Editing Laplace's equation (section)

===Green's function===
A [[Green's function]] is a fundamental solution that also satisfies a suitable condition on the boundary {{mvar|S}} of a volume {{mvar|V}}. For instance, 
<math display="block">G(x,y,z;x',y',z')</math>
may satisfy
<math display="block"> \nabla \cdot \nabla G = -\delta(x-x',y-y',z-z') \qquad \text{in } V,</math> 
<math display="block"> G = 0 \quad \text{if} \quad (x,y,z) \qquad \text{on } S.</math>

Now if {{math|''u''}} is any solution of the Poisson equation in {{mvar|V}}:
<math display="block"> \nabla \cdot \nabla u = -f,</math>

and {{math|''u''}} assumes the boundary values {{math|''g''}} on {{mvar|S}}, then we may apply [[Green's identities|Green's identity]], (a consequence of the divergence theorem) which states that

<math display="block"> \iiint_V \left[ G \, \nabla \cdot \nabla u - u \, \nabla \cdot \nabla G \right]\, dV = \iiint_V \nabla \cdot \left[ G \nabla u - u \nabla G \right]\, dV = \iint_S \left[ G u_n -u G_n \right] \, dS. \,</math>

The notations ''u<sub>n</sub>'' and ''G<sub>n</sub>'' denote normal derivatives on {{math|''S''}}. In view of the conditions satisfied by {{math|''u''}} and {{math|''G''}}, this result simplifies to

<math display="block"> u(x',y',z') =  \iiint_V G f \, dV - \iint_S G_n g \, dS. \,</math>

Thus the Green's function describes the influence at {{math|(''x''′, ''y''′, ''z''′)}} of the data {{math|''f''}} and {{math|''g''}}. For the case of the interior of a sphere of radius {{math|''a''}}, the Green's function may be obtained by means of a reflection {{harv| Sommerfeld| 1949}}: the source point {{math|''P''}} at distance {{math|''ρ''}} from the center of the sphere is reflected along its radial line to a point ''P''' that is at a distance

<math display="block"> \rho' = \frac{a^2}{\rho}. \,</math>

Note that if {{math|''P''}} is inside the sphere, then ''P&prime;'' will be outside the sphere. The Green's function is then given by
<math display="block"> \frac{1}{4 \pi R} - \frac{a}{4 \pi \rho R'}, \,</math>
where {{mvar|R}} denotes the distance to the source point {{mvar|''P''}} and {{math|''R''′}} denotes the distance to the reflected point ''P''′. A consequence of this expression for the Green's function is the '''[[Poisson integral formula]]'''. Let {{mvar|ρ}}, {{mvar|θ}}, and {{mvar|φ}} be [[spherical coordinates]] for the source point {{math|''P''}}.  Here {{mvar|θ}} denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice. Then the solution of the Laplace equation with Dirichlet boundary values {{math|''g''}} inside the sphere is given by {{harv|Zachmanoglou|Thoe|1986|loc=p. 228}}
<math display="block">u(P) =\frac{1}{4\pi} a^3\left(1-\frac{\rho^2}{a^2}\right) \int_0^{2\pi}\int_0^{\pi} \frac{g(\theta',\varphi') \sin \theta'}{(a^2 + \rho^2 - 2 a \rho \cos \Theta)^{\frac{3}{2}}} d\theta' \, d\varphi'</math> 
where
<math display="block"> \cos \Theta = \cos \theta \cos \theta' + \sin\theta \sin\theta'\cos(\varphi -\varphi')</math>
is the cosine of the angle between {{math|(''θ'', ''φ'')}} and {{math|(''θ''′, ''φ''′)}}.  A simple consequence of this formula is that if {{math|''u''}} is a harmonic function, then the value of {{math|''u''}} at the center of the sphere is the mean value of its values on the sphere. This mean value property immediately implies that a non-constant harmonic function cannot assume its maximum value at an interior point.