94 7 Fields, Spatial Differential Operators
Application in Electrostatics
For static electric fieldsE, the curl vanishes:∇×E=0. TheE-field is the negative
gradient of electrostatic potentialφ(r), which obeys the Laplace equation:
Eμ=−∇μφ⇒Δφ= 0. (7.44)Usually, in applications to specific problems, the electrostatic potential has to obey
certain boundary conditions, e.g.φmust be constant on an electrically conducting
(metal) surface.
7.3.5 Conventional Classification of Vector Fields
Based on the previous discussions, vector fields can be classified, as shown in the
Table7.1.
This conventional classification scheme does not include the information on the
symmetric traceless part∇vof the gradient of the vector field. There are, however,
many applications in physics, where∇vmatters. Examples shall be presented later.
7.3.6 Second Spatial Derivatives of Spherically Symmetric
Scalar Fields
From the examples presented in Sect.7.2.2, on might assume, that∇v is nonzero
only when the pertaining potential function involves special directions, e.g. specified
by constant unit vectors in the examples (ii), the 2D version of (iii), in (iv) and (v).
However, also the generalspherical potential, which depends on the position vectorr
only via its magnituder=
√
rκrκ, leads to non-vanishing values for∇v. Of course,
as discussed before, the antisymmetric part of∇v, associated with the vorticity, is
zero when a scalar potential exists.
Table 7.1The conventional classification of vector fields
∇×v= 0 ∇×v = 0
∇·v= 0 vorticity and source free source free vorticity field
Laplace field with vector potentialA
v=∇Φ,ΔΦ= 0 v=∇×A
∇·v = 0 vorticity free Poisson field general vector field
with source densityρ
v=∇Φ,ΔΦ=ρ v=∇Φ+∇×A