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.
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.