• Support PF! Buy your school textbooks, materials and every day products Here!

Jacobian of the transformation

  • Thread starter Niles
  • Start date
  • #1
1,868
0

Homework Statement


Hi

I wish to perform an integral of the form

[tex]
\int_0^a {\int_0^b {f\left( {x - y} \right)dxdy} }
[/tex]

What I do first is to define s := x-y, and ds = dx. Then we get

[tex]
\int_0^a {\int_{-y}^{b-y} {f\left( {s} \right)dsdy} }
[/tex]

Then I can define t := x+y, so dt = dy. Then I get

[tex]
\int_{x}^{x+a} {\int_{-y}^{b-y} {f\left( {s} \right)dsdt} }
[/tex]

I also have to multiply by 2, since it is the Jacobian of the transformation. But look at the limits: It doesn't seem to make things easier. Where am I going wrong?
 
Last edited:

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,833
956


Well, what, exactly do you want to do with that? You obviously can't "do" the integral without knowing what f is. However, your limits of integration in the final integral makes no sense. From the first form, since the limits of integration are constants, the integral does not depend on x or y but your last form for the integral clearly does.

The region of integration in the x,y plane is a rectangle with sides parallel to the x and y axes. In the u,v plane it is still a rectangle but with sides at 45 degrees to the u and v axes. Depending on what a and b are, you would probably have to divide the area into two separate integrations. In any case, I don't see that change of variable helping.
 
  • #3
1,868
0


The reason why I am asking is because in my book they write the following

[tex]
I = \int_0^a {dt_1 \int_0^a {dt_2 \,F(x,t_1 - t_2 )} } F(y,t_1 - t_2 ) = a\int_0^a {dt\,\,F(x,t)F(y, - t)}
[/tex]

I cannot quite see how the second equality-sign comes about. I thought it came from substitution, as above.
 

Related Threads on Jacobian of the transformation

  • Last Post
Replies
10
Views
4K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
3
Views
12K
  • Last Post
Replies
1
Views
2K
Replies
3
Views
6K
Replies
4
Views
2K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
1
Views
1K
Replies
3
Views
2K
Replies
3
Views
773
Top