Let $U \subset \mathbb{R^3}$ be an compact domain. Let $(u_1,u_2,u_3)$ an coordinate system on $U$ and assume there is a parametrization of $U$ such that $$U: \vec{r}\left(u_{1}, u_{2}, u_{3}\right)=x_{1}\left(u_{1}, u_{2}, u_{3}\right) \hat{i}+x_{2}\left(u_{1}, u_{2}, u_{3}\right) \hat{j}+x_{3}\left(u_{1}, u_{2}, u_{3}\right) \hat{k}, \quad\left(u_{1}, u_{2}, u_{3}\right) \in \tilde{U}$$
while $\tilde{U}$ is a domain in $\mathbb{R^3}$ and assume $\vec{r}\left(u_{1}, u_{2}, u_{3}\right)\in C^1$. Lets define the matrix $g_{3x3}$ such that $$g_{i j}=\left\langle\frac{\partial \vec{r}}{\partial u_{i}}, \frac{\partial \vec{r}}{\partial u_{j}}\right\rangle, \quad 1 \leq i \leq 3,1 \leq j \leq 3$$
Show that $$V_{U}=\iiint_{\tilde{U}} \sqrt{\det(g)} d u_{1} d u_{2} d u_{3}$$
I am not really sure how to approach this problem, I understand how the matrix $g$ is looks like, and I can see it is a symmetric but what does it represent? I also know there is a relation between determinant to volume but I don't see how those relations can help me here.
One has \begin{equation} g = J^T J \end{equation} where $J = \frac{\partial r}{\partial u}$ is the jacobian matrix of $r$. Hence \begin{equation} \det(g) = \det(J)^2 \qquad \Longrightarrow \qquad \sqrt{\det(g)} = |\det(J)| \end{equation} The formula for $V_U$ is simply the change-of-variables formula in the Lebesgue integral in ${\mathbb R}^d$.