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
    Your answer should be ready to integrate; among other things, all substitutions should be made, and you should determine the correct limits.
  • Use the Divergence Theorem to find another way to do the problem.
    This time, complete the computation.

Learning Outcomes