- #1

arildno

Science Advisor

Homework Helper

Gold Member

Dearly Missed

- 10,089

- 135

1. INTRODUCTION

Many students become frustrated when they first meet the Dirac Delta function, typically in a course involving electrostatics, or Laplace transforms.

As it is commonly presented, the Dirac function seems totally meaningless:

Either, it is "defined" as:

[tex]\int_{-\infty}^{\infty}\delta(x)dx=1,\delta(x)=0,x\neq0[/tex]

or, in conjunction with an arbitrary function f:

[tex]\int_{I}\delta(x)f(x)dx=f(0),0\in{I},\int_{I}\delta(x)f(x)dx=0,0\notin{I}[/tex]

where I is any interval on the real axis.

What sort of function can do these sort of things??

The student's confusion is not markedly lessened when:

the lecturer praises the Dirac function's ability to sample a function value, then the confusion increases when a mathematician scoffs at the whole "definition" and says "it is just a formalism, you've got to think of it as a distribution", and when the lecturer is confronted by this, he backs off a bit and says "yeah, the mathematician is right, but we won't get into distributions in this course, and, anyways, the theory of distributions is unnecessary for PRACTICAL purposes, and that's what this course is all about.."

What shall the poor student think of all of this?

My aim in creating this thread is to make an ultra-short intro to the mathematical way of looking at it.

It won't be very technical, but fairly rigorous, yet, I hope easily understood.

In the next post, I'll look into functionals, and the last post will provide the link between the functional perspective and the common way of "defining" the Dirac function.

Many students become frustrated when they first meet the Dirac Delta function, typically in a course involving electrostatics, or Laplace transforms.

As it is commonly presented, the Dirac function seems totally meaningless:

Either, it is "defined" as:

[tex]\int_{-\infty}^{\infty}\delta(x)dx=1,\delta(x)=0,x\neq0[/tex]

or, in conjunction with an arbitrary function f:

[tex]\int_{I}\delta(x)f(x)dx=f(0),0\in{I},\int_{I}\delta(x)f(x)dx=0,0\notin{I}[/tex]

where I is any interval on the real axis.

What sort of function can do these sort of things??

The student's confusion is not markedly lessened when:

the lecturer praises the Dirac function's ability to sample a function value, then the confusion increases when a mathematician scoffs at the whole "definition" and says "it is just a formalism, you've got to think of it as a distribution", and when the lecturer is confronted by this, he backs off a bit and says "yeah, the mathematician is right, but we won't get into distributions in this course, and, anyways, the theory of distributions is unnecessary for PRACTICAL purposes, and that's what this course is all about.."

What shall the poor student think of all of this?

My aim in creating this thread is to make an ultra-short intro to the mathematical way of looking at it.

It won't be very technical, but fairly rigorous, yet, I hope easily understood.

In the next post, I'll look into functionals, and the last post will provide the link between the functional perspective and the common way of "defining" the Dirac function.

Last edited by a moderator: