Tensors for Physics

84 7 Fields, Spatial Differential Operators

r=|r 1 −r 2 |only, when these particles are effectively round and do not posses any
internal directional properties, like electric dipole or quadrupole moments, which
influence their interaction.

7.2 Vector Fields, Divergence and Curl or Rotation

Letv(r)be a vector field with Cartesian componentsvμ(r). Here the symbolvcan be

associated with “vector”, in general, or with the flow velocity field of hydrodynamics.

Thevectorfieldmightalsobeassociatedwiththedisplacementinducedbyadeforma-

tion of an elastic solid. The mathematical considerations to be presented are invariant

with respect to different interpretations in physics.

Vector fields can be visualized as a field of arrows. At each pointr, an arrow can

be drawn, whose length and direction is determined byv(r). Firstly, some examples

are considered.

7.2.1 Examples for Vector Fields

(i) Homogeneous Field

As previously mentioned, a vector field of the typev=const., where the vector

everywhere has constant length and direction, is referred to as a homogeneous field.

Let the direction be specified by the constant unit vectore. Then one has, apart from

a numerical factor,

vμ=eμ=const.

This field is the gradient of the simple potential functionΦ=rνeν=x(Fig.7.1).

Fig. 7.1 Homogeneous and

linearly increasing vector

fields