Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.5 SURFACES

x

y

z

S

O

T

P

r(u, v)

∂r

∂u

∂r

∂v

v=c 2

u=c 1

Figure 10.4 The tangent planeTto a surfaceSat a particular pointP;

u=c 1 andv=c 2 are the coordinate curves, shown by dotted lines, that pass

throughP. The broken line shows some particular parametric curver=r(λ)

lying in the surface.

10.5 Surfaces

AsurfaceSin space can be described by the vectorr(u, v) joining the originOof

a coordinate system to a point on the surface (see figure 10.4). As the parameters

uandvvary, the end-point of the vector moves over the surface. This is very

similar to the parametric representationr(u) of a curve, discussed in section 10.3,

but with the important difference that we requiretwoparameters to describe a

surface, whereas we need only one to describe a curve.

In Cartesian coordinates the surface is given by

r(u, v)=x(u, v)i+y(u, v)j+z(u, v)k,

wherex=x(u, v),y=y(u, v)andz=z(u, v) are the parametric equations of the

surface. We can also represent a surface byz=f(x, y)org(x, y, z)=0.Either

of these representations can be converted into the parametric form in a similar

manner to that used for equations of curves. For example, ifz=f(x, y)thenby

settingu=xandv=ythe surface can be represented in parametric form by

r(u, v)=ui+vj+f(u, v)k.

Any curver(λ), whereλis a parameter, on the surfaceScan be represented

by a pair of equations relating the parametersuandv, for exampleu=f(λ)

andv=g(λ). A parametric representation of the curve can easily be found by

straightforward substitution, i.e.r(λ)=r(u(λ),v(λ)). Using (10.17) for the case

where the vector is a function of a single variableλso that the LHS becomes a