In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if $$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \operatorname{PV}\int_{\mathbb{R}^2} \frac{(y_1,y_2,\nabla f(x-y,t)\cdot y)}{[\lvert y\rvert^2 + (x_3 - f(x-y,t)^2)]^{3/2}}\ dy$$ and for $\varepsilon $ positive and $x=(x_1,x_2)$, we define $$ v^1(x_1,x_2,f(x,t),t) =\lim_{\varepsilon \rightarrow 0} v\bigl(x_1-\varepsilon \partial_{x_1} f(x,t) , x_2 - \varepsilon \partial_{x_2} f (x,t), f(x,t) + \varepsilon , t\bigr),$$ then they say \begin{align*} v^1(x_1,x_2,f(x,t),t) ={} & v(x_1,x_2,f(x,t),t) \\ & {}+ \frac{\rho_2-\rho_1}{2}\frac{\partial_{x_1} f(x,t)(1,0,\partial_{x_1}f(x,t))}{1+(\partial_{x_1}f(x,t))^2 +(\partial_{x_2}f(x,t))^2 } \\ & {}+ \frac{\rho_2-\rho_1}{2}\frac{\partial_{x_2} f(x,t)(0,1,\partial_{x_2}f(x,t))}{1+(\partial_{x_1}f(x,t))^2 +(\partial_{x_2}f(x,t))^2 }. \end{align*}
I don't understand how they get that expression for $v^1$. Please, any help or idea is welcome.
