Polar coordinates; Jacobian and order of the variables

Question

I'm trying to understand the Jacobian a bit. I know the general form of the transformation with two variables:

$$\int\int_Ag(x,y)\,\mathrm dx\,\mathrm dy=\int\int_{T(A)}g\big(x(u,v),y(u,v)\big)\lvert J(u,n)\rvert \,\mathrm du\,\mathrm dv.$$

Say we'd like to change to polar coordinates;

$$\int\int_Ag(x,y)\,\mathrm dx\,\mathrm dy=\int\int_{T(A)}g\big(x(r,\theta),y(r,\theta)\big)\lvert J(r,\theta)\rvert \,\mathrm dr\,\mathrm d\theta.$$

How do we know the order of $(r,\theta)$? Could it also have been $(\theta,r)$? This would yield to the same Jacobian determinant but with the opposite sign.

Answer 1

Well it would lead to a sign difference but the magnitude would be the same. Switching order is fine.

Like in Linear Algebra. Elementary row operations change the sign of the determinant but the magnitude remains unchanged.