What is the Definition of the Delta Function?

[Leo Liu]

TL;DR Summary

See the title please.

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?

 

[anuttarasammyak]

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.

 

[PeroK]

Summary:: See the title please.

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.

 

[neilparker62]

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 !

 

[Leo Liu]

There are multiple approaches to delta function. Here I would start from
Heaviside Step Funiiton
https://en.wikipedia.org/wiki/Heaviside_step_function

Thank you. This concept isn't hard to grasp at all. I will look into its connection with Dirac delta.

 

[wrobel]

$\delta$-function is a linear continuous functional on the space of $C^\infty(\mathbb{R})$-functions with compact support. By definition $\delta(\varphi)=\varphi(0)$.
The formulas like $H'=\delta$ are understood in the generalized sense
$$-\int_{\mathbb{R}}H(x)\varphi'(x)dx=\varphi(0)$$
For details see any textbook on functional analysis. For example https://www.springer.com/gp/book/9783540586548