Given another function

for

defined on a triangle with vertices .

Now, given

for is a reference function defined on a reference triangle with vertices .

I want to evaluate the integral

,

but using a transformation, so that I can calculate

instead.

Now, I can use a transformation

where is given by

.

From my understanding, I should be able to do something like the following

,

where is the Jacobian of the transformation . I am slightly confused by this because I don't see from where, exactly, it came.

I am unsure if the Gradients are in the same variable. My guess is that they aren't, so I will have to use a chain rule there. I am having difficulty with this transformation. I guess it is not clear to me if the Jacobian is on the transformation that I have shown. It seems as though it is wrong. Any help would be greatly appreciated.

NOTE: This is in application to Finite Elements, but I felt it belonged in Calculus, since that is where I am confused.