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### THE FISHING NET

#### Essentials

##### 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 $d\SS$, since most likely different choices were made for $d\rr_i$ 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 $d\rr=dx\,\ii+dy\,\jj+dz\,\kk$; 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.
• 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 $\grad\cdot\FF=0$, 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 $d|z|=|dz|$.

##### Subsidiary ideas

• The geometry of flux.

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##### 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 $d\rr_1$ and $d\rr_2$ (and which is which!), and discuss the integration limits in each case.
• An alternative approach to this problem is to determine $\dS$ geometrically, compute $\FF\cdot\nn$ 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 $d\rr$ 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.

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