Tensors for Physics

7.1 Scalar Fields, Gradient 79

the equipotential surfaces are coaxial cylinders. In this case, the values of the

potential are determined by a function of a single variable, viz.:ρ.

3. Spherical Geometry. Of special importance are potential functions which depend
   onthepositionvectorrjustviaitsmagnitude,i.e.viathedistancer=|r|=

√

r·r

of the pointrfrom the centerr=0. The equipotential surfaces of such aspherical

potentialΦ=Φ(r)are concentric spheres. In this special case, again, a function

depending on one variable only, suffices to quantify the value of the potential.

7.1.2 Differential Change of a Potential, Nabla Operator

The change dΦof the value of a potential functionΦ, when one goes from the
positionrto an adjacent positionr′=r+dris given by the differenceΦ(r+dr)−
Φ(r). Taking into account that one is dealing with a function depending on the three
components of the position vector, one has explicitly

dΦ=Φ(r 1 +dr 1 ,r 2 +dr 2 ,r 3 +dr 3 )−Φ(r 1 ,r 2 ,r 3 ).

It is assumed that the potential function can be expanded in a power series with
respect to the increment dr.Differential changeimplies that the magnitude of dris
small enough, such that terms nonlinear in drcan be disregarded. Then

dΦ=

∂Φ

∂r 1

dr 1 +

∂Φ

∂r 2

dr 2 +

∂Φ

∂r 3

dr 3 =

∂Φ

∂rμ

drμ (7.1)

is obtained. In 3D, the quantity∂∂Φrμhas three components and it is a vector, since the

scalar product with the vector drμyields the scalar dΦ.
The partial differentiation with respect to the Cartesian components of the position
vector is frequently denoted by thenabla operator

∇μ:=

∂

∂rμ

. (7.2)

Also the symbol∂μis used for the spatial partial derivative. Here the nabla operator
is preferred. Nabla applied on a scalar field is also referred to as thegradient field,
and sometimes denoted by gradΦ.

7.1.3 Gradient Field, Force.

A geometric interpretation of the gradient field∇μΦ(r)is obtained as follows. Con-
sider the case, where the differential change dris tangential to an equipotential