Editing Laplace's equation (section)

===Fundamental solution===
A [[fundamental solution]] of Laplace's equation satisfies
<math display="block"> \Delta u = u_{xx} + u_{yy} + u_{zz} = -\delta(x-x',y-y',z-z'),</math>
where the [[Dirac delta function]] {{math|''δ''}} denotes a unit source concentrated at the point {{math|(''x''′, ''y''′, ''z''′)}}. No function has this property: in fact it is a [[Distribution (mathematics)|distribution]] rather than a function; but it can be thought of as a limit of functions whose integrals over space are unity, and whose support (the region where the function is non-zero) shrinks to a point (see [[weak solution]]). It is common to take a different sign convention for this equation than one typically does when defining fundamental solutions. This choice of sign is often convenient to work with because −Δ is a [[positive operator]]. The definition of the fundamental solution thus implies that, if the Laplacian of {{math|''u''}} is integrated over any volume that encloses the source point, then
<math display="block"> \iiint_V \nabla \cdot \nabla u \, dV =-1.</math>

The Laplace equation is unchanged under a rotation of coordinates, and hence we can expect that a fundamental solution may be obtained among solutions that only depend upon the distance {{mvar|r}} from the source point. If we choose the volume to be a ball of radius {{mvar|a}} around the source point, then Gauss's [[divergence theorem]] implies that
<math display="block"> -1= \iiint_V \nabla \cdot \nabla u \, dV = \iint_S \frac{du}{dr} \, dS = \left.4\pi a^2 \frac{du}{dr}\right|_{r=a}.</math>

It follows that 
<math display="block"> \frac{du}{dr} = -\frac{1}{4\pi r^2},</math>
on a sphere of radius {{mvar|r}} that is centered on the source point, and hence
<math display="block"> u = \frac{1}{4\pi r}.</math>

Note that, with the opposite sign convention (used in [[physics]]), this is the [[potential]] generated by a [[point particle]], for an [[inverse-square law]] force, arising in the solution of [[Poisson equation]]. A similar argument shows that in two dimensions
<math display="block"> u = -\frac{\log(r)}{2\pi}.</math>
where {{math|log(''r'')}} denotes the [[natural logarithm]]. Note that, with the opposite sign convention, this is the [[potential]] generated by a pointlike [[Potential flow|sink]] (see [[point particle]]), which is the solution of the [[Euler equations (fluid dynamics)|Euler equations]] in two-dimensional [[incompressible flow]].