Jump to content
Sign in to follow this  
bingliantech

one question about the substitution in multiple integrals

Recommended Posts

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.

 

Share this post


Link to post
Share on other sites

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 by John

Share this post


Link to post
Share on other sites

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.

Share this post


Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now
Sign in to follow this  

×
×
  • Create New...

Important Information

We have placed cookies on your device to help make this website better. You can adjust your cookie settings, otherwise we'll assume you're okay to continue.