Imagine that you hold a net into a flowing river. How much water per unit time flows through the net? This is the basic question we will deal with in this section.
Section 4.7 showed how we can use vector valued functions of two variables to give a parametrization of a surface in space. For instance, the function with domain and parametrizes a sphere of radius centered at the origin. Section 4.7 also gives examples of how to write parametrizations based on other geometric relationships like when one coordinate can be written as a function of the other two. In Subsection 4.7.2, we set up a Riemann sum based on a parametrization that would measure the surface area of our curved surfaces in space.
In Figure 4.8.1, you can see a surface plotted using a parametrization . The red lines represent curves where varies and is held constant, while the yellow lines represent curves where varies and is held constant. The vector in red is which measures the direction and magnitude of change in the coordinates of the surface when just is varied. Similarly, the vector in yellow is which measures the direction and magnitude of change in the coordinates of the surface when just is varied. You can see that the parallelogram that is formed by and is tangent to the surface. The area of this parallelogram offers an approximation for the surface area of a patch of the surface.
From Section 1.4, we also know that (plotted in green) will be orthogonal to both and and its magnitude will be given by the area of the parallelogram.
In this activity we will explore the parametrizations of a few familiar surfaces and confirm some of the geometric properties described in the introduction above.
Draw your vector results from c on your graphs and confirm the geometric properties described in the introduction to this section. Namely, and should be tangent to the surface, while should be orthogonal to the surface (in addition to and ).
As we saw in Section 4.7, we can set up a Riemann sum of the areas for the parallelograms in Figure 4.8.1 to approximate the surface area of the region plotted by our parametrization. Equation (4.7.2) shows that we can compute the exact surface by taking a limit of a Riemann sum which will correspond to integrating the magnitude of over the appropriate parameter bounds. What if we wanted to measure a quantity other than the surface area? In the next section, we will explore 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 will be very valuable, especially the vector .
Consider a velocity vector field of a fluid and a flat porous surface through which the fluid is flowing. The porous surface needs to be oriented, so suppose that is a unit normal vector pointing in the direction of the orientation of .
The volume of fluid that flows through per unit time is the volume of a skewed cylindrical shape with base and height , where is the angle between and .
Suppose that is a flat surface with unit normal vector and is a constant vector field. Write for the angle betweeen and . The flux, or per-unit-time rate at which is flowing through , can be computed as either of the following expressions.
A fluid passes through a blob of area 8 in the -plane. The fluid flows with constant velocity vector . How much fluid flows through the blob in one unit of time?
Solution.
We have and . Project onto to get ; i.e., . So the flux is units of volume per unit of time.
In Figure 4.8.4, we illustrate the situation that we wish to study in the remainder of this section. We have a piece of a surface, shown by using shading. There is also a vector field, perhaps representing some fluid that is flowing. We are interested in measuring the flow of the fluid through the shaded surface portion.
We donβt care about the vector field away from the surface, so we really would like to just examine what the output vectors for the points on our surface. So instead, we will look at Figure 4.8.5.
The central question we would like to consider is βHow can we measure the amount of a three dimensional vector field that flows through a particular section of a curved surface?β, 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 the next figure, we have split the vector field along 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 purple.
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 Figure 4.8.6. 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 parametrization of the surface, then the vector varies smoothly across our surface and gives a consistent way to describe which direction we choose as βthroughβ the surface. If we define a positive flow through our surface as being consistent with the yellow vector in Figure 4.8.6, 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.
In this activity, you will compare the net flow of different vector fields through our sample surface. In Figure 4.8.7 you can select between five different vector fields. Once you select a vector field, the vector field for a set of points on the surface will be plotted in blue. Each blue vector will also be split into its normal component (in green) and its tangential component (in purple). The yellow vector defines the direction for positive flow through the surface.
Look at each vector field and order the vector fields from greatest flow through the surface to least flow through the surface. Remember that a negative net flow through the surface should be lower in your rankings than any positive net flow.
Now that we have a better conceptual understanding of what we are measuring, we can set up the corresponding Riemann sum to measure the flux of a vector field through a section of a surface. Let be the section of our surface and suppose that is parametrized by with and . The domain of is a region of the -plane, which we call , and the range of is .
In our classic calculus style, we slice our region of interest into smaller pieces. Specifically, we slice into equally-sized subintervals with endpoints and into equally-sized subintervals with endpoints . This divides into rectangles of size by . We index these rectangles as . Every has area (in the -plane) of . The partition of into the rectangles also partitions into corresponding pieces which we call . From Section 4.7 (specifically (4.7.1)) the surface area of is approximated by .
We want to measure the total flow of the vector field, , through , which we approximate on each and then sum to get the total flow. In other words, the flux of through is
For each , we approximate the surface by the tangent plane to at a corner of that partition element. This corresponds to using the planar elements in Figure 4.8.8, which have surface area . The vector can be used to measure the orthogonal direction (and thus define which direction we mean by positive flow through ) on the partition element. This means that
In Figure 4.8.8, you can change the number of sections in your partition and see the geometric result of refining the partition. The next activity asks you to carefully go through the process of calculating the flux of some vector fields through a cylindrical surface.
Figure 4.8.10 shows a plot of the vector field and a right circular cylinder of radius and height (with open top and bottom). Consider the vector field going into the cylinder (toward the -axis) as corresponding to a positive flux.
Based on your parametrization, compute ,, and . Confirm that these vectors are either orthogonal or tangent to the right circular cylinder. Is your orthogonal vector pointing in the direction of positive flux or negative flux?
In 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 .
For each of the three surfaces given below, compute , graph the surface, and compute for four different points of your choosing. You should make sure your vectors are orthogonal to your surface.
For each of the three surfaces in part c, use your calculations and Theorem 4.8.9 to compute the flux of each of the following vector fields through the part of the surface corresponding to the region in the -plane.
Your result for should be a scalar expression times . Explain why the outward pointing orthogonal vector on the sphere is a multiple of and what that scalar expression means.
If we used the sphere of radius 4 instead of , explain how each of the flux integrals from part d would change. You do not need to calculate these new flux integrals, but rather explain if the result would be different and how the result would be different.
Note that throughout this section, we have implicitly assumed that we can parametrize the surface in such a way that gives a well-defined normal vector. Technically, this means that the surface be orientable. Most βreasonableβ surfaces are orientable. However, there are surfaces that are not orientable. Perhaps the most famous is formed by taking a long, narrow piece of paper, giving one end a half twist, and then gluing the ends together. Try doing this yourself, but before you twist and glue (or tape), poke a tiny hole through the paper on the line halfway between the long edges of your strip of paper and circle your hole. After gluing, place a pencil with its eraser end on your dot and the tip pointing away. Think of this as a potential normal vector. Keep the eraser on the paper, and follow the middle of your surface around until the first time the eraser is again on the dot. Is your pencil still pointing the same direction relative to the surface that it was before?