EXAMPLE 5 If F(x, y, z) = 5y2i + (10xy + 2e5z)j + 10yeszk, find a function f such that Vf = F. SOLUTION If there is such a function f, then (1) fx ( X, y, z ) = (2) fy(x, y, z) = 10xy + 25z ) fz ( x, y, z ) = Integrating (1) with respect to x, we get ( 4) f ( x, y, z ) = + g( y, z) where g(y, z) is a constant with respect to x. Then differentiating (4) with respect to y, we have fy ( x, y, z) = + gy ( ), z) and comparison with (2) gives gy ( ) , Z) = Thus g(y, z) = 2yesz + h(z) and we write (4) as f ( x, y, Z) = + 2yesz + h(z). Finally, differentiating with respect to z and comparing with (3), we obtain h' ( z ) = and therefore h(z) = K, a constant. Give the desired function. f ( x, y, Z ) = It is easily verified that Vf = F