In this exercise, we use the notation of the proof of Theorem 1 and prove C F3(x, y, z)k

Chapter 17, Problem 35

(choose chapter or problem)

In this exercise, we use the notation of the proof of Theorem 1 and prove C F3(x, y, z)k dr = S curl(F3(x, y, z)k) dS 12 In particular, S is the graph of z = f (x, y) over a domain D, and C is the boundary of S with parametrization (x(t), y(t), f (x(t), y(t))). (a) Use the Chain Rule to show that F3(x, y, z)k dr = F3(x(t), y(t), f (x(t), y(t)) fx (x(t), y(t))x (t) + fy (x(t), y(t))y (t) dt and verify that C F3(x, y, z)k dr = C0 F3(x, y, z)fx (x, y), F3(x, y, z)fy (x, y) dr where C0 has parametrization (x(t), y(t)). (b) Apply Greens Theorem to the line integral over C0 and show that the result is equal to the right-hand side of Eq. (12). 36.