Is it possible to execute line integrals of non-conservative vector fields on curves defined by implicit relation such as $\sin(xy)=x+y$? I want to execute the line integral (analytically) of a vector field over the curve defined by implicit function $\sin(xy)=x+y$ for some $x=a$ & $y=c$ to $x=b$ & $y=d$.
The difficulties I face executing this problem are:


*

*There can no explicit parametrization of $x$ and $y$ wrt to some parameter $t$. Ted Shifrin has pointed to me in chat that this problem has become unsolvable for non-conservative vector fields for this reason. But, for conservative vector fields, the problem becomes trivial to execute.

*Even if we consider some conservative vector field, this problem is problematic some values of $x$, $y$ has multiple corresponding values. See the pictures. 
Then, to use the fundamental theorem of line integral we need to divide the curve into several parts where the $\sin(xy)=x+y$ relation becomes a function.


*And last of all , is this problem really (analytically) unsolvable for non-conservative vector fields? I can always numerically execute the integral (of course, approximately). But, that's not my point here. Is there any novel technique present in the to-date literature to somehow overcome the barrier of the non-conservative vector fields?
Any help would be appreciated.
 A: Of course in the case of your example you would have to resort to a numerical integration in the end, even if you had a "magical formula" solving your problem. Maybe the following is of help: 
Assume you have a simple arc $\gamma$ beginning at some point $(x_0,y_0)$ and ending at some point $(x_1,y_1)$. This arc is (part of) the zero set of a given function $f(x,y)$. Connect $(x_1,y_1)$ with $(x_0,y_0)$ by a simple polygonal path $\sigma$ (e.g., two vertical and one horizontal segments), such that $\gamma+\sigma=\partial B$ for a certain simply connected compact domain $B$ in the plane. In order to compute the integral
$$W:=\int_\gamma P(x,y)\,dx+Q(x,y)\,dy$$ we now use Green's theorem:
$$W=\int_B(Q_x-P_y)\>{\rm d}(x,y)-\int_\sigma P\,dx+Q\,dy\ .$$
Here the first integral has to be computed numerically; e.g., using a Monte Carlo method: Produce random points $(x_k,y_k)$ in a rectangle containing $B$ and keep only the points $(x_k,y_k)\in B$ (they in particular would have to satisfy $f(x_k,y_k)<0$). The second integral is easy to compute since we have an explicit parametrization of $\sigma$.
