Why is the non-symmetric function, the Jacobian used in changes of variables?

Question

For example, if we take the polar coordinates $x(r, \theta) = r\cos \theta$ and $y(r, \theta) = r \sin \theta$ the Jacobian is

$$\frac{\partial(x, y)}{\partial(r,\theta)} = \frac{\partial x}{\partial r}\frac{\partial y}{\partial\theta}-\frac{\partial x}{\partial\theta}{\frac{\partial y}{\partial r}} = r\cos^2(x) + r\sin^2(x) = r$$

Hence $dxdy = r\space drd\theta$ This is not symmetric with respect to the variables like one would expect though. Expressing $x$ and $y$ as functions of $r$ and $\theta$ in that order is an arbitrary convention. We could just as easily write $x(\theta,r) = r\cos\theta$ and $y(\theta, r) = r\sin\theta$ in which case the Jacobian would be $-r$ and $dx dy$ would transform into $-r\space drd\theta$.

You might be interested in my answer to Why can't we convert the area element $dA$ to polar by multiplying the polar expressions for $dx$ and $dy$?, which touches upon the idea of the order mattering. — Mark S., Jan 01 '21 at 02:28

score 3 · Accepted Answer · answered Jan 01 '21 at 02:20

The change-of-variables formula is with the absolute value of the determinant of the Jacobian, not with just the Jacobian itself. So in either case we have $$ dx dy = | \det J |dr d\theta = r dr d\theta.$$

See, for example, https://en.wikipedia.org/wiki/Integration_by_substitution#Substitution_for_multiple_variables

Why is the non-symmetric function, the Jacobian used in changes of variables?

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