§1.6 Green’s functions; Helmholtz Theorem Christopher Crawford PHY

Outline Helmholtz theorem – L/T projection 2 nd derivatives – there’s really only one! Longitudinal / transverse projections of the Laplacian Longitudinal / transverse separation of a vector field Scalar / Vector potentials Green’s function G(x,y) Gradient & Laplacian of 1/r potential – point charge Definition of Green’s function– `tent’ function Expansion in delta functions– `pole’ function The Laplacian as a linear operator – `X-ray’ operator The inverse Laplacian operator– `shrink-wrap’ operator Particular solution of Poisson’s equation 2

2 nd derivatives: only one! All combinations of vector derivatives: the differential chain 3

L/T separation of E&M fields 4

Scalar and vector potentials Scalar potential (Flow) conservative or irrotational field integral formulation source: divergence (charge) gauge invariance Vector potential (Flux) solenoidal or incompressible field integral formulation source: curl (current) gauge invariance 5

Potential and field of a point source 6 Gradient DivergenceFlux Planar angle Solid angle

Green’s function G(r,r’) 7 The potential of a point-charge A simple solution to the Poisson’s equation Zero curvature except infinite at one spot

Multiple poles 8

Infinitely dense poles 9

General solution to Poisson’s equation Expand f(x) as linear combination of delta functions Invert linear Lapacian on each delta function individually 10

Green’s functions as propagators 11 Action at a distance: G(r’,r) `carries’ potential from source at r' to field point (force) at r In quantum field theory, potential is quantized G(r’,r) represents the photon (particle) that carries the force How do you measure the `shape’ of the proton?