I'm quite confused about the representation of the Dirac Delta distribution. I've got two general questions about that topic:
- I read different literature and found contradictory statements: Let $\overline{x}=(x,y,z), \overline{x_0}=(x_0,y_0,z_0)\in\mathbb{R}^3$. Now I'm interested in the representation of cylindrical coordinates. Transforming $\overline{x},\overline{x_0}$ in cylindrical coordinates yields $\overline{r}=(r,\theta,z),\overline{r_0}=(r_0,\theta_0,z_0)$. Now I found \begin{align} \delta(\overline{r}-\overline{r_0})&=\frac{1}{r}\delta(r-r_0)\delta(\theta-\theta_0)\delta(z-z_0),\\ \delta(\overline{r}-\overline{r_0})&=\frac{1}{r_0}\delta(r-r_0)\delta(\theta-\theta_0)\delta(z-z_0) \ \ \text{and}\\ \delta(\overline{x}-\overline{x_0})&=\frac{1}{r}\delta(r-r_0)\delta(\theta-\theta_0)\delta(z-z_0). \end{align} All statements were handled without any proof, so I just can't check the right one. My intuition would say that the left side needs cartesian coordinates, and the factor $\frac{1}{r}$ would be the price for the transformation. What's the right use ofe representation?
2.Scaling $y=cx$ yields $\delta(cx)=\frac{1}{|c|}\delta(x)$. eevery proof uses the integral representation and means the equation in the distributionals sense. But I hought the integral representation is only of symbolical meaning because of an analogy to regular distributions. That means there is only the implicit definition of the Delta distribution by $\delta(\varphi):=\varphi(0)$ and one can't prove it by using integral representation. But with the implicit definition I can't see a way to get the claimed equation. Any ideas?