Question

Suppose F = F ( x, y, z ) is a gradient field with F = ∇f, S is a level surface of f, and C is a curve on S. What is the value of the line integral ∫C F ·dr?

Answer 1

if F=gradient F, it's conservative which means your curve can be anything and you will still do the same amount of work. So, it's just f(end)-f(start) meaning final point minus initial point

Answer 2

I hope that helps