Surface integrals of vector fields

Review of Orientations.

Let \(S\) be an orientable surface. We recall the following important definitions which will be used to define the next integral.

• An orientation of \(S\) is a choice of unit normal vector \(\vec{n}\) at each point of \(S\) which varies continuously over \(S\text{.}\)
• If \(\vec{r}(u,v)\) is a parameterization of \(S\text{,}\) then a normal vector to \(S\) is the vector \(\vec{N}=\vec{r}_{u} \times \vec{r}_{v}\text{.}\) We note that the vector \(\vec{N}\) is not necessarily a unit vector and does not necessarily have to point in the same direction as an orientation, see next comment.
• The parameterization \(\vec{r}(u,v)\) of \(S\) is said to be orientation-preserving if \(\vec{n} =\vec{N}/||\vec{N}||\) and orientation-reversing if \(\vec{n} =-\vec{N}/||\vec{N}||\text{.}\)

Defining a Surface Integral of a Vector Field.

We define the integral \(\int \int_{S} \vec{F}(x,y,z)\cdot d\vec{S}\) of a vector field over an oriented surface \(S\) to be a scalar measurement of the flow of \(\vec{F}\) through \(S\) in the direction of the orientation. It is defined as follows:

• Let \(\vec{n}\) denote the orientation of \(S\) and let \(\vec{r}(u,v)\) be an orientation-preserving parameterization of \(S\) over some \(uv\)-region \(D\text{.}\)
• Divide \(D\) up into small rectangles of lengths \(\Delta u\) in the \(u\)-direction and \(\Delta v\) in the \(v\)-direction and area \(\Delta A=\Delta u \Delta v\text{.}\) Note: rectangles may not perfectly fit in \(D\text{,}\) but later in the process we will take \(\Delta u ,\Delta v\rightarrow 0\text{,}\) so they will approximate \(D\) better and better.
• If we apply \(\vec{r}(u,v)\) to the edges of these rectangles, they each create a “patch” on the surface \(S\text{,}\) with all the rectangles creating a lattice, similar to a quilt. We create a Riemann sum over these patches.

• For a given patch, the vectors \(\vec{r}_{u} \Delta u\) and \(\vec{r}_{v} \Delta v\) approximate the length of the sides in the \(u\)-direction and \(v\)-direction. It follows that the area of a given patch can be approximated by

  \begin{equation*} ||\vec{r}_{u} \Delta u\times \vec{r}_{v} \Delta v|| =||\vec{r}_{u} \times \vec{r}_{v}|| \Delta u \Delta v=||\vec{r}_{u} \times \vec{r}_{v}|| \Delta A=||\vec{N}|| \Delta A \end{equation*}

• For a given point \((u_{i}^*,v_{j}^*)\text{,}\) the dot product \(\vec{F}(\vec{r} (u_{i}^* , v_{j}^*))\cdot \vec{n} (u_{i}^* , v_{j}^*)\) gives the flow of \(\vec{F}\) through \(S\) in the direction of \(\vec{n}\text{.}\) Therefore, for a given patch, if \((u_{i}^*,v_{j}^*)\) is a point in that patch, we can approximate the flow of \(\vec{F}\) through that patch in the direction of \(\vec{n}\) as

  \begin{equation*} \left(\vec{F}(\vec{r} (u_{i}^* , v_{j}^*))\, \cdot \vec{n}\right) ({\rm Area \, of \, Patch}) = \vec{F}(\vec{r} (u_{i}^* , v_{j}^*))\, \cdot \vec{n}\,||\vec{N}||\Delta A. \end{equation*}

• Fixing a point \((u_{i}^*,v_{j}^*)\) in each patch, we can construct our Riemann some over all the patches as

  \begin{equation*} \sum_{j=1}^{m} \sum_{i=1}^{n} \vec{F} (\vec{r} (u_{i}^* , v_{j}^*))\cdot \vec{n}\, ||\vec{N}||\Delta A, \end{equation*}

  and we can now define our integral as the limit of this double sum. Specifically:

  \begin{equation*} \int \int_{S} \vec{F} \cdot d\vec{S} =\lim_{n,m\rightarrow \infty} \sum_{j=1}^{m} \sum_{i=1}^{n} \vec{F} (\vec{r} (u_{i}^* , v_{j}^*))\cdot \vec{n} \,||\vec{N}||\Delta A. \end{equation*}

Surface Integral Formula.

If \(\vec{r}(u,v)\) is an orientation-preserving parameterization of \(S\) is over the \(uv\)-region \(D\text{,}\) \(\vec{N}\) is the normal vector for \(\vec{r}(u,v)\text{,}\) and \(dA=du\,dv\) or \(dv\,du\) depending upon how the bounds for \(D\) are set up, then \(\int \int_{S} \vec{F} \cdot d\vec{S}\) is calculated using the following formula:

Note however that \(\vec{n}\) is a normal vector and \(\vec{r}(u,v)\) is orientation-preserving, and so in particular, \(\vec{n} =\vec{N} /||\vec{N}||\text{.}\) In particular, this formula simplifies to:

Surface Integral of Vector Fields Depend on Orientation.

By the way it was defined, it is clear that the integral \(\int \int_{S} \vec{F}(x,y,z)\cdot d\vec{S}\) depends on orientation. Specifically, if \(-S\) denotes the surface \(S\) with opposite orientation, then \(\int \int_{-S} \vec{F}(x,y,z)\cdot d\vec{S}=-\int \int_{S} \vec{F}(x,y,z)\cdot d\vec{S}\text{.}\) In particular, if a given parameterization is orientation-reversing, we need to make sure we add a negative sign when re-writing the integral as a regular double integral over the \(uv\)-region \(D\text{.}\)

Formulas for Special Cases.

Since we have a standard formula for \(\vec{N}\) when either \(S\) is the graph of a function or has cross-sections which are concentric circles, we get the following special cases:

• When \(S\) is the graph of \(z=g(x,y)\) and parameterized by the orientation-preserving parameterization \(\vec{r}(u,v)=u\vec{i} +v\vec{j} +g(u,v)\vec{k}\text{,}\) we have

  \begin{equation*} \int \int_{S} \vec{F} (x,y,z)\cdot d\vec{S} =\int \int_{D} \vec{F} (u,v,g(u,v))\, \cdot \left( -g_u\vec{i} -g_v\vec{j} +\vec{k}\right) \,dA. \end{equation*}

• When \(S\) is has cross sections which are concentric circles centered on the \(z\)-axis with radius \(g(z)\) at height \(z\) and is parameterized by the orientation-preserving parameterization \(\vec{r}(u,v)=g(v)\cos{(u)}\vec{i} +g(v)\sin{(u)} \vec{j} +v\vec{k}\text{,}\) we have

  \begin{align*} \amp \int \int_{S} \vec{F} (x,y,z)\cdot d\vec{S} =\\ \amp \int \int_{D} \vec{F} (g(v)\cos{(u)},g(v)\sin{(u)},v) \cdot \left(g(v)\cos{(u)}\vec{i} +g(v)\sin{(u)} \vec{j} -g(v)g'(v)\vec{k} \right)\,dA. \end{align*}

Set up and Calculation.

To set up and evaluate \(\int \int_{S} \vec{F} (x,y,z)\cdot d\vec{S}\text{,}\) we take the following steps:

• Find a parameterization \(\vec{r}(u,v)\) for \(S\text{.}\) This includes finding the \(uv\)-bounds for \(D\text{.}\)
• Find the normal vector \(\vec{N}=\vec{r}_u \times \vec{r}_v\text{.}\)
• Perform the composition \(\vec{F}(\vec{r}(u,v))\text{.}\)
• Evaluate the dot product \(\vec{F} (\vec{r}(u,v)) \cdot \vec{N}\text{.}\)
• Check if the parameterization is orientation-preserving or reversing. Then invoke the integral formula and evaluate using standard double-integration techniques making sure to negate the double integral if the parameterization is orientation-reversing.