Surface Area

The surface area of a parametric surface is the analogue to the arclength of a parametric curve. It adds a unit of measurement to a surface.

Definition §

Surface Area ^definition

Let $O$ be a 3D parametric surface that is $C^1$ smooth (except possibly for a finite number of points), given by the parametric equation $\Phi :D\to O$ where $\Phi (s,t)=(u(s,t),v(s,t),w(s,t))$. Then the surface area, $S$ is given by:

$$S={\iint }_D\Vert {\vec{T}}_s\times {\vec{T}}_t\Vert \text{d}s\text{d}t={\iint }_D\Vert \frac{\partial \Phi }{\partial s}\times \frac{\partial \Phi }{\partial t}\Vert \text{d}s\text{d}t$$

• ${\vec{T}}_{s,t}$: The tangent vectors to the surface relative to $s$ and $t$ respectively.
• $S:{\mathbb{R}}^2\to \mathbb{R}$

Surface Area Of A Generic Bivariate Function §

We can use the surface area formula to find the standard surface area of a bivariate function, $f(x,y)$.

Start by taking the function as a parameterisation of $x,y$:

$$\varphi (x,y)=(x,y,f(x,y))$$

The surface area is given by:

$${\iint }_S\text{d}s={\iint }_D\Vert {\varphi }_x\times {\varphi }_y\Vert \text{d}x\text{d}y$$

And the partial derivates would be:

$${\varphi }_x=\left(1,0,\frac{\partial f}{\partial x}\right){\varphi }_y=\left(0,1,\frac{\partial f}{\partial y}\right)$$

Taking the cross product:

$$\Vert {\varphi }_x\times {\varphi }_y\Vert =\Vert \det \left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 0& \frac{\partial f}{\partial x}\\ 0& 1& \frac{\partial f}{\partial y}\end{array}\right]\Vert =\Vert \left(-\frac{\partial f}{\partial x},-\frac{\partial f}{\partial y},1\right)\Vert$$

$$=\sqrt{{\left(\frac{\partial f}{\partial x}\right)}^2+{\left(\frac{\partial f}{\partial y}\right)}^2+1}$$

So the generic surface area of a bivariate function is given by:

$$SA={\iint }_D\sqrt{{\left(\frac{\partial f}{\partial x}\right)}^2+{\left(\frac{\partial f}{\partial y}\right)}^2+1}\text{d}x\text{d}y$$