G Some Tools of Vector Analysis
669
i.e., this Green’s function satisfies
∇
2 G P (x, x
) = δ
3 (x − x
) .
(G.50)
For the solution to Eq. (G.45) in three-dimensional space, we have
φ(r) = φ 0 (r) −
1
4π
1
|r − r |
f (r
)dV
.
(G.51)
The function φ 0 (r) is a solution of the homogeneous equation, and is determined by
boundary conditions on φ.
G.11 Derivative Decomposition of a Vector Field
Every differentiable vector field, E, in a simply connected region, can be expressed
in terms of the gradient of a scalar field, φ, plus the curl of a divergenceless field, A:
E = −∇φ + ∇ × A .
(G.52)
(The minus sign is put in for later convenience.) In fact, we can give explicit forms
for how to find the fields φ and A:
φ(r) =
1
4π
∇ · E(r )
| r − r |
dV
,
A(r) =
1
4π
∇ × E(r )
| r − r |
dV
.
(G.53)
These expressions can be found by taking the divergence and curl of the
left and right-hand sides of Eq. (G.52) and applying our solution to Poisson’s
equation (G.51).