Tensors for Physics

116 8 Integration of Fields

Remark: in physics, forcesFassociated with a potential are given by the negative

gradient of the potential function. Thus the mechanical potentialΦis computed as

Φ(r)=−

∫r

r 1

Fμdrμ+const. (8.11)

8.1 Exercise: Compute Path Integrals along a Closed Curve

Consider a special closed path composed of the curveC 1 withrgiven by

{x, 0 , 0 }, −ρ≤x≤ρ,

and the curveC 2 withrdetermined by

{x,y, 0 }, x=ρcosφ, y=ρsinφ, 0 ≤φ≤π.

The curveC 1 is a straight line,C 2 is a semi-circle with the constant radiusρ,cf.

Fig.8.3. The differential drneeded for the integration is equal to dx{ 1 , 0 , 0 }and

ρdφ{−sinφ,cosφ, 0 }for the curvesC 1 andC 2 , respectively. Compute the loop

integralI=

∮

Cvμdrμalong the closed curve defined here for the following three
vector fields:
(i) homogeneous field, wherev=e=const., witheparallel to thex-axis;
(ii) radial field, wherev=r;
(iii)solid-like rotation field, wherev=w×r, with the constant axial vectorw
parallel to thez-axis.
Hint: guess whetherI=0orI =0 is expected for these vector fields, before
you begin with the explicit calculation. Denote the line integrals along the curves
C 1 andC 2 byI 1 andI 2. The desired integralIalong the closed curve is the sum
I 1 +I 2.

Fig. 8.3 Closed curve

composed of a semi-circle

and a straight line