Calculus 3 Line integrals

Question

Given F=<2yz,2xz+2,2xy+2z>, find a function f so that F=∇f(x,y,z)

Use your answer to evaluate ∫CF⋅dr along the curve C: x=t^2, y=t+2, z=2t−2,  0≤t≤2

Answer 1

It is easy to check that curl F = 0, so a potential function V will exist.

From the way that the second order terms in F are arranged , it is easy to see that V has a term 2 xyz.

From there it is not too hard to guess the other terms to fill out V as:

V = 2 xyz + 2y + z²

The end points on the curve are (0,2,-2) and (4,4,2)

The line integral is thus V(4,4,2) - V(0,2,-2) = 68