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MathGroup Archive 1997

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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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