Motivating Questions
- How can we measure how much of a vector field flows through a surface in space?
- How can we calculate the amount of a vector field that flows through common surfaces, such as the graph of a function \(z=f(x,y)\text{?}\)
Section 4.8 Flux Integrals
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 \(\vr(s,t)=\langle 2\cos(t)\sin(s),
2\sin(t)\sin(s),2\cos(s)\rangle\) with domain \(0\leq t\leq 2
\pi\) and \(0\leq s\leq \pi\) parametrizes a sphere of radius \(2\) 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 \(\vr(s,t)=\langle{f(s,t),g(s,t),h(s,t)}\rangle\text{.}\) The red lines represent curves where \(s\) varies and \(t\) is held constant, while the yellow lines represent curves where \(t\) varies and \(s\) is held constant. The vector in red 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 just \(s\) 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 just \(t\) is varied. You can see that the parallelogram that is formed by \(\vr_s\) and \(\vr_t\) 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 \(\vr_s\times \vr_t\) (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.
Preview Activity 4.8.1.
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.
(a)
Use the ideas from Section 4.7 to give a parametrization \(\vr(s,t)\) of each of the following surfaces. Be sure to specify the bounds on each of your parameters.
(i)
A sphere centered at the origin of radius 3.
(ii)
A right circular cylinder centered on the \(x\)-axis of radius 2 when \(0\leq x\leq 3\text{.}\)
(iii)
The first octant portion of the plane \(x+2y+3z=6\text{.}\)
(b)
Draw a graph of each of the three surfaces from the previous part. Label the points that correspond to \((s,t)\) points of \((0,0)\text{,}\) \((0,1)\text{,}\) \((1,0)\text{,}\) and \((2,3)\text{.}\)
(c)
For each parametrization from part a, calculate \(\vr_s\text{,}\) \(\vr_t\text{,}\) and \(\vr_s \times \vr_t\text{.}\)
(d)
For each parametrization from part a, find the value for \(\vr_s\text{,}\)\(\vr_t\text{,}\) and \(\vr_s \times \vr_t\) at the \((s,t)\) points of \((0,0)\text{,}\) \((0,1)\text{,}\) \((1,0)\text{,}\) and \((2,3)\text{.}\)
(e)
Draw your vector results from c on your graphs and confirm the geometric properties described in the introduction to this section. Namely, \(\vr_s\) and \(\vr_t\) should be tangent to the surface, while \(\vr_s \times \vr_t\) should be orthogonal to the surface (in addition to \(\vr_s\) and \(\vr_t\)).
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 \(\vr_s \times \vr_t\) 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 \(\vr_s \times \vr_t\text{.}\)
Subsection 4.8.1 Constant vector field through a flat surface
Consider a velocity vector field \(\vF\) of a fluid and a flat porous surface \(S\) through which the fluid is flowing. The porous surface needs to be oriented, so suppose that \(\mathbf{n}\) is a unit normal vector pointing in the direction of the orientation of \(S\text{.}\)
A picture of the flat surface S and a vector F emanating from it.
The volume of fluid that flows through \(S\) per unit time is the volume of a skewed cylindrical shape with base \(S\) and height \(|\vF|\cos\theta\text{,}\) where \(\theta\) is the angle between \(\vF\) and \(\mathbf{n}\text{.}\)
Flux of a constant vector field through a flat surface.
Suppose that \(S\) is a flat surface with unit normal vector \(\mathbf{n}\) and \(\vF\) is a constant vector field. Write \(\theta\) for the angle betweeen \(\mathbf{n}\) and \(\vF\text{.}\) The flux, or per-unit-time rate at which \(\vF\) is flowing through \(S\text{,}\) can be computed as either of the following expressions.
\begin{equation*}
\operatorname{area}(S) (\mathbf{n} \cdot \vF) = \operatorname{area}(S) |\vF| \cos(\theta).
\end{equation*}
Example 4.8.3.
A fluid passes through a blob of area 8 in the \(xy\)-plane. The fluid flows with constant velocity vector \(\langle 1, -2, 3 \rangle\text{.}\) How much fluid flows through the blob in one unit of time?
Solution.
We have \(\vF = \langle 1, -2, 3 \rangle\) and \(\mathbf{n} = \langle 1, 0, 0 \rangle\text{.}\) Project \(\vF\) onto \(\mathbf{n}\) to get \(\langle 0, 0, 3 \rangle\text{;}\) i.e., \(\vF \cdot \mathbf{n} = 3\text{.}\) So the flux is \(3 \cdot 8 = 24\) units of volume per unit of time.
Subsection 4.8.2 The Idea of the Flux of a Vector Field through a Surface
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 don’t care about the vector field away from the surface, so we really would like to just examine what the output vectors for the \((x,y,z)\) points on our surface. So instead, we will look at Figure 4.8.5.
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 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 \(\vr_s \times \vr_t\) 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 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.
Activity 4.8.2. Visualizing flux through a surface.
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.
Subsection 4.8.3 Measuring the Flux of a Vector Field through a Surface
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 \(Q\) be the section of our surface and suppose that \(Q\) is parametrized by \(\vr(s,t)\) with \(a\leq s\leq b\) and \(c \leq t \leq d\text{.}\) The domain of \(\vr\) is a region of the \(st\)-plane, which we call \(D\text{,}\) and the range of \(\vr\) is \(Q\text{.}\)
In our classic calculus style, we slice our region of interest into smaller pieces. Specifically, we slice \(a\leq s\leq b\) into \(n\) equally-sized subintervals with endpoints \(s_1,\ldots,s_n\) and \(c \leq t \leq d\) into \(m\) equally-sized subintervals with endpoints \(t_1,\ldots,t_n\text{.}\) This divides \(D\) into \(nm\) rectangles of size \(\Delta{s}=\frac{b-a}{n}\) by \(\Delta{t}=\frac{d-c}{m}\text{.}\) We index these rectangles as \(D_{i,j}\text{.}\) Every \(D_{i,j}\) has area (in the \(st\)-plane) of \(\Delta{s}\Delta{t}\text{.}\) The partition of \(D\) into the rectangles \(D_{i,j}\) also partitions \(Q\) into \(nm\) corresponding pieces which we call \(Q_{i,j}=\vr(D_{i,j})\text{.}\) From Section 4.7 (specifically (4.7.1)) the surface area of \(Q_{i,j}\) is approximated by \(S_{i,j}=\vecmag{(\vr_s \times
\vr_t)(s_i,t_j)}\Delta{s}\Delta{t}\text{.}\)
We want to measure the total flow of the vector field, \(\vF\text{,}\) through \(Q\text{,}\) which we approximate on each \(Q_{i,j}\) and then sum to get the total flow. In other words, the flux of \(\vF\) through \(Q\) is
\begin{equation*}
\text{Flux}=\sum_{i=1}^n\sum_{j=1}^m\vecmag{\vF_{\perp
Q_{i,j}}}\cdot S_{i,j}\text{,}
\end{equation*}
where \(\vecmag{\vF_{\perp Q_{i,j}}}\) is the length of the component of \(\vF\) orthogonal to \(Q_{i,j}\text{.}\)
For each \(Q_{i,j}\text{,}\) we approximate the surface \(Q\) by the tangent plane to \(Q\) at a corner of that partition element. This corresponds to using the planar elements in Figure 4.8.8, which have surface area \(S_{i,j}\text{.}\) The vector \(\vw_{i,j}=(\vr_s \times \vr_t)(s_i,t_j)\) can be used to measure the orthogonal direction (and thus define which direction we mean by positive flow through \(Q\)) on the \(i,j\) partition element. This means that
\begin{equation*}
\vF_{\perp Q_{i,j}} =\vecmag{\proj_{\vw_{i,j}}\vF(s_i,t_j)}
= \frac{\vF(s_i,t_j)\cdot \vw_{i,j}}{\vecmag{\vw_{i,j}}}
\end{equation*}
Combining these pieces, we find that the flux through \(Q_{i,j}\) is approximated by
\begin{align*}
\text{Flux through } Q_{i,j} \amp= \vecmag{\vF_{\perp
Q_{i,j}}}\cdot S_{i,j}
= \left(\frac{\vF_{i,j}\cdot \vw_{i,j}}{\vecmag{\vw_{i,j}}} \right)
\left(\vecmag{\vw_{i,j}}\Delta{s}\Delta{t}\right)\\
\amp = \left(\vF_{i,j} \cdot (\vr_s \times \vr_t)\right)
\left(\Delta{s}\Delta{t}\right)\text{,}
\end{align*}
where \(\vF_{i,j} = \vF(s_i,t_j)\text{.}\) Therefore we may approximate the total flux by
\begin{equation*}
\text{Total Flux}=\sum_{i=1}^n\sum_{j=1}^m \left(\vF_{i,j}\cdot \vw_{i,j}\right) \left(\Delta{s}\Delta{t}\right)\text{.}
\end{equation*}
Taking the limit as \(n,m\rightarrow\infty\) gives the following result.
Theorem 4.8.9.
Let a smooth surface \(Q\) be parametrized by \(\vr(s,t)\) over a domain \(D\text{.}\) The total flux of a smooth vector field \(\vF\) through \(Q\) is given by
\begin{equation*}
\iint_D \vF \cdot (\vr_s \times \vr_t)\, dA\text{.}
\end{equation*}
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.
Activity 4.8.3. Checking the Visualization for Flux.
(a)
Figure 4.8.10 shows a plot of the vector field \(\vF=\langle{y,z,2+\sin(x)}\rangle\) and a right circular cylinder of radius \(2\) and height \(3\) (with open top and bottom). Consider the vector field going into the cylinder (toward the \(z\)-axis) as corresponding to a positive flux.
(i)
Reasoning graphically, do you think the flux of \(\vF\) throught the cylinder will be positive, negative, or zero?
(ii)
Parametrize the right circular cylinder of radius \(2\text{,}\) centered on the \(z\)-axis for \(0\leq z \leq 3\text{.}\) Be sure to give bounds on your parameters.
(iii)
Based on your parametrization, compute \(\vr_s\text{,}\) \(\vr_t\text{,}\) and \(\vr_s \times \vr_t\text{.}\) 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?
(iv)
Use your parametrization to write \(\vF\) as a function of \(s\) and \(t\text{.}\)
Hint.
The \(x\) coordinate is given by the first component of \(\vr\text{.}\)
(v)
Compute the flux of \(\vF\) through the parametrized portion of the right circular cylinder.
(vi)
Does your computed value for the flux match your prediction from earlier?
(vii)
Use Figure 4.8.11 to make an argument about why the flux of \(\vF=\langle{y,z,2+\sin(x)}\rangle\) through the right circular cylinder is zero.
(b)
How would the results of the flux calculations be different if we used the vector field \(\vF=\left\langle{y,z,\cos(xy)+\frac{9}{z^2+6.2}}\right\rangle\) and the same right circular cylinder? Explain your reasoning.
(c)
How would the results of the flux calculations be different if we used the vector field \(\vF=\langle{y,-x,3}\rangle\) and the same right circular cylinder? Explain your reasoning.
Subsection 4.8.4 Efficiently Calculating Flux Integrals
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 \(z=f(x,y)\text{.}\)
Activity 4.8.4. Flux Through Surfaces of the Form \(z=f(x,y)\).
In this activity, we will look at how to use a parametrization of a surface that can be described as \(z=f(x,y)\) to efficiently calculate flux integrals.
(a)
Suppose that \(S\) is a surface given by \(z=f(x,y)\text{.}\) Find a parametrization \(\vr(s,t)\) of \(S\text{.}\)
Hint.
Use \(s=x\) and \(t=y\text{.}\)
(b)
Show that the vector orthogonal to the surface \(S\) has the form
\begin{equation*}
\vr_s \times \vr_t=\left\langle -\frac{\partial{f}}{\partial{x}},-\frac{\partial{f}}{\partial{y}},1 \right\rangle\text{.}
\end{equation*}
(c)
For each of the three surfaces given below, compute \(\vr_s
\times \vr_t\text{,}\) graph the surface, and compute \(\vr_s
\times \vr_t\) for four different points of your choosing. You should make sure your vectors \(\vr_s \times
\vr_t\) are orthogonal to your surface.
(i)
\(z=x^2+y^2\)
(ii)
\(x+2y+z=-4\)
(iii)
\(z=x^2-y^2\)
(d)
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 \(D\) in the \(xy\)-plane.
(i)
\(\vF=\langle{x,y,z}\rangle\) with \(D\) given by \(0\leq x,y\leq 2\)
(ii)
\(\vF=\langle{-y,x,1}\rangle\) with \(D\) as the triangular region of the \(xy\)-plane with vertices \((0,0)\text{,}\) \((1,0)\text{,}\) and \((1,1)\)
(iii)
\(\vF=\langle{z,y-x,(y-x)^2-z^2}\rangle\) with \(D\) given by \(0\leq x,y\leq 2\)
Spheres and portions of spheres are another common type of surface through which you may wish to calculate flux.
Activity 4.8.5. Calculating the Flux through a Sphere.
For this activity, let \(S_R\) be the sphere of radius \(R\) centered at the origin.
(a)
Parametrize \(S_R\) using spherical coordinates. Give your parametrization as \(\vr(s,t)\text{,}\) and be sure to state the bounds of your parametrization.
Hint.
Use \(s=\theta\) and \(t=\phi\text{.}\)
(b)
Use your parametrization of \(S_R\) to compute \(\vr_s \times \vr_t\text{.}\)
(c)
Your result for \(\vr_s \times \vr_t\) should be a scalar expression times \(\vr(s,t)\text{.}\) Explain why the outward pointing orthogonal vector on the sphere is a multiple of \(\vr(s,t)\) and what that scalar expression means.
(d)
Use your parametrization of \(S_2\) and the results of part b to calculate the flux through \(S_2\) for each of the three following vector fields.
(i)
\(\vF_1=\langle{x,y,z}\rangle\)
(ii)
\(\vF_2=\langle{-y,x,-1}\rangle\)
(iii)
\(\vF_3=\langle{x-y,y+x,z-1}\rangle\)
(e)
Use computer software to plot each of the vector fields from part d and interpret the results of your flux integral calculations.
(f)
If we used the sphere of radius 4 instead of \(S_2\text{,}\) 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.
Remark 4.8.12.
Note that throughout this section, we have implicitly assumed that we can parametrize the surface \(S\) in such a way that \(\vr_s\times \vr_t\) gives a well-defined normal vector. Technically, this means that the surface be orientable. Most “reasonable” 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?
Subsection 4.8.5 Summary
- A flux integral of a vector field, \(\vF\text{,}\) on a surface in space, \(S\text{,}\) measures how much of \(\vF\) goes through \(S_1\text{.}\)
- Let the smooth surface, \(S\text{,}\) be parametrized by \(\vr(s,t)\) over a domain \(D\text{.}\) The total flux of a smooth vector field \(\vF\) through \(S\) is given by\begin{equation*} \iint_D \vF \cdot (\vr_s \times \vr_t)\, dA\text{.} \end{equation*}
- If \(S_1\) is of the form \(z=f(x,y)\) over a domain \(D\text{,}\) then the total flux of a smooth vector field \(\vF\) through \(S_1\) is given by\begin{equation*} \iint_D \vF(x,y,f(x,y)) \cdot \left\langle -\frac{\partial{f}}{\partial{x}},-\frac{\partial{f}}{\partial{y}},1 \right\rangle\, dA\text{.} \end{equation*}
