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Equation data
Category:2. First-Order Partial Differential Equations
Subcategory:2.1. Linear Equations
{\rm div}{\mathbf U}=-\frac{q}{4\pi}\,{\delta (r)} \hfill \break 
where $\mathbf r=\{x,y,z\},$ \space $ r=\sqrt {x^2+y^2+z^2}$. Bounary condion is $ {\mathbf U \rightarrow    0}  $ when 
$ r \rightarrow \infty $. $
Particular solution is [1,2]: $$ {\mathbf U}=q\frac {z^2-2x^2-2y^2}{4\pi r^5}\mathbf r $$
in spherical coordinates: $$ {\mathbf U}=\frac {q}{4\pi r^2}\{2-3\cos \theta,0,0\}. $$
For equition (1) the particular potential solution is known [3]:
$${\mathbf U}_p={\rm grad} \frac {q}{4\pi r}=\frac {q}{4\pi r^3}\mathbf r$$
 or in spherical coordinates: $${\mathbf U}_p=\frac {q}{4\pi r^2}\{1,0,0\}.$$
Novelty:New solution(s) / integral(s)
References:1. A.Ivanchin. Nepotenzialnoe vihrevoe reshenie zadachi ob electrone. Rotational Processes in Geology and Physics. Moscow. 2007, pp.211-218. (In russian). 2. A.Ivanchin. arXiv:0902.1286 [pdf]
3. G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers, McGraw Hill, New York, 1968
Author/Contributor's Details
Last name:Ivanchin
First name:Alexander
Statistic information
Submission date:Sun 08 Mar 2009 07:30
Edits by author:0

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