distance from the source point. If we choose the volume to be a ball of radius around the source point, then Gauss' divergence theorem implies that It follows that on a sphere of radius that is centered on the source point, and hence 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 where denotes the natural logarithm. Note that, with the opposite sign convention, this is the potential generated by a pointlike sink (see point particle), which is the solution of the Euler equations in two-dimensional incompressible flow. Green's function A Green's function is a fundamental solution that also satisfies a suitable condition on the boundary of a volume . For instance, may satisfy Now if is any solution of the Poisson equation in : and assumes the boundary values on , then we may apply Green's identity, (a consequence of the divergence theorem) which states that The notations un and Gn denote normal derivatives on . In view of the conditions satisfied by and , this result simplifies to Thus the Green's function describes the influence at of the data and . For the case of the interior of a sphere of radius , the Green's function may be obtained by means of a reflection : the source point at distance from the center of the sphere is reflected along its radial line to a point P that is at a distance Note that if is inside the sphere, then P′ will be outside the sphere. The Green's function is then given by where denotes the distance to the source point and denotes the distance to the reflected point P′. A consequence of this expression for the Green's function is the Poisson integral formula'. Let , , and be spherical coordinates for the source point . Here 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 inside the sphere is given by where is the cosine of the angle between and . A simple consequence of this formula is that if is a harmonic function, then the value of at the center of