Parameterized surface

A parameterized surface in 3D space is the image of a function

$\mathbf{r}(u,v)=⟨x(u,v),y(u,v),z(u,v)⟩$

with $(u,v)$ ranging over a domain $D\subseteq {\mathbb{R}}^2$. Two parameters trace out a 2D object in 3D space.

Two parameters → two degrees of freedom on the surface. The surface itself lives in 3D — you can walk around it, view from different angles. Don’t say “2D object” in a way that suggests it sits in 2D space. Same caution as for curves, which have one parameter but live in 3D.

Standard parameterizations

Graph $z=f(x,y)$. Use $x,y$ as parameters:

$\mathbf{r}(x,y)=⟨x,y,f(x,y)⟩.$

Cylinder $x^2+y^2=a^2$, $0\le z\le h$. Parameters $\varphi ,z$:

$\mathbf{r}(\varphi ,z)=⟨a\cos \varphi ,a\sin \varphi ,z⟩,0\le \varphi \le 2\pi ,0\le z\le h.$

Sphere of radius $a$. Parameters $\theta$ (polar angle from $+z$) and $\varphi$ (azimuthal), using the engineering/physics convention matching Spherical coordinates:

$\mathbf{r}(\theta ,\varphi )=⟨a\sin \theta \cos \varphi ,a\sin \theta \sin \varphi ,a\cos \theta ⟩,0\le \theta \le \pi ,0\le \varphi \le 2\pi .$

(Mathematicians sometimes swap $\theta$ and $\varphi$. The labels follow the convention of whatever source is being used — always check before substituting.)

Cone $z=\sqrt{x^2+y^2}$ up to $z=h$. Parameters $r,\varphi$:

$\mathbf{r}(r,\varphi )=⟨r\cos \varphi ,r\sin \varphi ,r⟩,0\le r\le h,0\le \varphi \le 2\pi .$

Plane through ${\mathbf{P}}_0$ spanned by $\mathbf{u},\mathbf{v}$:

$\mathbf{r}(s,t)={\mathbf{P}}_0+s\mathbf{u}+t\mathbf{v}.$

The same surface admits many parameterizations. Choose the one that simplifies your integral.

Tangent vectors and normal

Hold $v$ fixed and let $u$ vary: $\mathbf{r}(u,v_0)$ traces a curve on the surface (a “$u$-curve”). The partial derivative ${\mathbf{r}}_u=\partial \mathbf{r}/\partial u$ is a tangent vector to that curve. Similarly ${\mathbf{r}}_v$ is tangent to the “$v$-curve.”

At each point, ${\mathbf{r}}_u$ and ${\mathbf{r}}_v$ span the Tangent plane — the 2D linear approximation of the surface at that point. See that note for the plane equation, the implicit-surface form ($\mathbf{n}=\nabla F$), and the graph-of-function form (first-order Taylor expansion).

The Cross product ${\mathbf{r}}_u\times {\mathbf{r}}_v$ is:

• Perpendicular to both tangent vectors — normal to the surface.
• Magnitude $|{\mathbf{r}}_u\times {\mathbf{r}}_v|$ equals the area of the infinitesimal parallelogram spanned by ${\mathbf{r}}_{u}du$ and ${\mathbf{r}}_{v}dv$ — the local area-element factor.

So ${\mathbf{r}}_u\times {\mathbf{r}}_v$ is the vector area element: direction normal, magnitude $=$ area-element factor. Compactly, $d\mathbf{S}=({\mathbf{r}}_u\times {\mathbf{r}}_v)dudv$.

Smoothness

The surface is smooth at $(u,v)$ if ${\mathbf{r}}_u\times {\mathbf{r}}_v\ne 0$. The nonzero condition rules out edges and corners (places where the surface “collapses” along a direction).

The graph shortcut

For the graph parameterization $\mathbf{r}(x,y)=⟨x,y,f(x,y)⟩$:

${\mathbf{r}}_x=⟨1,0,f_x⟩,{\mathbf{r}}_y=⟨0,1,f_y⟩,$

${\mathbf{r}}_x\times {\mathbf{r}}_y=⟨-f_x,-f_y,1⟩.$

This has positive $z$-component, so points upward (which is usually the convention for “outward” or “upward” orientation; flip the sign for downward).

The graph shortcut is the workhorse for any surface that can be written as $z=f(x,y)$. The vector area element is $d\mathbf{S}=⟨-f_x,-f_y,1⟩dxdy$ (upward) or its negative (downward).

Orientation

A choice of unit normal at each point — continuous over the surface. Most surfaces are two-sided and admit two opposite orientations.

• Sphere: outward normal $\hat{\mathbf{r}}/|\mathbf{r}|$, or inward.
• Graph: upward (positive $z$-component), or downward.

Some surfaces (the Möbius strip) are non-orientable — they have no consistent global choice of normal. Vector Calculus and Complex Analysis stays with orientable surfaces; the standard ones are.

In context

Parameterized surfaces feed into:

• Scalar surface integral ${\iint }_{S}fdS$.
• Flux integral ${\iint }_S\mathbf{F}\cdot d\mathbf{S}$.
• The right-hand side of Stokes’ theorem and Divergence theorem.

The pattern matches that of parameterized curves: parameterize, compute the appropriate “differential factor” from derivatives of the parameterization, integrate over the parameter domain.