Mathematical Methods for Physics and Engineering - Matematica.NET

10.5 SURFACESzT∂r∂uSu = c 1P∂r∂vr(u, v)v = c 2OyxFigure 10.4 The tangent plane T to a surface S at a particular point P ;u = c 1 and v = c 2 are the coordinate curves, shown by dotted lines, that passthrough P . The broken line shows some particular parametric curve r = r(λ)lying in the surface.10.5 SurfacesAsurfaceS in space can be described by the vector r(u, v) joining the origin O ofa coordinate system to a point on the surface (see figure 10.4). As the parametersu and v vary, the end-point of the vector moves over the surface. This is verysimilar to the parametric representation r(u) of a curve, discussed in section 10.3,but with the important difference that we require two parameters to describe asurface, whereas we need only one to describe a curve.In Cartesian coordinates the surface is given byr(u, v) =x(u, v)i + y(u, v)j + z(u, v)k,where x = x(u, v), y = y(u, v) andz = z(u, v) are the parametric equations of thesurface. We can also represent a surface by z = f(x, y) org(x, y, z) =0.Eitherof these representations can be converted into the parametric form in a similarmanner to that used for equations of curves. For example, if z = f(x, y) thenbysetting u = x and v = y the surface can be represented in parametric form byr(u, v) =ui + vj + f(u, v)k.Any curve r(λ), where λ is a parameter, on the surface S can be representedby a pair of equations relating the parameters u and v, for example u = f(λ)and v = g(λ). A parametric representation of the curve can easily be found bystraightforward substitution, i.e. r(λ) =r(u(λ),v(λ)). Using (10.17) for the casewhere the vector is a function of a single variable λ so that the LHS becomes a345