Tensors for Physics

8.1 Line Integrals 115

or

I=Φ(r 2 )−Φ(r 1 ). (8.7)

For the case of a vector field obtained as a gradient of a potential, the line integral is
given by the difference of the potential between the end and the start points of the
curve, irrespective of its path in between these points. As consequence, the integral
along a closed curve vanishes:

∮

C

vμ(r)drμ= 0. (8.8)

For (8.8) to be valid, the region in which the curveClies, has to be compact, it should
not have any holes.

8.1.5 Computation of the Potential for a Vector Field

For a vector fieldvwhich obeys the integrability condition (7.35), or equivalently
∇×v=0, the pertaining potential function can be computed with the help of a
line integral. A convenient integration path can be chosen, starting from an arbitrary
pointr 1 and ending at the variable positionr. Then the path integralIis a function
ofr. More specifically, one has

I=I(r)=

∫r

r 1

vμdrμ=Φ(r)−Φ(r 1 ), (8.9)

or equivalently

Φ(r)=

∫r

r 1

vμdrμ+const. (8.10)

Obviously, the potential functionΦis only determined up to a constant const.

A Simple Example: Homogeneous Vector Field

As a simple special case the constant homogeneous vector fieldv = const. is
considered. Then one has

I(r)=vμ

∫r

r 1

drμ=v·(r−r 1 ),

and consequently

Φ(r)=v·r+const.