ABSTRACT
A harmonic potential or function is a solution of the Laplace equation, and it has a singularity at a point multipole, where Poisson’s equation is satis”ed. ¡e analytic (generalized) functions, used as solutions of the Laplace equation in the plane (in space), are thus particular harmonic functions in two (three) dimensions. ¡ere are general properties of harmonic potentials that are common to both plane and spatial cases and extend to any higher dimension, for example, the boundary conditions (Section 9.1) that render the solution of the Laplace and Poisson equation unique (1) in a compact region, that is, in a region of ”nite extent; and (2) if the region is noncompact, that is, includes the point-at-in”nity, an additional asymptotic condition is needed for the unicity of solution of the Laplace and Poisson equations. ¡e proof of (1) and (2) uses the Green integral identities that can also be applied to (3) the ”eld (Section 9.2) due to a distribution of sources in a bounded region. ¡e ”eld due to a distribution of sources in a unbounded region can be obtained via the in›uence or Green function, which is a ramp function/logarithmic potential/inverse distance potential/or inverse power distance potential in, respectively, one/ two/three/or more dimensions N = 1/2/3/≥ 4; this includes the logarithmic (Newtonian) potential in two (three) dimensions (Section 9.4). ¡e properties of harmonic functions that apply in any dimension (Section 9.3) include (4) the theorem of the extrema that a function harmonic in a region takes the maximum and the minimum value on the boundary; for example, a function harmonic (analytic) in a three-(two-) dimensional region has the maximum (minimum) modulus on the boundary surface (curve); (5) the mean value theorem that a function harmonic on a hypersphere has at the center the mean of the values on the boundary; for example, if a function is harmonic (analytic) in a sphere (disk), the value at the center is the mean over the spherical surface (circular boundary); (6) the theorem of constancy (nullity) that at a function harmonic in a region and constant (zero) on the boundary is also constant (zero) in the interior; for example, if a function is harmonic (analytic) in a three-(two-) dimensional region and is constant (zero) on the boundary, then it is constant (zero) in the interior as well. All these properties can be interpreted in terms of (a) the temperature in steady heat conduction, the scalar potential for irrotational, incompressible ›ow, the gravity ”eld and the electrostatic ”eld; and (b) the vector potential for the magnetostatic ”eld and rotational ›ow. ¡ey lead to the Newton/Coulomb (Biot-Savart) law for the force [Section 9.4 (Section 9.5)] in the gravity/electric (magnetic) ”eld that is irrotational (solenoidal). ¡e vector potential exists in three dimensions (Section 9.5) and reduces to a scalar ”eld function in two dimensions, and its extension beyond three dimensions requires multivectors (Notes 9.20 through 9.52) that are considered in the context of tensor calculus (Notes 9.6 through 9.52) in order to address questions of unicity and invariance. ¡e scalar potential applies (Section 9.4) in one, two, and three dimensions, and extends to any higher dimension, for example, as concerns (7) the multipolar expansion that leads to generalized or hyperspherical Legendre polynomials (Section 9.6); (8) the speci”cation of the potential and the ”eld of multidimensional multipoles in terms of hyperspherical and hypercylindrical coordinates (Section 9.7); (9) the hypersphere theorem (Section 9.8) that generalizes the circle (sphere)
theorem [Section I.24.6 (Section 6.6)] to a hypersphere in a space of any dimension; and (10) the method of images also extends (Section 9.9) from straight lines (Chapter I.16)/planes (Sections 8.7 and 8.8) [circles (Section I.24.7)/spheres (Section 8.9)] to hyperplanes (hyperspheres).
