Volume Element in Terms of g

A volume element in general coordinates  
\[(u_1, u_2, u_3)\]
  is  
\[dV = \left| \frac{\partial \mathbf{r}}{\partial u_1} \cdot (\frac{\partial \mathbf{r}}{\partial u_2} \times \frac{\partial \mathbf{r}}{\partial u_3}) \right| du_1 du_2 du_3\]
.
From this result  
\[\left| \frac{\partial \mathbf{r}}{\partial u_1} \cdot (\frac{\partial \mathbf{r}}{\partial u_2} \times \frac{\partial \mathbf{r}}{\partial u_3}) \right|^2=g \rightarrow \left| \frac{\partial \mathbf{r}}{\partial u_1} \cdot (\frac{\partial \mathbf{r}}{\partial u_2} \times \frac{\partial \mathbf{r}}{\partial u_3}) \right|= \sqrt{g}\]
.
Hence  
\[dV = \sqrt{g} du_1 du_2 du_3\]
.

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