Can someone explain to me this part of the proof of the jacobian?

I dont know what they're talking about...I can follow the rest (the cross product bla bla bla bla bla) but I dont know how they're getting these two vectors...I figured it has something to do with partial differentials but Im still confused. If anyone could provide any insight Id be appreciative.

The notation is a bit screwy, but here's what I think they're doing.

So suppose the surface is parametrised by [itex]\vec T(u,v)[/itex].
Take a small rectangle in the domain with dimensions [itex]\Delta u, \Delta v[/itex], the bottom left corner being the point [itex](u_0,v_0)[/itex].
The image of this rectangle is a patch of area, which can be approximated by the parallelogram formed by the vectors:

[tex]\vec T(u_0+\Delta u,v_0)-\vec T(u_0,v_0)[/tex]
and
[tex]\vec T(u_,v_0+\Delta v)-\vec T(u_0,v_0)[/tex] (a picture helps here).

These vectors are in turn approximated by
[tex]\frac{\partial}{\partial u}\vec T(u_0,v_0)\Delta u[/tex]
and
[tex]\frac{\partial}{\partial v}\vec T(u_0,v_0)\Delta v[/tex]
respectively.

So area patch is about [itex]|\vec T_u \times \vec T_v|\Delta u \Delta v[/itex] and you can figure out the rest.

It might help (me, anyways) if you would say what you're trying to prove. The Jacobian is a number associated with a matrix; it doesn't make any more sense to ask about a proof of the Jacobian than it does to ask about a proof of the number 2.

Oh, I see it better now. THanks a lot, just wanted to say that before I go to bed. If i need further clarification Ill post the fool proof. THakns a lot guys