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 ____________________________________________________________________