Surface integral

A surface integral is a double integral over a 2D surface embedded in ${\mathbb{R}}^3$. Important extensions of surface integrals are:

• Circulation
• Flux

The differential element $dS$ is given by the norm of the cross product:

$$dS=|\frac{\partial \vec{r}}{\partial u}\times \frac{\partial \vec{r}}{\partial v}|dudv$$ $$d\vec{S}=\frac{\partial \vec{r}}{\partial u}\times \frac{\partial \vec{r}}{\partial v}dudv$$

which is precisely why we care about parametrisation. The normal vector is given by:

$${\vec{n}}_s=\frac{{\vec{T}}_u\times T_v}{|{\vec{T}}_u\times {\vec{T}}_v|}$$

Computations

Steps

1. Determine the position vector $\vec{r}$ pointing to the surface $S$ and region $R$.
2. Find the partial derivatives of $\vec{r}$ with respect to $u$ and $v$. Should get two vectors out of this.
3. Take the cross product of the partials.
4. Compute the unit normal vector.
5. Compute $d\vec{S}={\vec{n}}_{S}dS$.

Differential elements

The differential element $dS$ for a sphere with radius $r$ (in spherical coordinates) is:

$$d\vec{S}=r^2\sin \theta d\phi d\theta \hat{r}$$

For a cylinder with radius $R$ (in cylindrical coordinates):

$$dS=Rd\theta dz$$

See also

• Line integral, the single-variable equivalent