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I have been looking everywhere but I am unable to prove $$\delta(\vec{x}-\vec{a}) = \frac{1}{fgh}\delta(x_u-a_u)\delta(x_v-a_v) \delta(x_w-a_w)$$

Where $f,g,h$ are scale factors for an orthogonal system $u,v,w$. If $\vec{a}$ lies on a degenerate coordinate then $$\delta(\vec{x}-\vec{a}) = \frac{1}{fg\int hdw}\delta(x_u-a_u)\delta(x_v-a_v) \delta(x_w-a_w)$$

I know that the delta function is a generalized function, and is generally used in the form $$\int_{r_0\in V} f(\vec{r})\delta (\vec{r}-\vec{r_0})dV = f(\vec{r_0})$$

But I am unsuccesful in using this to prove the above expressions.

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3  
You need the change of variables formula for integrals. More generally this is a fact about pullbacks of a distribution by functions. For a treatment, see section 7.2 of Friedlander and Joshi, Introduction to the theory of distributions. –  Willie Wong Apr 27 '11 at 9:18
    
@Willie For the latter part of your comment, could you post it as an answer. –  samanwita Apr 30 '11 at 19:36

2 Answers 2

up vote 2 down vote accepted

(Reposting as an answer by request of the OP.)

This is a fact about pullbacks of a distribution by functions. For a treatment, see section 7.2 of Friedlander and Joshi, Introduction to the theory of distributions.

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$$F(\vec{a})=\int \delta(\vec{x}-\vec{a})\,F\,dx\,dy\,dz=\int\delta(\vec{x}-\vec{a})\,F\,|J|\,du\,dv\,dw$$

where $J$ is the Jacobian, and

$$\int\delta(x_u-a_u)\delta(x_v-a_v)\delta(x_w-a_w)\, F\,du\,dv\,dw=F(\vec{a}) \; .$$

In other words,

$$\delta(\vec{x}-\vec{a}) |J|=\delta(x_u-a_u)\delta(x_v-a_v)\delta(x_w-a_w) \; .$$

For orthogonal coordinates, $J=fgh$.

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