# On the relationship between the volume element in curvilinear coordinates and the Jacobian

Suppose we have a curvilinear coordinate system in $\mathbb{R}^3$, $(q_1, q_2, q_3)$, with $x_i = f_i(q_1, q_2, q_3)$. We define the local basis for this coordinate system by $\hat{e_i} = \frac{\partial\vec r}{\partial q_i}$, and the scale factors in this coordinate system as $h_i = \|\hat{e_i}\|$. Then it is known that the volume element in our coordinate system is $$dV = h_1h_2h_3dq_1dq_2dq_3$$ but we also know that the volume element, under a coordinate transformation, is transformed into $$dxdydz = \left|\frac{\partial(x,y,z)}{\partial(q_1,q_2,q_3)}\right|dq_1dq_2dq_3$$ where $\frac{\partial(x,y,z)}{\partial(q_1,q_2,q_3)}$ is the Jacobian determinant.

Is my understanding correct in that $h_1h_2h_3$ should equal the absolute value of the Jacobian determinant, or am I misinterpreting something? If my understanding is correct, how can I prove this? I know that if $F = (f_1, f_2, f_3)$, then $$\begin{pmatrix} \hat{e}_1 \\ \hat{e}_2 \\ \hat{e}_3 \end{pmatrix} = (JF)^\text{T}\begin{pmatrix}\hat{i} \\ \hat{j} \\ \hat{k}\end{pmatrix}$$ (where $JF$ is also the Jacobian matrix, just different notation), but I have no clue how I might use this to show the above equality (assuming it's true).

• Your formula is only valid when the $\hat e_i$ are mutually orthogonal (perpendicular). – Ted Shifrin Feb 23 '18 at 22:23
• I had that thought. Upon further thinking, the if the $\hat{e}_i$ are orthogonal, then quantity $h_1h_2h_3$ is equal to the $\|\hat{e}_1\cdot(\hat{e}_2\times\hat{e}_3)\|$, which is also the formula for the volume of a parallelepiped, which is analogous to the Jacobian factor in multivariable substitution (which is also a "volume"). Could you elaborate on this in an answer? – user3002473 Feb 23 '18 at 22:32

The column vectors of your Jacobian matrix will be the vectors $\hat e_i$ (by the way, I thought it was customary to use $\hat v$ only for unit vectors $v$!). The determinant of that matrix will be (up to sign) the product of the lengths of those column vectors if and only if the vectors $\hat e_i$ form an orthogonal set. (In general, the absolute value of the determinant is the volume of the parallelogram whose edges are the column vectors.)