Undergrad Jacobian matrix generalization in coordinate transformation

[mertcan]

hi, I always see that jacobian matrix is derived for just 2 dimension ( ıt means 2x2 jacobian matrix) in books while ensuring the coordinate transformation. After that kind of derivation, books say that you can use same principle for higher dimensions. But, I really wonder if there is a proof which ensure that jacobian matrix is compatible with higher dimensions? I am asking because there are proofs which ensure the jacobian matrix only in 2 dimensions not higher dimensions.. Thanks in advance... I am looking forward to your mathematical demonstrations...

 

[andrewkirk]

Jacobian matrices have several useful properties.
For which one of those properties are you seeking a proof?

 

[mertcan]

Jacobian matrices have several useful properties.
For which one of those properties are you seeking a proof?

While doing integral in terms of space-time coordinates, and if you want to change coordinates, we should use jacobian matrix and determinant rule. I know the proof of why we should use jacobian determinant rule if there are 2 coordinates, but I do not know the proof of how this jacobian determinant rule fits with the higher dimensions or high order coordinates ?

 

[mertcan]

I hope my question has become explicit after my last post...

 

[mertcan]

Is there someone who can answer my question ? :D I am really looking forward to your answers ...

 

[andrewkirk]

The proof you are seeking is that of the 'change of variable formula for integrals using Jacobians'. It is long and complex and I expect it will only appear in fairly advanced vector calculus texts.

This Stack Exchange Q&A gives an intuitive overview of the proof (in the first answer) and also contains a reference to a text in which the full proof can be found (in the last answer).