Tensors for Physics

8.1 Line Integrals 113

Fig. 8.1 Two curvesC 1 and
C 2 starting and ending at the
same points

∫

C 1

f(r)drμ−

∫

C 2

f(r)drμ=

∮

C

f(r)drμ= 0.

HereC=C 1 −C 2 is a closed curve.
On the other hand, when

∫

C 1 f(r)drμ=

∫

C 2 f(r)drμholds true, in a special case,
for arbitrary curvesC 1 andC 2 within the regionB, then one has

∮

C

f(r)drμ= 0.

This then applies to any closed curve withinB, provided thatBis asimply connected
region, i.e. when there are no “holes” inB.

8.1.3 Line Integrals for Scalar and Vector Fields

(i) Scalar Fields

As mentioned before, the line integral (8.1) is a vector, when the field functionfis
a scalar.
A simple example isf=1. Then one obtains

Iμ=

∫p 2

p 1

rμ(p)dp=rμ(p 2 )−rμ(p 1 ),

which is the vector pointing fromr 1 tor 2 , cf. Fig.8.2.
The line integral, forf=1, should not be confused with the arc lengths, which
is given by

s=

∫

C

|dr|=

∫p 2

p 1

(

drμ

dp

drμ

dp

) 1 / 2

dp.

The quantitysis a scalar.