Past exam of the mathematical methods II course of the University of Oxford / 2015 resit / 8 / b / Solution

Set $A=10x^4-2x{y}^3$ and $B=-3x^2{y}^2$. Their mixed derivatives agree: $A_y=-6x{y}^2=B_x$. Both components are smooth on all of ${\mathbb{R}}^2$, a simply connected space, so the line integral is path independent.

Choose a horizontal segment from $(0,0)$ to $(2,0)$ followed by a vertical segment to $(2,1)$. On the first, $y=0$ and $dy=0$; on the second, $x=2$ and $dx=0$. ThusThe selected path and its orientation are shown below. No contribution comes from a differential whose coordinate is held constant on a segment.