Consider f and c below. f(x, y, z) = yzexzi + exzj + xyexzk, c. r(t) = (t2 + 2)i + (t2 − 1)j + (t2 − 4t)k, 0 ≤ t ≤ 4 (a) find a function f such that f = ∇f.

We're looking for a scalar function such that its gradient is equal to the given vector-valued function :



That is, we require





Integrating the first equation with respect to gives



Differentiating with respect to gives



but from this step we can see that it's not possible for to be a function of and , independent of . So there is no such function .