Flux Integrals

This section relies on parameterized surfaces, which was first introduced in . While some of the activities in this section may be too much for a single student to do in a class setting, we suggest that different cases of the vector fields and surfaces can be split for small group work. Students can then present answers to the larger group.

showed how we can use vector-valued functions of two variables to give a parameterization of a surface in space. For instance, the function \(\vr(s,t)=\langle 2\cos(t)\sin(s), 2\sin(t)\sin(s),2\cos(s)\rangle\) with domain \(0\leq t\leq 2 \pi\) and \(0\leq s\leq \pi\) parameterizes a sphere of radius \(2\) centered at the origin. also gives examples of how to write parameterizations based on other geometric relationships like when one coordinate can be written as a function of the other two. In , we set up a Riemann sum based on a parameterization that would measure the surface area of our curved surfaces in space.

In , we plot a surface using a parametrization \(\vr(s,t)=\langle{f(s,t),g(s,t),h(s,t)}\rangle\). The magenta curves represent curves where \(s\) varies and \(t\) is held constant, while the yellow curves represent curves where \(t\) varies and \(s\) is held constant. The vector in magenta is \(\vr_s=\frac{\partial \vr}{\partial s}=\langle{f_s,g_s,h_s}\rangle\) which measures the direction and magnitude of change in the coordinates of the surface when only \(s\) is varied. Similarly, the vector in yellow is \(\vr_t=\frac{\partial \vr}{\partial t}=\langle{f_t,g_t,h_t}\rangle\) which measures the direction and magnitude of change in the coordinates of the surface when only \(t\) is varied. We also plot the parallelogram that is formed by \(\vr_s\) and \(\vr_t\), which is tangent to the surface. The area of this parallelogram offers an approximation for the surface area of a patch of the surface.

From , we also know that \(\vr_s\times \vr_t\) (plotted in green) will be orthogonal to both \(\vr_s\) and \(\vr_t\) and its magnitude will be given by the area of the parallelogram.

As we saw in , we can set up a Riemann sum of the areas for the parallelograms in to approximate the surface area of the region plotted by our parametrization. Equation shows that we can compute the exact surface area by taking a limit of a Riemann sum which will correspond to integrating the magnitude of \(\vr_s \times \vr_t\) over the appropriate parameter bounds. What if we wanted to measure a quantity other than the surface area? Our focus in this section we will be the exploration of a specific case of this question: How can we measure the amount of a three dimensional vector field that flows through a particular section of a surface? The geometric tools we have reviewed in this section, especially the vector \(\vr_s \times \vr_t\), will be valuable.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

In this section we will look at how to measure the amount of a vector field that flows through a surface in space. illustrates a plot that demonstrates this idea. Our definition of divergence in looked at measuring the amount of vector field flowing out of a small region on a 2D plane. In this subsection, we will set up a precise measurement of this same measurement but over a region of a curved surface in 3D.

As with understanding line integrals of vector-valued functions in , we don't care about the output of the vector field at points away from the surface. We would really would like to examine the output vectors for the points on our surface. To do this, we will look at , which plots the output of our vector field at an array of points on our surface.

The central question we would like to consider is How can we measure the amount of a three dimensional vector field that flows through a particular section of a curved surface?, so we only need to consider the amount of the vector field that flows through the surface. Any portion of our vector field that flows along (or tangent) to the surface will not contribute to the amount that goes through the surface. In , we have split the vector field for points on our surface into two components. One component, plotted in green, is orthogonal to the surface. The component that is tangent to the surface is plotted in magenta.

In order to measure the amount of the vector field that moves through the plotted section of the surface, we must find the accumulation of the lengths of the green vectors in . Notice that some of the green vectors are moving through the surface in a direction opposite of others. In other words, we will need to pay attention to the direction in which these vectors move through our surface and not just the magnitude of the green vectors.

If we have a parameterization of the surface, then the vector \(\vr_s \times \vr_t\) varies smoothly across our surface and gives a consistent way to describe which direction we choose as through the surface. If we define a positive flow through our surface as being consistent with the yellow vector in , then there is more positive flow (in terms of both magnitude and area) than negative flow through the surface. Thus, the net flow of the vector field through this surface is positive.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Now that we have developed a conceptual understanding of what we are trying to measure, we can set up a corresponding Riemann sum to measure the flux of a vector field through a section of a surface. In the following theorem, we utilize a parameterization to calculate both the normal vector to the surface at a location as well an measure of how the surface area is chaninging at that point. You can look at the proof that follows this theorem to see precisely how the elements discussed give rise to this method of calculation.

In the next example, we will look at how is used on a part of a cone and make sense of the vector \(\vw(t,s)=\vr_s \times \vr_t\) and the scalar \(\vF \cdot (\vr_s \times \vr_t)\).

The next activity asks you to carefully go through the process of calculating the flux of some vector fields through a cylindrical surface.

In the exercises for this section, we will look at some computational ideas to help us more efficiently compute the value of a flux integral. In many cases, the surface we are looking at the flux through can be written with one coordinate as a function of the others. For simplicity, we consider \(z=f(x,y)\). Additionally, there will be exercises that guide you through common surfaces like spheres and cylinder surfaces.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.