Order of rows in the Jacobian arbitrary?

[geoduck]

I just wanted to verify which order you put the rows of the Jacobian. If your initial variables are (x,y) and change to new variables u(x,y)=f(x,y) and v(x,y)=g(x,y), then you'll get a Jacobian. If this Jacobian is negative, would you change your definition to u(x,y)=g(x,y) and v(x,y)=f(x,y), since dxdy is assumed positive, so dudv needs to be positive?

So can you always get away with putting absolute values on the Jacobian?

My worry is that region of integration might be negative, but the measure positive, when making a change of variables.

An example is $\int_{-5}^{6} f(x)dx$ with $u(x)=-x$.

Then $\int_{-5}^{6} f(x)dx=\int_{5}^{-6}f(-u) (-du)$.

So it would have been a mistake to make the Jacobian positive:

$\int_{-5}^{6} f(x)dx \neq \int_{5}^{-6}f(-u) (|-1|du)$

Can I say that in all integrals, we automatically put the lowest limit on the bottom, and the highest limit on the top, and make sure all Jacobians are positive? Would that work every time?

 

[UltrafastPED]

The Jacobian determinant contains information about the local region; as it is a function of the local coordinates, it _may_ change sign over a region.

This is discussed here: https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant#Jacobian_determinant

So if the Jacobian determinant is negative everywhere for your region of integration, then the orientation has been reversed ... so you would then reverse the integration order.

If it is positive everywhere, no change required.

If it is ever zero, your coordinate space has collapsed, and you will need to reconsider how to proceed ... thus you would not ordinarily expect to find a Jacobian determinant that changes sign during an integration unless:

(a) you are doing problems from a text on real analysis or differential geometry ...
(b) you are working a "real" problem, from the real world

or

(c) your professor likes trick questions!