one question about the substitution in multiple integrals

[bingliantech]

I know the Jacobian of the coordinate transformation measures how much the transformation is expanding or contracting the area around a point in G as G is transformed into R.

My question is why we need the Jacobian in the transformation,the deltaA which represent small area go into almost zero in the integral,and the deltaS which represnt the transformed small area also go into zero,why we need the Jacobian to measure the multiple.

 

[John]

While the integral is defined in this context as a limit in terms of increasing numbers of increasingly small units of n-dimensional volume, each unit of volume never actually becomes zero. Rather, we ultimately "arrive at" infinitesimally small units of volume, but each of these infinitesimal units still has a particular shape that changes depending on what coordinate system we're using. The Jacobian serves as the ratio of these infinitesimal volumes in one coordinate system to those in another.

Edited April 2, 2014 by John

[Bignose]

John's answer is pretty good. Another way to think about it is that, sure the final destination of both limits is the the volume going to zero, but the Jacobian helps make sure that both trajectories on their way to the voluem going to zero are the same.

 

For a very simplified example,

 

lim of x as x goes to 0 = 0 as well as

lim of x^2 as x goes to 0 = 0.

 

But, the trajectory each makes on its way to zero are obviously different. The Jacobian in effect modifies the trajectory so that it isn't just the final limit that is the same, but the trajectory into that limit.