Tensors for Physics

114 8 Integration of Fields

Fig. 8.2 Line integral yields
a relative position vector

(ii) Vector Fields

Now letfbe the componentvνof a vector fieldv=v(r). Then the line integral

Iμν=

∫

C

vν(r)drμ (8.4)

is a second rank tensor. In general, it can be decomposed into an isotropic part which
is proportional to its trace times the unit tensorδμν, an antisymmetric part, and a
symmetric traceless part, cf. Chap. 6.
In some applications, the trace

I≡Iμμ=

∫

C

vμ(r)drμ (8.5)

is needed. The scalar quantityIis referred to asthe curve integral of a vector field.

8.1.4 Potential of a Vector Field

Now consider the special case where a vector fieldvis given by the gradient of a
scalar potential fieldΦ=Φ(r),

vμ=∇μΦ.

The scalar line integral (8.5) of such a vector field is computed according to

I=

∫

C

vμ(r)drμ=

∫p 2

p 1

∂Φ

∂rμ

drμ

dp

dp=

∫p 2

p 1

dΦ

dp

dp=Φ(r(p 2 ))−Φ(r(p 1 )),

(8.6)