Surface Integrals

Normal Vector to a Surface

• Parametric form
  If $\sigma$ is parametrized by $\boldsymbol{r}(u,v) = x(u,v)\hat{i} + y(u,v)\hat{j} + z(u,v)\hat{k}$, a normal vector to $\sigma$ is $\boldsymbol{N} = \dfrac{\partial \boldsymbol{r}}{\partial u} \times \dfrac{\partial \boldsymbol{r}}{\partial v}$.
• Implicit form
  If $\sigma$ is expressed as $G(x,y,z) = 0$, a normal vector at $(x,y,z)$ on $\sigma$ is $\boldsymbol{N} = \nabla G(x,y,z)$.
• Graph form
  If $\sigma$ is given by $z = g(x,y)$, an upward-pointing normal vector is $\boldsymbol{N} = \langle -g_x(x,y), -g_y(x,y), 1 \rangle$.

A surface is smooth iff it has a non-zero normal vector at each point. A surface is piecewise smooth iff it is composed of finitely many smooth pieces.

Surface Integral of a Scalar Function

Let $f$ be a continuous scalar function defined on a piecewise smooth surface $\sigma$. Partition $\sigma$ into $n$ subregions, where the $k$th subregion has area $\Delta S_k$. Choose a sample point $(x_k^*, y_k^*, z_k^*)$ in the $k$th subregion. The surface integral of $f$ over $\sigma$ is

provided the limit exists and does not depend on the partition or the choice of sample points.

Surface Integral over a Parametric Surface

Let $\sigma$ be a smooth parametric surface given by $\boldsymbol{r}(u,v) = x(u,v)\hat{i} + y(u,v)\hat{j} + z(u,v)\hat{k}$, where $(u,v)$ varies over a region $R$ in the $uv$-plane. If $f$ is continuous on $\sigma$,

Surface Integral over a Graph

Let $\sigma$ be a surface given by $z = g(x,y)$, and let $R$ be its projection onto the $xy$-plane. If $g$, $g_x$, $g_y$ are continuous on $R$ and $f$ is continuous on $\sigma$,