Let us assume that you simply have a point: $(x_1, x_2).$ You also have a transformation $H,$ that maps this single point to a new point: $(y_1, y_2).$ So $y_1 = h_1(x_1, x_2),$ and $y_2 = h_2(x_1, x_2).$
My question is simple:
Given a new point of $(x_1 + dx_1, x_2),$ what is the new transformed point equal to, in terms of any partials necessary? (aka, partials of $h_1$ w.r.t $x_1,$ or $x_2,$ and $h_2$ w.r.t $x_1$ and $x_2$).
$H$ is differentiable.
So my question is, let us get the first transformed point, $y_1$ in this case. This is equal to $h_1(x_1 + dx_1, x_2)$. What is this actually equal to, in terms of partials? Similarly, get the transformed point $y_2$, in this case will be $h_2(x_1 + dx_1, x_2)$. What is this in terms of the partials?
Edit 2: I am trying to understand bottom part of page 2 of this paper PDF. To be precise, I am trying to understand how the author got $(x_1+dx_1,x_2)→(y_1+a,y_2+b).$ I think it should be $(x_1+dx_1,x_2)→(y_1+a,y_2+c).$ (Page two, towards the bottom).