Scalar and Vector Potential Theory

Abstract

This provides a specialised representation for harmonic functions, which proves to be particularly convenient for solving certain boundary-value problems. Physically speaking, it models the properties of continuous electrostatic charge distributions over closed conductors, so providing an easy entry into the theory. Thus, if charges are introduced on a smooth, closed, conducting surface ∂B, we posit a continuous charge density σ(q) at every q ⊂ ∂B. It is convenient to write dq for the area element at q, in which case σ(q)dq defines the charge strength associated with dq. This generates an electrostatic potential g(p,q)σ(q)dq at any point p of space, where

$$[{\text{g(}}\underline {\text{p}} {\text{,}}\underline {\text{q}} {\text{) = g(}}\underline {\text{q}} {\text{,}}\underline {\text{p}} {\text{) = |}}\underline {\text{p}} {\text{ - }}\underline {\text{q}} {{\text{|}}^{{\text{ - 1}}}}$$

((1))

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