In this video, I explain what the Dirac delta REALLY is - and no, it's not a function, at least in the usual sense!

I always felt unsatisfied by the informal definition of the Dirac delta given in physics or engineering courses, but when I finally learned the rigorous definition, it seemed disconnected to the informal one and not very enlightening. I decided to make this video to connect the two definitions and show how intuition and rigor can (and should!) coexist.

Technical footnote: Sometimes the Dirac delta is presented as a distribution or "generalized function", i.e. a continuous linear functional on the space of smooth compactly supported functions. Distribution theory is very powerful, and the Dirac delta appears when trying to make sense of differentiation of non-differentiable functions, or solving differential equations in a "weak" sense. Despite this, distributions can get quite technical (even the existence of smooth compactly supported functions is a bit technical), so I felt it best to avoid covering them in this video.

Submitted as part of the Summer of Math Exposition 2 (#SoME2) contest.

Timestamps:
00:00 - Introduction
00:25 - Informal Definition
1:45 - Measures
3:08 - The Dirac measure
3:58 - Integration with respect to measures
4:53 - Explaining the sifting property
5:35 - Why infinite at zero?
7:10 - Linear functionals
8:45 - A rigorous definition

свернуть