Past exam of the mathematical methods II course of the University of Oxford / 2021 resit / 9 / c / Solution

An exact differential is an expression $Pdx+Qdy$ that equals $df=f_{x}dx+f_{y}dy$ for a single-valued differentiable function $f$ on the domain. Its line integral is an endpoint difference by the fundamental theorem for line integrals.

Integrating $f_x=P$ with respect to $x$ gives $f=2x^5-x^2{y}^3+g(y)$. Matching $f_y=-3x^2{y}^2+g^{\prime }(y)$ to $Q$ requires $g^{\prime }(y)=0$. HenceThe arbitrary additive constant cancels. This constructs the vector-field potential globally and provides a second proof of path independence.