Surface integrals of scalar functions

Defining a Surface Integral of a Scalar Function.

The integral \(\int \int_{S} f(x,y,z)\,dS\) of a scalar function over a general surface \(S\) is defined as a natural generalization of the standard double integral of a function of two variables over a region in the plane. It is defined as follows:

1. Suppose that \(S\) is parameterized by \(\vec{r}(u,v)\) over some \(uv\)-region \(D\text{.}\)
2. 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.
3. 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.

4. 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*}

5. 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} f(\vec{r} (u_{i}^* , v_{j}^*))\,||\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} f\,dS =\lim_{n,m\rightarrow \infty} \sum_{j=1}^{m} \sum_{i=1}^{n} f(\vec{r} (u_{i}^* , v_{j}^*)) \, ||\vec{N}||\,\Delta A. \end{equation*}

Surface Integral Formulas.

If \(S\) is parameterized by \(\vec{r}(u,v)\) over the \(uv\)-region \(D\) and \(\vec{N}\) 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 \(\int \int_{S} f \,dS\) is calculated using the following formula:

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 \(\vec{r}(u,v)=u\vec{i} +v\vec{j} +g(u,v)\vec{k}\text{,}\) we have

  \begin{equation*} \int \int_{S} f(x,y,z)\,dS =\int \int_{D} f(u,v,g(u,v))\, \sqrt{(g_u)^2 +(g_v)^2 +1} \,dA. \end{equation*}

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

  \begin{equation*} \int \int_{S} f(x,y,z)\,dS =\int \int_{D} f(g(v)\cos{(u)},g(v)\sin{(u)},v) \, |g(v)| \, \sqrt{1+(g'(v))^2} \,dA. \end{equation*}

General Set up and Calculation.

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

1. Find a parameterization \(\vec{r}(u,v)\) for \(S\text{.}\) This includes finding the \(uv\)-bounds for \(D\text{.}\)
2. Find the normal vector \(\vec{N}=\vec{r}_{u} \times \vec{r}_{v} \) and its magnitude.
3. Perform the composition \(f(\vec{r}(u,v))\text{.}\)
4. Invoke the integral formula and evaluate using standard double integration techniques.