t) 2(x, y) = a(x, y) д (iv) The triple integral lll 5(x, y, z) dx dydz can never be negative. (v) When there is a change of variables from (x, y) to (s, t), partial derivatives obey the chain rule: дf дх дf ds до дх