Tagged Questions
0
votes
0answers
36 views
Generalization of $\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}$. Examples?
I've taken the following from Wikipedia Dirac Composition
$\delta(g(x)) = \frac{\delta(x-x_0)}{|g'(x_0)|}$
It is natural therefore to define the composition $δ(g(x))$ for continuously ...
2
votes
0answers
63 views
Delta function and integrating over level sets?
Consider the three-dimensional integral
$$
\int_{\mathbb R^3} d^3x\,f(x)\delta(g(x))
$$
where $\delta$ is the dirac delta, $f,b:\mathbb R^3\to\mathbb R$ and $g(x) = 0$ on some surface $S$. Is there ...
0
votes
1answer
127 views
Justification of use of delta functions/rigorous proof of Green's function for Poisson equation
I'm looking at the proofs of Helmholtz's theorem, but I'm having trouble justifying their interchange of integration and differentiation and subsequent treatment of the "integrand" as a delta ...
3
votes
0answers
273 views
Property of derivative of Dirac delta function in $\mathbb{R}^n$
With reference to Property of Dirac delta function in $\mathbb{R}^n$, is there a similar formula for $\langle g^*\delta', f \rangle$ (or even $\langle g^*\delta^{(n)}, f \rangle$)? By similar I mean a ...
0
votes
0answers
106 views
Integration methods for functions with Delta distributions
Which Monte-Carlo methods are available for computing a multidimensional integral with Delta distributions (in case one cannot sample them explicitly)?
PS: I also asked a similar question at ...
6
votes
3answers
787 views
Property of Dirac delta function in $\mathbb{R}^n$
How does one prove the following identity?
$$\int _Vf(\pmb{r})\delta (g(\pmb{r}))d\pmb{r}=\int _S\frac{f(\pmb{r})}{|\text{grad} g(\pmb{r})|}d\sigma$$
where $S$ is the surface inside $V$ where ...
