Parametric surface

Fixing \(v=v_0\) and letting \(u\) vary traces a curve on the surface; fixing \(u=u_0\) traces another. These coordinate curves form a grid, and their tangent vectors are the partial derivatives \(\mathbf r_u\) and \(\mathbf r_v\). It is the same idea as a parametric curve, with one more parameter.

If \(\mathbf r_u\times\mathbf r_v\ne\mathbf 0\), that cross product is a normal vector to the surface, and the tangent plane at \(\mathbf r(u_0,v_0)\) is the plane through that point with that normal. When \(\mathbf r_u\times\mathbf r_v=\mathbf 0\) the parametrization is singular: at the apex of the cone \(\langle v\cos u,\,v\sin u,\,v\rangle\) this happens at \(v=0\). Moreover, \(\lVert\mathbf r_u\times\mathbf r_v\rVert\) is the area of the parallelogram spanned by the tangent vectors, the key fact for surface area.

Standard parametrizations: the graph \(z=f(x,y)\) is \(\langle x,\,y,\,f(x,y)\rangle\), with normal \(\langle -f_x,\,-f_y,\,1\rangle\); the cylinder of radius \(a\) is \(\langle a\cos u,\,a\sin u,\,v\rangle\); the sphere of radius \(a\) is \(\langle a\sin\varphi\cos\theta,\,a\sin\varphi\sin\theta,\,a\cos\varphi\rangle\) with \(0\le\varphi\le\pi\) and \(0\le\theta\le2\pi\).

A typical mistake is to think the parametrization is unique. The same surface admits many, and what matters is the set of points, not the formula. Another is forgetting the domain \(D\): without it, \(\langle\cos u,\sin u,v\rangle\) is an infinite cylinder, not a can. This concept uses the parametric curve and the cross product, and prepares the surface integral.