Multivariable Delta Function Integral

[bologna121121]

Homework Statement

I have to find this integral:

\int \delta (( \frac{p^{2}}{2m} + Cz ) - E ) p^{2} dp dz

where E, m, and C can be considered to be constants.

Homework Equations

I'm semi-familiar with delta functions, i.e. i know that:

\int \delta (x - a) dx = 1

and that you can usually change the variable of integration to match the variable in the delta function, if it's not written explicitly as above.

The Attempt at a Solution

My problem is that I don't really know how to work with this in two dimensions, with both variables appearing inside the delta function. I thought maybe there might be a way to split it into two different delta functions, with one variable appearing in each? But this is just a guess, and I can't really find any supporting evidence. Thanks in advance.

 

[sunjin09]

Homework Statement

I have to find this integral:

\int \delta (( \frac{p^{2}}{2m} + Cz ) - E ) p^{2} dp dz

where E, m, and C can be considered to be constants.

Homework Equations

I'm semi-familiar with delta functions, i.e. i know that:

\int \delta (x - a) dx = 1

and that you can usually change the variable of integration to match the variable in the delta function, if it's not written explicitly as above.

The Attempt at a Solution

My problem is that I don't really know how to work with this in two dimensions, with both variables appearing inside the delta function. I thought maybe there might be a way to split it into two different delta functions, with one variable appearing in each? But this is just a guess, and I can't really find any supporting evidence. Thanks in advance.

First you need to know how to scale a delta function, i.e., δ(a*x)=1/a*δ(x); then you integrate z first, and treat everything else as constants, the result is very simple.

 

[bologna121121]

Ah...I guess that's pretty obvious. Thank you very much.