Surface Area

The central idea

For a regular parametric surface $\mathbf r(u,v)$ covering $S$ once, $A(S)=\iint_U\|\mathbf r_u\times\mathbf r_v\|dudv$. For a $C^1$ graph $z=f(x,y)$ over $D$, this becomes $A=\iint_D\sqrt{1+f_x^2+f_y^2}dA$. The factor measures local area stretching from parameter space.

Definitions, hypotheses, and notation

A surface may require multiple nonoverlapping parameter patches; areas add across their interiors, while shared edges have zero area. Coordinate singularities such as a sphere's poles do not invalidate the area integral when they occupy a measure-zero parameter set and the rest of the coverage is controlled.

For an implicit surface, solving locally as a graph over a coordinate plane gives a corresponding area factor. The best projection is one where the relevant normal component does not vanish; otherwise the graph representation becomes vertical and must be changed or split.

Conceptual meaning

A small parameter rectangle maps to a tangent parallelogram. Its area is the norm of the cross product of the two edge vectors. A tilted graph patch has more area than its horizontal projection, reflected by a factor at least one.

A dependable method and decision rule

1. Choose a parametrization and domain that cover the target surface once.
2. Compute two tangent vectors in a consistent parameter order.
3. Take the norm of their cross product.
4. Integrate over parameter space or the graph projection.
5. Check degeneracies, overlap, units, and projection lower bounds.

Fully worked example

Common mistakes and why they fail