Vector calculus eq. needs translation

[Goddar]

Homework Statement

Hi, this is not part of a problem but just an equation I'm having a hard time to decipher (for the reference the original one is in "Statistical Mechanics" by Pathria, eq. 3.7.16)
We define: r = |r₂–r₁|,
Where bold letters are vectors, and we basically integrate a function of 2 vectors over a volume V. So here's the beast:
\int\intg(r₂–r₁)dr₁dr₂=V\intg(r)(4\pir²dr

(the integral on the right side runs now from 0 to \pi. Sorry the pi's shouldn't appear like raised powers)
I understand the idea, roughly, but can't make the math rigorous: mainly, i don't know what to do with dr₁dr₂ to obtain the right side...
Thanks for helping

 

[fzero]

He's made a change of variables to \mathbf{R} = \mathbf{r}_1+\mathbf{r}_2 and \mathbf{r} = \mathbf{r}_2-\mathbf{r}_1. I don't have the text, but presumably g is a function of | \mathbf{r}_2-\mathbf{r}_1| only. However I believe that the integral on the RHS must still vary from 0 to \infty.

 

[Goddar]

The RHS integral is indeed from 0 to infinity, my mistake.. But g is a function of r₂–r₁ so while a scalar, it's a function of a vector (the vector-difference of the r's).
As defined here however, r is not a vector but the scalar |r₂–r₁|.
If i define a vector R = r₂+r₁,
for instance, i can express the integral as:
\int\intg(r₂–r₁)(dR–dr₁)dr₁
But i still can't see how to fill the gap...

 

[fzero]

The RHS integral is indeed from 0 to infinity, my mistake.. But g is a function of r₂–r₁ so while a scalar, it's a function of a vector (the vector-difference of the r's).
As defined here however, r is not a vector but the scalar |r₂–r₁|.
If i define a vector R = r₂+r₁,
for instance, i can express the integral as:
\int\intg(r₂–r₁)(dR–dr₁)dr₁
But i still can't see how to fill the gap...

If we change variables to

<br /> \mathbf{R} = \mathbf{r}_1+\mathbf{r}_2, \mathbf{r} = \mathbf{r}_2-\mathbf{r}_1,<br />

then

d\mathbf{r}_1d\mathbf{r}_2 = d\mathbf{R} d\mathbf{r}.

I don't have the book handy, so you might want to define g if you need help getting further along.

 

[Goddar]

g is not given explicitly, it's kept general as a function of r. But i think your answer is precisely where I'm lost:
it seems like dRdr should be the equivalent of (dr₁)²–(dr₂)² to me...
component-wise, i can't seem to make sense of these expressions. Then if you're right, the integration over dR should yield a factor of V and switching from dr to dr would give the integrand a factor of 4πr and send the limit of integration to infinity?

 

[fzero]

g is not given explicitly, it's kept general as a function of r. But i think your answer is precisely where I'm lost:
it seems like dRdr should be the equivalent of (dr₁)²–(dr₂)² to me...
component-wise, i can't seem to make sense of these expressions.

You might want to read up on the Jacobian matrix, since that's the correct way to compute the new measure after a change of variables. http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant

Then if you're right, the integration over dR should yield a factor of V and switching from dr to dr would give the integrand a factor of 4πr and send the limit of integration to infinity?

Yes.