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.