7.2 Vector Fields, Divergence and Curl or Rotation 85
(ii) Linearly Increasing Field
Letv(r)be given by
vμ=eμeνrν=xeμ. (7.16)In this case, the direction of the vector is still parallel to a constant unit vector, viz.
toe. This field can be derived from the potential function
Φ=( 1 / 2 )(eνrν)^2 =( 1 / 2 )x^2 , (7.17)which is that of an one-dimensional harmonic oscillator.
(iii) Radial and Cylindrical Fields
The vector field
vμ=rμ (7.18)has radial symmetry. It is the gradient ofΦ=( 1 / 2 )rνrν=( 1 / 2 )r^2 , which has the
functional form of the potential of an isotropic harmonic oscillator, in 3D.
The 2D version of a radial vector field is given by
vμ=rμ⊥=rμ−eμeνrν, (7.19)where the constant unit vector is perpendicular to the plane, in which the vector
arrows lie. In this case the potential function
Φ=( 1 / 2 )r⊥νrν⊥=( 1 / 2 )(x^2 +y^2 ) (7.20)is of the type of a 2D isotropic harmonic oscillator. Here the components of the
position vector in the plane perpendicular toehave been denoted byxandy(Fig.7.2).
Fig. 7.2 Cylindrical vector field