## Student handout: 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.

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