How to compute the surface an N-sphere using delta functions

[charlesmartin14]

Homework Statement

I am trying to understand how to compute the surface an N-sphere , for large N, to leading order (and exactly)

Given a vector J with norm N, with N large, how does one compute the volume integral ? That is, what representation of the delta function. And what is the exact result ?

Relevant Equations

various delta function representations

Problem Statement: I am trying to understand how to compute the surface an N-sphere , for large N, to leading order (and exactly)

Given a vector J with norm N, with N large, how does one compute the volume integral ? That is, what representation of the delta function. And what is the exact result ?
Relevant Equations: various delta function representations

.

 

[charlesmartin14]

∫dJδ(J²-N)≈exp(N/2(1+ln2π))

The area of an N-sphere goes like 2π^N/2 so I know this is close but I am haven't remembered the trick yet how to get the exact result

 

[haruspex]

Problem Statement: I am trying to understand how to compute the surface an N-sphere , for large N, to leading order (and exactly)

Given a vector J with norm N, with N large, how does one compute the volume integral ? That is, what representation of the delta function. And what is the exact result ?
Relevant Equations: various delta function representations

.

Is this what you are after?
https://en.m.wikipedia.org/wiki/N-sphere#Recurrences

 

[charlesmartin14]

No I was thinking more to use a relation like

$$\delta[g(x)]=\dfrac{\delta(x-x_{0})}{|g'(x)|_{x=x_{0}}}$$

or maybe the simpler relation

$$\delta[(x^{2}-a^{2})]=\dfrac{1}{2|a|}[\delta(x+a)+\delta(x-a)]$$

so that we can reduce

$$\delta[(\mathbf{J}^{2}-N)]=\dfrac{1}{2N^{1/2}}[\delta(\mathbf{J}+\sqrt{N})+\delta(\mathbf{J}-\sqrt{N})]$$

Which should give 2 identical values when integrated over $\int\;d\mathbf{J}$. Then we need to represent $\int\;d\mathbf{J}$ using

$$d\mathbf{J}=\Pi_{i=1}^{N}dj_{i}$$

and then compute the integral as a product of N identical integrals over $dj_{i}$

OR

I suppose one could try to do the $\int\;d\mathbf{J}$ integral in N-dim spherical coordinates, and then the relation (above on wikipedia) might be useful

EVENTUALLY

I want to add some constraints on the $\mathbf{J}$ vectors, such as specifying an arbitrary vector $\mathbf{K}$, and asking what is

$$\int\;d\mathbf{J}\delta(\mathbf{J}^{2}-N)\delta(\dfrac{1}{N}\mathbf{J}^{T}\mathbf{K}-E)=?$$

So I would like to work it all out, in gross detail, using the delta function forms