for the Jacobian transformation from a region S to R does this limit expression represent the slope of the tangent line to the side in the transformed region R?

so you can approximate the area of the new region R by the cross product parallelogram since in the limit that delta u goes to 0, the curves of a parellelogram becomes straight lines? but it seems that this would only work in specific cases where region S or R is similar to a parallelogram?

[latex]r_{u}^{} =lim_{\Delta u\rightarrow 0}\frac{r(u_o +\Delta u,{v_{o}})-r(u_{o},v_{o})}{\Delta u}[/latex]

according to my vertice method, you don't have to transform the lines of a trapezoid into a new trapezoid, you can just calculate the vertices

I should look at Spivack or Hubbard- I found a copy of the Loomis text. thanks very much!

(btw how do you get the Latex to post directly on the forum?)

(I'm having bad health problems)