Jacobian in change of coordinates

[Simfish]

So in the change of coordinates equation for multiple integrals, we have the Jacobian which allows us to change our bounds of integration.

It's the connection between multivariable calculus and linear algebra.

is this possible since the linear transformation allows a one-to-one-mapping between one coordinate system and the other coordinate system? And is this precisely the reason why we can take the Jacobian of a multiple integral?

 

[quasar987]

I don't understand your question, but I will just say that it might be insightful for you to know that if S is a region of R^n and T is a linear map from R^n to R^n, then the volume of T(S) equals the volume of S times the determinant of T (as a matrix). (And also that a linear map sends little cubes to little parallelepipedes)

 

[mjsd]

So in the change of coordinates equation for multiple integrals, we have the Jacobian which allows us to change our bounds of integration.

It's the connection between multivariable calculus and linear algebra.

is this possible since the linear transformation allows a one-to-one-mapping between one coordinate system and the other coordinate system? And is this precisely the reason why we can take the Jacobian of a multiple integral?

hard to see what you are really asking but it is true that linear analysis has a lot to do with Jacobian and Integration. It comes from the realm of Measure theory where you get all those results for Lebesgue integral on R^n and product measures...

 

[mathwonk]

in linear algebra a determinant measures the volume change under a linear map.

in non linear calculus, integrating the (different) determinant functions at different points measures volume change under a non linear map.