EXAMPLE 5 If F(x, y, z) = 4y2i + (8xy + 4e4z)j + 16ye4zk, find a function f such that ∇f = F. SOLUTION If there is such a function f, then

If there is such a scalar function f, then

Integrate both sides of the first equation with respect to x :

Differentiate both sides with respect to y :

Integrate both sides with respect to y :

Plug this into the equation above with f , then differentiate both sides with respect to z :

Integrate both sides with respect to z :

So we end up with

answer: 48

just remember to use pemdas:

parentheses

expressions

multiplication

division

addition

subtraction