The change of variable $(u,v)\rightarrow (m(u,v),n(u,v))$. Under the change of variable, the double integral were written as
$$ \int_{a_1}^{b_1} du \int_{a_2}^{b_2} dv f(u,v) = \int_{c_1}^{d_1} dm \int_{c_2}^{d_2} dn f(m,n) |\frac{\partial(u,v)}{\partial(m,n)}| $$ However, for the treatment of Jacobian $|\frac{\partial(u,v)}{\partial(m,n)}|$, there are two different definitions.
The first is simply the determinant of the Jacobian matrix $\det (\frac{\partial(u,v)}{\partial(m,n)}) $. This is shown in Paul's Online Notes
The second is the absolute value of the determinant of the Jacobian matrix $|\det (\frac{\partial(u,v)}{\partial(m,n)}) |$. This is shown in LibreTexts Mathematics and Wikipedia
On the wolfram's website, the Jacobian is also simply the determinant of the Jacobian matrix without the absolute value. However, it was mentioned that "$f$ (the integrand) is a global orientation-preserving diffeomorphism".
Later, one realized that it has to do with orientation transformation and orientation preserving integral or non orientation preserving integral. However, one thought the orientation preserving transformation meant the sign of $\det (\frac{\partial(u,v)}{\partial(m,n)})$, i.e. the change of variable, not the $f$ the integrand.
Consider the case where $(u,v)\rightarrow (u,n=-v)$, a reflection, then
Case 1 would suggest $\int_{a_1}^{b_1} du \int_{a_2}^{b_2} dv f(u,v) = \int_{a_1}^{b_1} du \int_{-a_2}^{-b_2} dn f(u,n) \cdot (-1) $
Case 2 would suggest $ \int_{a_1}^{b_1} du \int_{a_2}^{b_2} dv f(u,v) =\int_{a_1}^{b_1} du \int_{-a_2}^{-b_2} dn f(u,n) $
This led to a difference in sign.
Does there necessarily need an absolute value around the determinant of the Jacobian matrix for the change of variable?
- The case of 1d change of variable were different since there wasn't an orientation? But what about the integration order $\int_a^b$ and $\int_{b}^a$ itself?
