Understanding the Dirac Delta Function in Spherical Coordinates

[element1945]

Homework Statement

Justify the following expretion, in spherical coordinates;

delta (vector r) = (1 / r^2 * sin (theta) ) * delta(r) * delta(theta) * delta(phi)

Homework Equations

The Attempt at a Solution

I don't know what it means... please help?

 

[lanedance]

are you sure you don't mean
$$r^2{\sin{\theta}.dr.d\theta.d\phi$$

this an expression for a volume integrand, over spherical coordinates $$(r, \theta, \phi)$$

the delta represents each coordinate integral, whilst the $$r^2\sin{\theta}$$ factor comes from the jacobian, based on the coordinate transform from cartesian to spherical coordinates

in simple terms try drawing the volume element formed by the infintesimals, (approximating a infintesiaml cube in the limit..)
$$dV = d\textbf{r} = r^2\sin{\theta}.dr.d\theta.d\phi$$

and you will see where the $$r^2{\sin{\theta}$$ terms comes from

 

[John Creighto]

I'l presume also aside from using the Jacobian for the coordinate transom one should start with:

$$\delta(x-x_o,y-y_o,z-z_o)=\delta(x-x_o)\delta(y-y_o)\delta(z-z_o)$$

 

[lanedance]

Cheers John, I've re-read the question - missed the meaning of delta first time round...

element1945 can you elaborate on the problem at all? also do you understand what the 1 dimensional delta function is?