Change of Variables Theorem

Question

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 1

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

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

Answer 2

These pages will help.

Here there is the theorem for $\mathbb R^2$.

Answer 3

$$ \mathbb{E} F(Y)= \mathbb{E} F(g(X)) = \int F(g(x)) f_X(x) \, \text{d} x = [y=g(x), x=g^{-1}(y)]=\\ =\int F(y) f_X(g^{-1}(y)) | \det \mathcal{J}g^{-1}(y)| \, \text{d}y$$ $\Longrightarrow f_Y(y) =f_X(g^{-1}(y)) $