Parametric Surfaces

The central idea

A parametric surface is a map $\mathbf r:U\subset\mathbb R^2\to\mathbb R^3$, $\mathbf r(u,v)=\langle x(u,v),y(u,v),z(u,v)\rangle$. At a regular point, $\mathbf r_u$ and $\mathbf r_v$ are linearly independent and span the tangent plane; $\mathbf r_u\times\mathbf r_v$ is an oriented normal and its norm is the area scale factor.

Definitions, hypotheses, and notation

Regularity is a property of the parametrization at a parameter point, not necessarily a geometric singularity of the surface. Spherical coordinates degenerate at poles even though a sphere is smooth. A different chart can cover such points. Globally, many surfaces require multiple charts to avoid overlap or degeneration.

For a graph $z=f(x,y)$, the standard parametrization $\langle x,y,f(x,y)\rangle$ gives cross product $\langle-f_x,-f_y,1\rangle$ in one order. This links parametric normals, tangent-plane formulas, and the graph surface-area factor.

Conceptual meaning

Two parameter directions create a coordinate grid on the surface. A tiny parameter rectangle maps to an approximate parallelogram spanned by $\mathbf r_u du$ and $\mathbf r_v dv$. Their cross product captures both local orientation and stretched area.

A dependable method and decision rule

1. State the parameter domain and identify repeated or degenerate coordinates.
2. Compute both tangent vectors in the displayed parameter order.
3. Take their cross product and check it is nonzero where regularity is claimed.
4. Use the cross product for a normal, tangent plane, area, or oriented surface element.
5. Reverse factor order when the opposite orientation is required.

Fully worked example

Common mistakes and why they fail