Surface Integrals of Vector Fields

Math Online's 1000th Page - 7:04 PM CST on April 4th, 2015

Surface Integrals of Vector Fields

We have already computed surface integrals of scalar functions of a surface $\delta$. We will now take a look at surface integrals of vector fields over a surface $\delta$.

We will explain surface integrals of vector fields with the term known as flux. To visualize what flux really is, imagine a smooth orientable surface $\delta$ (that is also permeable) in $\mathbb{R}^3$ whose normal field is $\hat{N}$, and suppose that this surface is submerged in a fluid with an associated velocity field, $\mathbf{V}$. The rate for which fluid passes through the surface $\delta$ in accordance to the velocity field is the flux of that field over the surface $\delta$.

Let $P$ be a point on $\delta$ and let $dS$ be the area element around $P$. The fluid that passes through this area element $dS$ from time $t$ to time $t + dt$ forms a cylinder. The base of this cylinder is the area element $dS$, while the height of this cylinder is $\| \mathbf{V}(P) \| \: dt \cos \theta$ where $\theta$ is the angle between $\hat{N}(P)$ and $\mathbf{V}(P)$.

The total rate at which fluid passes through $dS$ can be obtained with the following surface integral:

\begin{align} \quad \iint_{\delta} \mathbf{V} \cdot \hat{N} \: dS \end{align}
Definition: Let $\delta$ be an orientable surface with unit normal $\hat{N}$ and let $\mathbf{F}$ be a continuous vector field. Then the Flux of $\mathbf{F}$ across $\delta$ is $\iint_{\delta} \mathbf{F} \cdot \hat{N} \: dS$.

The notation $\iint_{\delta} \mathbf{F} \cdot d\vec{S}$ is also used to denote the flux of a vector field, where $d \vec{S}$ is the Vector Area Element can be obtained from the regular area element $dS$ with the formula $d\vec{S} = \hat{N} \cdot dS$.

Let's now try to find a formula for computing surface integrals over vector fields.

Suppose that the surface $\delta$ is parameterized as $\vec{r}(u, v) = (x(u, v), y(u, v), z(u, v))$ for $(u, v) \in D$. Recall from the Surface Integrals page that $\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}$ is normal to the surface $\delta$, and that:

\begin{align} \quad \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v} \end{vmatrix} = \frac{\partial (y, z)}{\partial (u, v)} \vec{i} + \frac{\partial (z, x)}{\partial (u, v)} \vec{j} + \frac{\partial(x, y)}{\partial (u, v)} \vec{k} \end{align}

The corresponding area element for the surface $\delta$ was then $dS = \biggr \| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \biggr \| \: du \: dv$. We also have that:

\begin{align} \quad \hat{N} = \frac{\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}}{ \biggr \| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \biggr \|} \end{align}

Thus the vector area element for our surface integral will be (choosing the sign to reflect the appropriate orientation of $\delta$):

\begin{align} \quad d \vec{S} = \hat{N} dS = \pm \frac{\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}}{ \biggr \| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \biggr \|} \biggr \| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \biggr \| \: du \: dv = \pm \left ( \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \right ) \: du \: dv \end{align}

Thus if $\mathbf{F}(x, y, z) = P(x, y, z) \vec{i} + Q(x, y, z) \vec{j} + R(x, y, z) \vec{k}$ is a continuous vector field on $\mathbb{R}^3$ then the flux inward/outward of $\mathbf{F}$ over the surface $\delta$ with unit normal $\hat{N}$ and parameterized by $\vec{r}(u, v) = (x(u, v), y(u, v), z(u, v))$ for $(u, v) \in D$ is:

\begin{align} \quad \iint_{\delta} \mathbf{F} \cdot \hat{N} \: dS = \pm \iint_D (P(x, y, z), Q(x, y, z), R(x, y, z)) \cdot \left (\frac{\partial (y, z)}{\partial (u, v)}, \frac{\partial (z, x)}{\partial (u, v)}, \frac{\partial (x, y)}{\partial (u, v)} \right ) \: du \: dv \\ \quad \iint_{\delta} \mathbf{F} \cdot \hat{N} \: dS = \pm \iint_D \left (P(x, y, z) \frac{\partial (y, z)}{\partial (u, v)} + Q(x, y, z) \frac{\partial (z, x)}{\partial (u, v)} + R(x, y, z) \frac{\partial (x, y)}{\partial (u, v)} \right ) \: du \: dv \end{align}
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