Tensors for Physics

118 8 Integration of Fields

8.2.2 Examples for Parameter Representations of Surfaces

(i) Plane

Leteandube two orthogonal unit vectors. Then

r=pe+qu

represents the plane spanned by the vectorseandu. With thex- andy-axes of a
coordinate system chosen parallel toeandu, the plane is represented byx=p,
y =q,z=0. The vector normal to the plane is parallel to thez-direction. The
(p,q)-mesh covering the plane are orthogonal straight lines parallel to thex- and
y-axes.
Alternatively, theplanar polar coordinatesρandφcan be used as parameters to
represent the plane. Here the position vector within thex–y-plane is expressed as
r={ρcosφ, ρsinφ, 0 }, and consequently

∂r

∂ρ

={cosφ,sinφ, 0 }=eρ,

∂r

∂φ

={−ρsinφ, ρcosφ, 0 }=ρeφ. (8.15)

Unit vectors inρ- andφ-directions are denoted byeρandeφ, see Fig.8.5. These
vectors are orthogonal. The plane is covered by a mesh of straight lines starting at
the origin and concentric circles, corresponding toφ=const. andρ=const.

(ii) Cylinder Mantle

For a circular cylinder with constant radiusρ, its mantle surface is described by

r={ρcosφ, ρsinφ,z}, (8.16)

Fig. 8.5 Planar polar
coordinates with the vectors
eρandeφ