EXAMPLE 5 If F(x, y, z) = 3y$^2$x + (6xy + 4e$^{3z}$)j + 12ye$^{3z}$k, find a function f such that $
abla$f = F.
SOLUTION If there is such a function f, then
(1) f$_x$(x, y, z) =
(2) f$_y$(x, y, z) = 6xy + 4e$^{3z}$
(3) f$_z$(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
f$_y$(x, y, z) = + g$_y$(y, z)
and comparison with (2) gives
g$_y$(y, z) =
Thus g(y, z) = 4ye$^{3z}$ + h(z) and we write (4) as
f(x, y, z) = + 4ye$^{3z}$ + 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 $
abla$f = F.
