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:

  1. 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.

  2. Even if we consider some conservative vector field, this problem is problematic some values of $x$, $y$ has multiple corresponding values. See the pictures. enter image description hereenter image description here

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.

  1. 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.

  • $\begingroup$ To use the fundamental theorem of line integrals, you need to know the $(x,y)$-coordinates of the starting and ending point. Knowing just the $x$-coordinate is not enough. $\endgroup$ – Nick Feb 25 '18 at 14:32
  • $\begingroup$ Okay, I will edit it. @Nick I can approximately calculate the corresponding $y$ values for the $x$s. $\endgroup$ – Partha Sarker Feb 25 '18 at 15:19
  • $\begingroup$ @Nick But, That doesn't my other question. Is it really unsolvable for non-conservative vector fields? $\endgroup$ – Partha Sarker Feb 25 '18 at 15:20

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 $\sigma$.


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