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I am trying to derive $ dV = \sqrt{g} du^1 du^2 du^3 $ for some general curvilinear coordinate $(u^1,u^2,u^3)$ system in $\mathbb{R}^3$ where $g = \mathrm{det}[g_{ij}]$. I am using the following facts:

  1. In my coordinate system there exists a reciprocal pair of bases at each point: $$ \mathbf{e}_i = \frac{\partial \mathbf{r}}{\partial u^i }, \quad \mathbf{e}^i=\nabla u^i $$ such that $\mathbf{e}_i\cdot \mathbf{e}^j = \delta^j_i$.
  2. Any vector $\mathbf{a} \in \mathbb{R}^3$ can be expanded as $$\mathbf{a}=a^i \mathbf{e}_i = a_i \mathbf{e}^i $$ and our contravariant and covariant components are given by $a^i = \mathbf{a}\cdot \mathbf{e}^i$ and $a_i = \mathbf{a} \cdot \mathbf{e}_i$ respectively.
  3. The components of the metric tensor $\mathbf{g}$ in the basis $ \{ \mathbf{e}_i \}$ are given by $g_{ij} = \mathbf{e}_i \cdot \mathbf{e}_j $.
  4. For a matrix $A$, $\mathrm{det}A = \varepsilon_{ijk}A_{i1}A_{j2}A_{k3}$.

Okay, here we go:

$$ dV = |\mathbf{e}_1 \cdot (\mathbf{e}_2 \times \mathbf{e}_3 ) |du^1 du^2 du^3 \\ =\varepsilon_{ijk}(\mathbf{e}_1)_i (\mathbf{e}_2)_j (\mathbf{e}_3)_k du^1 du^2 du^3$$

where $(\mathbf{e}_i)_j$ denotes the $j$th component of the vector $\mathbf{e}_i$. Well using fact numbers 2 and 3 from above, $(\mathbf{e}_i)_j = \mathbf{e}_i \cdot \mathbf{e}_j =g_{ij}$, so we get

$$ dV = \varepsilon_{ijk} g_{1i} g_{2j} g_{3k} du^1 du^2 du^3 \\ = (\mathrm{det}g )du^1 du^2 du^3 $$

where I have used the definition of the determinant from fact number 4. I have obviously done something wrong, where does the square root come from?

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I worked out the problem myself so thought I'd post the answer:

My problem was that I assumed the triple product $ |\mathbf{e}_1 \cdot (\mathbf{e}_2 \times \mathbf{e}_3)| = \varepsilon_{ijk} (\mathbf{e}_1)_i (\mathbf{e}_2)_j (\mathbf{e}_3)_k $ where $i,j,k$ are labelling the components w.r.t. the basis $\{ \mathbf{e}_i \}$. This is incorrect because I have assumed that the basis $\{ \mathbf{e}_i \} $ is orthonormal and the dot product and cross product are the same as they are in a Cartesian basis.

In order to get around this I must express $\{ \mathbf{e}_i \}$ in terms of the orthonormal Cartesian basis which I shall call $ \{ \mathbf{x}_a \}$, where the indices $a,b,c,\ldots$ are used to label Cartesian components. We have

$$ \mathbf{e}_i = \sum_a c_{ia} \mathbf{x}_a $$

so we have

$$ dV = \varepsilon_{abc} (\mathbf{e}_1)_a (\mathbf{e}_2)_b (\mathbf{e}_3)_c du^1 du^2 du^3 \\ = \varepsilon_{abc}c_{1a}c_{2b}c_{3c}du^1 du^2 du^3 \\ = \mathrm{det}Cdu^1 du^2 du^3$$

where I have let $C$ be the matrix with the components $C_{ia}$. Consider the product $CC^\mathrm{T}$:

$$ (CC^\mathrm{T})_{ij} = \sum_{k} c_{ik} c_{jk} \\ = \mathbf{e}_i \cdot \mathbf{e}_j \\ = g_{ij}$$

so $CC^\mathrm{T} = g$, and hence $\mathrm{det}g=(\mathrm{det}C)^2$, so substituting this result in the above expression for $dV$ I get

$$ dV = \sqrt{g} du^1 du^2 du^3 $$

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