The following path integral arises as a 2D fluid dynamics problem when evaluating the advective flux of a concentration $C(x,y)$ by the 2D velocity field $[u(x,y),v(x,y)]$ across a path between $s_1=[x_1,y_1]$ and $s_2=[x_2,y_2]$ with a layer depth $H(x,y)$:
$$flux=\int_{s_1}^{s_2}C(x,y)H(x,y)[u(x,y),v(x,y)]\cdot\hat{\textbf{n}}\cdot ds$$ where $\hat{\textbf{n}}$ is the unit vector normal to the segment $s$.
Solving this the "right" way by parameterizing $x$ and $y$ using $s$ produces ungainly results, but it occurred to me that using:
$$ds=\sqrt{dx^2+dy^2}$$ and $$\hat{\textbf{n}}=\frac{[-dy,dx]}{\sqrt{dx^2+dy^2}}$$ reduces this to:
$$flux=-\int_{s_1}^{s_2}C(x,y)H(x,y)u(x,y)\cdot dy + \int_{s_1}^{s_2}C(x,y)H(x,y)v(x,y)\cdot dx$$
Can I somehow use this trick? If so, how would I evaluate the integration limits for the two definite integrals in terms of $(x,y)$? Note that both integrals contain a non-integrating variable that cannot be assumed constant.