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

Graphical or geometric meaning

Radial parameter curves hold $v$ fixed and climb the paraboloid; angular curves hold $u$ fixed and form horizontal circles. Their tangent directions span the surface except at $u=0$, where polar coordinates collapse all angles to one point.

Common mistakes and why they fail

Verification and reasonableness checks

• Substitute coordinates into the Cartesian surface equation.
• Verify the normal is perpendicular to both tangent vectors.
• Inspect parameter boundaries and coordinate degeneracies.

Two tangent directions determine local geometry

A parametrized surface $\mathbf r(u,v)$ has tangent vectors $\mathbf r_u$ and $\mathbf r_v$. Their cross product is normal and supplies the area scale $\|\mathbf r_u\times\mathbf r_v\|$, provided it is nonzero. Reversing parameter order reverses orientation but not scalar area. The parameter domain controls which patch is traced and whether parts overlap; a familiar surface equation does not guarantee one-to-one coverage. To find a tangent plane, evaluate both tangent vectors at the parameter pair corresponding to the point. For flux, choose the cross-product order that matches the required normal. For area, use its magnitude. Check boundary curves of the parameter domain to understand the patch before integrating.

Practice

1. Parametrize the plane $z=x+y$.
2. What vector area factor is used?
3. What does a zero cross product indicate?

Connections and next steps

• Surface Area
• Surface Integrals
• Tangent Planes