# What is Dirac delta function?

• I
Leo Liu
TL;DR Summary
I came across it in the derivation of Gauss' law of electric flux from Coulomb's law. I did some research on it, but the wikipedia page about it was slightly confusing. All I know about it is that it models an instantaneous surge by a distribution. However I am still perplexed by this concept. Can someone please explain it to me like I am a 5-year-old with some calculus or direct me to some useful resources other than wiki?

Last edited:
NTuft

Gold Member
There are multiple approaches to delta function. Here I would start from
Heaviside Step Funiiton
https://en.wikipedia.org/wiki/Heaviside_step_function
We can not differentiate Heaviside Step Function at 0 in usual healthy sense. But Dirac boldly invented a strange delta "function" as the result of its derivative including x=0.
$$H'(x)=\delta(x)$$

Say we give 1 Ns impulse to a 1kg body to make it move with 1m/s.
There are many ways to do it, e.g.
Applying Force 1 N during 1 second,
Force 10N during 1/10 second,
Force 100N during 1/100second,
-----
Force 1/x N during x second,
------

We can make time duration x as small as we like with increasing Force 1/x .

There is no limit of x to zero in this sequence because 1/x diverges to infinity. But Dirac say there exists force limit ##\delta(x)##, that is to say, in a instant of time, the infinite force is applied to generate 1 Ns momentum on the body. Delta function is a momentum maker in a instant of time.
$$\int_{-\infty}^x \delta(\xi) d\xi = H(x)$$
Heaviside step function shows thus made momentum by Delta function at x=0.

Last edited:
NTuft, DaveE, atyy and 1 other person
Homework Helper
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Can someone please explain it to me like I am a 5-year-old ...
The Dirac delta function is rarely taught at infant school. Especially these days.

berkeman and atyy
Homework Helper
The Dirac delta function is rarely taught at infant school. Especially these days.
What's the world coming to these days ?! A B C Dd !

atyy and PeroK
Leo Liu
$$-\int_{\mathbb{R}}H(x)\varphi'(x)dx=\varphi(0)$$