Equal number of variables
Change of variable from 2d cartesian to polar coordinate is easy because the number of variables remain the same.
$\iint_{R}f(x,y)dxdy=\iint_{S}f(x(r,\theta),y(r,\theta))\left|\frac{\partial (x,y)}{\partial (r,\theta)}\right| drd\theta$
Unequal number of variables
I try to evaluate an integral by a change of variable from the absolute barycentric coordinate in a square pyramid to Cartesian coordinate has the following formula:
$\vec{x} = \sum_{i = 1}^{5} w_{i}\vec{v}_{i}$ where:
- The constraint is: $1 = \sum_{i=1}^{5} w_{i}$
- $w_{i} \ge 0$
- $\vec{x} = (x, y, z)$
- $\vec{w} = (w_{1}, w_{2}, w_{3}, w_{4}, w_{5})$
- $\vec{v}_{i}$ are the 5 vertices of the square pyramid.
The correspondence between 3D Cartesian coordinate and the absolute barycentric coordinate is not one-to-one. See Servaes's comments.
The Jacobian matrix $J$ is 3 $\times$ 5 if I omit the partial derivatives for the constraint.
I can't use the usual formula for change of variable because the Jacobian $J$ is rectangular and I can't get the determinant of $J$.
Question
In general, how to perform a change of variable with unequal number of variables for a multivariable integral?
Update
Barycentric coordinates are only of use when dealing with simplices. There the number of variables is one greater than the dimension of the simplex, and the representation is unique. Concerning the Jacobian one may say the following:
Assume that we have the following situation: A simplex $S\subset{\mathbb R}^n$ has vertices ${\bf a}_i$ $(0\leq i\leq n)$. The points ${\bf x}\in S$ can then be written in the form $${\bf x}=\sum_{i=0}^n\lambda_i{\bf a}_i,\qquad \lambda_i\geq 0,\quad\sum_{i=0}^n\lambda_i=1\ .\tag{1}$$The $\lambda_i$ are the barycentric coordinates of ${\bf x}$. The representation $(1)$ then allows of the parametric representation $$\psi:\quad S_{\rm st}\to S, \qquad(\lambda_1,\ldots\lambda_n)\mapsto {\bf x}={\bf a}_0+\sum_{i=1}^n\lambda_i({\bf a}_i-{\bf a}_0)\ ,$$ where $S_{\rm st}:=\{(\lambda_1,\ldots\lambda_n)\,|\,\lambda_i\geq0, \ \sum_{i=1}^n\lambda_i\leq1\}$ is the standard simplex in ${\mathbb R}^n$. If we now are given a function $f: S\to{\mathbb R}$ in terms of the barycentric coordinates $\lambda_i$ then we can write $$\int_S f(\lambda_0,\ldots,\lambda_n)\>{\rm d}({\bf x})=J_\psi\int_{S_{\rm st}} f(\lambda_0,\ldots,\lambda_n){\rm d}(\lambda_1,\ldots,\lambda_n)\ .$$ Here the Jacobian $J_\psi=|{\rm det}(A)|$, where $A$ is the matrix with the differences ${\bf a}_i-{\bf a}_0$ in its columns. This is the same as the volume of the parallelepiped spanned by these vectors.
In order to integrate over a more complicated body $B$, as your pyramid, you have to present it in terms of inequalities describing the exact boundary of $B$. It is a hard mathematical problem to determine the exact bounding (hyper)planes of a $d$-dimensional body defined as convex hull of $n\geq d+1$ points in ${\mathbb R}^d$. Note that the so-called cross body with $2d$ vertices $\pm {\bf e}_i$ $(1\leq i\leq d)$ has $2^d$ facets. (In ${\mathbb R}^3$ this is the octahedron with $6$ vertices and $8$ faces).