8.2 Surface Integrals, Stokes 117
8.2 Surface Integrals, Stokes
8.2.1 Parameter Representation of Surfaces
Surfaces in 3D space can e.g. be described by a relation of the formz=z(x,y),
where the components of the position vector on the surface are denoted byx,y,z.
Implicitly, the surface can also be determined byΦ=Φ(x,y,z)=const., where
Φis a scalar function, by analogy to potentials, cf. Sect.7.1.1. Sometimes, it is
advantageous to use a parameter presentation of the form
rμ=rμ(p,q), (8.12)for the Cartesian components of the position vectorrlocated on the surface, with
the two parameterspandq.
For a constantq,e.g.q=q 0 , the relationrμ =rμ(p,q 0 )describes a curve
with the curve parameterp. Likewise, forp= p 0 , the relationrμ =rμ(p 0 ,q)
is a curve with the curve parameterq. Different valuesp =p 0 ,p 1 ,p 2 ,...and
q=q 0 ,q 1 ,q 2 ,...yield a(p,q)-mesh of curves on the surface, cf. Fig.8.4.
Provided that the tangential vectors
t
p
μ=∂rμ
∂p, t
q
μ=∂rμ
∂q, (8.13)
in thep- andq-directions are not parallel to each other, the curvesr=r(p,q=
const.)andr=r(p=const.,q)cover the surface. Subject to this condition, a vector
normal to the surface is inferred from
ελμνtμptνq=ελμν∂rμ
∂p∂rν
∂q= 0. (8.14)
Fig. 8.4 Schematic view of
a surface generated by a
mesh of curves