When proving the Jacobian matrix for change of variables, you are usually given the linear transformation approach, computing that "transformed" area in the x,y plane using the determinant, which I understand, but that certainly wouldn´t be the type of procedure I´d follow at first, so I tried to prove the same theorem using something easier to digest: The increment theorem.
So, if we have $x=x(u,v) $ and $y =y(u,v)$ the most natural thing to do to find the differentials $dx $ and $dy$ in terms of your new variables is to consider, again, by the increment theorem, the relations:
$dx = \frac{\partial x}{\partial u}du + \frac{\partial x}{\partial v}dv $
$dy = \frac{\partial y}{\partial u}du + \frac{\partial y}{\partial v}dv $
And then, just multiply these two expressions to find the "little area" $dA=dxdy$ in terms of u and v.But the problem is that, when multiplying, you dont get that minus you´ll get when computing the determinant of the Jacobian...So, what´s wrong in my logic?
Note: I discard the du and dv´s squared terms simply because they´re irrelevant in the final expression, that is, those terms do not contribute to the notion of "infinitelly small change" which is already implicit when you are working with differentials of 1st degree. But again, this is not formal so if I´m wrong with this I´d highly appreciate a response on this as well.
