Tensors for Physics

112 8 Integration of Fields

Here drμis the Cartesian component of the differential change dralong the curve
C. The line integral is also calledcurve integralorpath integral.
When the curve is determined by aparameter representation

rμ=rμ(p), p 1 <p<p 2 ,

the line integral (8.1) can be expressed as the ordinary Riemann integral

Iμ=

∫p 2

p 1

f(r(p))

drμ

dp

dp. (8.2)

The parameter valuesp 1 andp 2 correspond to the start and end points of the curve
C, i.e.r(pi)=ri,i= 1 ,2.

Remark: in some applications, it may be convenient to use piecewise different
parameter representations for the curveC. A simple example is a curve depicted
in the Fig.8.3.
The sign of a line integral changes, when the integration is performed backwards
along the curve considered. This is obvious in the parameter representation since

∫p 2

p 1

...dp=−

∫p 1

p 2

...dp.

Notice: the line integral is a vector, provided thatfis a scalar. When the function
fis the component of a tensor of rank,e.g.f≡gν 1 ...ν, the resulting line integral
Iμν 1 ...νis a component of a tensor of rank+1. Some examples are considered in
Sect.8.1.3.

Remark: in the literature, the term “line integral” is also used for an integral with
the scalar integration element dswheresis the arc length of the curve. In that case,
the integral is a tensor of the same rank as that of the integrandf. Such integrals are
not considered here.

8.1.2 Closed Line Integrals

The symbol

∮

is used when the line integration is performed along a closed curveC:

Iμ=

∮

C

f(r)drμ. (8.3)

This type of line integral is also calledloop integralorcontour integral.
Next two curvesC 1 andC 2 are considered, which have common start and end
points, see Fig.8.1. In general, one has

∫

C 1 f(r)drμ =

∫

C 2 f(r)drμand conse-
quently