Find the work done by the force field f(x, y, z) = y + z, x + z, x + y on a particle that moves along the line segment from (1, 0, 0) to (5, 3, 2).

Note that the vector field is irrotational, since



which means is conservative. This means there is some scalar potential function such that . If we can find such a function, then we only need to evaluate the difference of and (because the gradient theorem holds in this case).

We're looking for a function that satisfies





Integrate the first equation with respect to to get



Differentiate with respect to , then we must have






Differentiate with respect to , and we have to have






Now, by the gradient theorem, we have



where is the line segment.