Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 3 / b / ii / Solution

On the first segment only $z$ varies and $F_z=x^{2}y=0$. On the second only $y$ varies and $F_y=x^{2}z=0$. On the last segment $y=z=\pi$, so the line integral isThe agreement is the expected path independence of a conservative vector field.