real analysis - Change of Variables Theorem

I am searching for a proof of the following theorem:

THEOREM

Suppose $(X_1, \ldots, X_n)$ is a random vector with joint density function $f_{X_1, \ldots, X_n}(x_1, \ldots , x_n)$ and $g$ is a smooth transformation on the domain of $(X_1, \ldots, X_n)$. Then the joint density of $(Y_1, \ldots, Y_n)= g(X_1, \ldots, X_n)$ is
$$ f_{Y_1, \ldots, Y_n }(y_1, \ldots, y_n) = f_{X_1, \ldots, X_n}(g^{-1}(y_1, \ldots, y_n)) \cdot |\det \mathcal{J}(g^{-1}(y_1, \ldots , y_n))|.$$

Maybe someone can give me some hints or references to prove this theorem.

Answer

This is straightforward using the change of variable formula, and the characterization
of the law of a variable $X$ by the application

$$
f \ge 0 \to E[f(X)]
$$