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 means two degrees of freedom on the surface. The surface itself lives in 3D; you can walk around it, view from different angles. So don’t read “2D object” as sitting 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 there. That note has the plane equation, the implicit-surface form ($\mathbf{n}=\nabla F$), and the graph-of-function form (first-order Taylor expansion).

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

• Perpendicular to both tangent vectors, so 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 works for any surface you can write 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: no consistent global choice of normal. We stay 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.

Same pattern as parameterized curves: parameterize, compute the differential factor from derivatives of the parameterization, integrate over the parameter domain.