Re: Divergence and Dirac Delta Function

• To: mathgroup at smc.vnet.net
• Subject: [mg9140] Re: Divergence and Dirac Delta Function
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Tue, 14 Oct 1997 10:01:55 -0400
• Organization: University of Western Australia
• Sender: owner-wri-mathgroup at wolfram.com

Elvis Dieguez wrote:

> I am a physics student currently enrolled in an upper level
> Electrodynamics course and, while playing around with Mathematica 3.0
> for Students, I discovered that if I tell Mathematica to compute the
> divergence of:      unit vector(r) / r^2 it will return the value of
> zero.  However, I know that, at the very least in Electrodynamics, the
> correct value should be:  4*Pi*(DiracDelta Function).  Is it possible to
> get Mathematica to output this value without explictly telling the
> program that this is the value?

The package

In[1]:= << Calculus`DiracDelta`

adds rules so that you handle some derivatives and integrals involving
distributions.  For example,

?UnitStep

UnitStep[x] is a function that is 1 for x > 0 and 0 for x <  0.
UnitStep[x1, x2, ...] is 1 for (x1 > 0) && (x2 > 0) &&... and 0
for  (x1 < 0) || (x2 < 0) || ... .

?DiracDelta

DiracDelta[x] is a distribution that is 0 for x != 0 and
satisfies Integrate[DiracDelta[x], {x, -Infinity,Infinity}] =
1. DiracDelta[x1, x2, ...] is a distribution that is 0 for x1 != 0 ||
x2 != 0 || ... and satisfies Integrate[DiracDelta[x1, x2, ...],
{x1, -Infinity, Infinity}, {x2, -Infinity, Infinity}, ...] = 1.

To compute the divergence of a point charge with field proportional to
1/r^2 you effectively need to compute the radial derivative of a
step-function, e.g.,

In[2]:= D[UnitStep[r], r]

Out[2]= DiracDelta[r]

In other words, the DiracDelta arises because of a step discontinuity at
r=0.

You might want to also have a look at

http://www.pd.uwa.edu.au/Physics/Courses/Second_Year/ElectroMagnetism.html

In particular Chapter 3 and Appendix B.

Cheers,
Paul

____________________________________________________________________
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA  6907                     mailto:paul at physics.uwa.edu.au
AUSTRALIA                             http://www.pd.uwa.edu.au/~paul

God IS a weakly left-handed dice player
____________________________________________________________________

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