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SOME APPLICATIONS OF GREEN'S IDENTITIES MATH 2573H, FALL 2010
 

Summary: SOME APPLICATIONS OF GREEN'S IDENTITIES
MATH 2573H, FALL 2010
UNIVERSITY OF MINNESOTA
GREG W. ANDERSON
1. Goal of the notes
Let B(x0, a) denote the ball (solid sphere) of radius a centered at x0, and let
B(x0, a) denote the boundary of B(x0, a), i.e., the (hollow) sphere of radius a
centered at x0. We wish to verify that
(1) average of 1
|x-x1| on B(x0, a) =
1
a if x1 is inside B(x0, a),
1
|x1-x0| if x1 is outside B(x0, a).
We also assume that x1 is not on the boundary B(x0, a). (Otherwise something
different happens which we leave aside here.) Formula (1) has a striking physical
interpretation, to wit, the gravitational potential of a thin uniform spherical shell
of mass is constant inside the shell whereas outside the shell it is the same as if all
the mass were to be crushed to the center.
Formula (1) can proved by straightforward (somewhat brutal) calculation of

  

Source: Anderson, Greg W. - School of Mathematics, University of Minnesota

 

Collections: Mathematics