Yeah, the Jacobian should work. I don't see where a divided by 0 has an opportunity to occur, but man the det of the Jacobian matrix doesn't look very fun to compute or work with. You should end up with

[assuming I did this right]

Update: Yep, just finished it and this works. It works out super nice, particularly the Jacobian simplifies to and it happens to be exactly what you need. They're independent, weird o.O I'm guessing you can extend this result to an arbitrary number of independent gammas.