All Questions
8 questions
2
votes
1
answer
156
views
Dirac Delta applied to the gradient of a function
The supplementary section of a paper I am reading uses the "substitution" property of the Dirac delta function :
$$\mathbf{v}\left(\mathbf{x}_j\right)\delta\left(\mathbf{x}-\mathbf{x}_j\...
- 49
1
vote
0
answers
52
views
What is the relationship between $\partial_x^2\frac1r$ and $\delta^3(r)$? [closed]
We have the equation
\begin{equation}
\nabla^2\frac1r=-4\pi\delta^3(r).
\end{equation}
I first encountered this equation in electrodynamics. So what is $\partial_x^2\frac1r$ then? It looks like the ...
- 41
1
vote
1
answer
115
views
Divergence not defined
I’m currently working on the practice problems in Introduction to Electrodynamics by Griffiths. I got confused by the solution to this problem.
What does “ill-defined divergence” even mean? I ...
- 353
0
votes
1
answer
162
views
Representing $\frac{\mathbf{r-r'}}{|\mathbf{r-r'}|^3}$ in polar coordinates
In his book introduction to electrodynamics, Griffiths uses derives the identity
$$\nabla \cdot \frac{\mathbf{\hat{r}}}{r^2} = 4\pi\delta^3(\mathbf{r})$$
Using the formula for divergence in polar ...
- 323
2
votes
0
answers
395
views
Dirac delta function representations in physics
The most common representation of the Dirac delta function in physics is
$$\delta(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty}dk \,e^{ikx}.$$
My question is in which sense is it a faithful representation ...
- 578
12
votes
1
answer
2k
views
How can I compute the derivative of delta function using its Fourier definition?
I am wondering if it's possible to compute the derivative of the Dirac Delta function using the definition obtained from Fourier transformation: $$\delta(x-x')=\frac{1}{2\pi}\int e^{-ik(x-x')}dk.$$
...
- 185
5
votes
1
answer
5k
views
Second derivative of Dirac delta expression
I have come across the expression
$$ \int f(x) \delta(x-a) \delta''(x-a) \mathrm dx$$
where the prime represents the derivative.
Usually with derivatives of the Dirac delta distribution I'd partially ...
- 9,197
14
votes
4
answers
22k
views
How do you do an integral involving the derivative of a delta function?
I got an integral in solving Schrodinger equation with delta function potential. It looks like
$$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$
I'm trying to solve this by ...
- 325