Let $\varSigma$ be a domain in $\mathbb{R}^n$, and consider the Laplace equation on $\varSigma$: That is $$ \triangle u =0, $$ in a well-known weak formulation of Elliptic equations (I omit the boundary condition $u=f$ for some function $f$).
My question is that how a change of variables, transforms this equation to another equation. To be more precise, let $ \theta$ be a map from $\varSigma$ to another domain (the map $ \theta: \varSigma \to \mathbb{R}^{n+1}_+ $ would be a good choice; where $\mathbb{R}^{n+1}_+ = \{ (x,t): x \in \mathbb{R}^{n} , t>0 \}$ is the upper half plane). Then the Laplace equation on $\varSigma$ would change to a new equation of type $$ \mathrm{div} (A \nabla v )=0 ,$$ where $v= u \circ \theta $, and the matrix $A(x)$ comes from the differentiation matrix $S=D \theta$; that is $S_{ij} = \frac{\partial \theta_i}{\partial x_j}$. I saw somewhere an expression of $A$ with respect to $S$; But I can't compute things and find a proof. This is the expression: $$ A= (S^T)^{-1} J(S) S^{-1},$$ where $J(S)$ is the Jacobean.
Any suggestions, proofs, and references are welcomed.