## Small Group Activity: The Fishing Net

Vector Calculus II 2021
Students compute surface integrals and explore their interpretation as flux.
What students learn
• Practice computing surface integrals;
• Geometric interpretation of flux;
• Geometric interpretation of divergence theorem.
• Media

A fishing net $S$ is in the shape of a triangular trough, as shown in the picture. The triangular sides are at $x=0$ and $x=5$, the rectangular sides are at $45^\circ$ to the vertical, and the bottom is at $z=0$; all lengths are measured in $\hbox{cm}$. There is no netting across the top, which is at $z=1$. Water is draining out of the net; the motion of the water is described by the vector field $\boldsymbol{\vec{F}} = \rho \left( a e^{\kappa z^2}\boldsymbol{\hat{y}} - b\,\boldsymbol{\hat{z}} \right)$ where $a=3\,{\hbox{cm}\over\hbox{s}}$, $b=5\,{\hbox{cm}\over\hbox{s}}$, $\kappa=2\,{\hbox{cm}^{-2}}$, and $\rho$ is the (constant) density of the water in $\hbox{g}\over\hbox{cm}^3$. The goal of this problem is to find the best way to evaluate the flux $\int\!\!\int \boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{S}}$ of water down through $S$.

• Set up the above surface integral, but do not evaluate it
• Use the Divergence Theorem to find another way to do the problem.
This time, complete the computation.

#### Main ideas

• Practice doing surface integrals
• The Divergence Theorem

#### Prerequisites

• Ability to do flux integrals
• Definition of divergence
• Statement of Divergence Theorem This lab can be used prior to covering the Divergence Theorem in class with either a minimal introduction or a restatement of the last question based on the assumption that the given vector field doesn't “lose” anything going through the net.

#### Warmup

• Perhaps a reminder about what the Divergence Theorem is.

#### Props

• whiteboards and pens
• a model of the fishing net, made from any children's building set

#### Wrapup

• Reiterate that the Divergence Theorem only applies to closed surfaces.
• Emphasize that the Divergence Theorem is one of several astonishing theorems relating what happens inside to what happens outside.
• Have several students show how they computed $d\boldsymbol{\vec{S}}$, since most likely different choices were made for $d\boldsymbol{\vec{r}}_i$ and hence the limits.

### Details

#### In the Classroom

• By now the groups should be working well. Sit back and watch!
• The main thing to watch out for is whether students choose the correct signs, both for the normal vectors and the limits of integration. Reiterate that one should always write $d\boldsymbol{\vec{r}}=dx\,\boldsymbol{\hat{x}}+dy\,\boldsymbol{\hat{y}}+dz\,\boldsymbol{\hat{z}}$; there should never be minus signs in this equation. The signs will come out right provided one integrates in the direction of the vectors chosen.
• Most students will realize quickly that there is no flux through the triangular sides. (encourage supporting this with geometry and a calculation).
• Some students will try to do the surface integrals! Point out that this isn't possible --- and that the instructions say not to.
• Student may be surprised at first when they calculate $\nabla\cdot\boldsymbol{\vec{F}}=0$, especially since they (correctly) won't think that the surface integrals will add to zero. Use this to motivate the “missing top”.
• Some students incorrectly think that $d|z|=|dz|$.
• It can be tough for people to take the image and develop an equation for the sides of the net. This is healthy.

#### Subsidiary ideas

• The geometry of flux.

#### Enrichment

• The surface integrals can in fact be done --- provided one adds them up prior to evaluating the integrals.
• This lab provides a good opportunity for students to visualize the flux: It's easy to see that the flux of the horizontal component of this vector field must be zero geometrically. (It's even easier to see that the vertical flux must be zero.)
• During the wrapup (or the following lecture), draw a picture such as the one above of one of the rectangular faces, showing all 4 possible choices for $d\boldsymbol{\vec{r}}_1$ and $d\boldsymbol{\vec{r}}_2$ (and which is which!), and discuss the integration limits in each case.
• An alternative approach to this problem is to determine $d\boldsymbol{\vec{S}}$ geometrically, compute $\boldsymbol{\vec{F}}\cdot\boldsymbol{\hat{n}}$ explicitly, and then do the integral using “standard” (increasing) limits. There is nothing wrong with this approach, but we would discourage the use of the $d\boldsymbol{\vec{r}}$ notation here for fear of making sign errors.
• One could show students the remarkable trick for integrating $e^{-x^2}$ from $0$ to $\infty$, by squaring and evaluating in polar coordinates.

Keywords
Learning Outcomes