Student handout: The Valley

Surfaces/Bridge Workshop 2023
Students compute vector line integrals and explore their properties.
What students learn
  • Practice with gradient and line integrals;
  • Exploration of path independence for vector line integrals.
You are in a valley whose height is given by \(h = a x^2 + a y^2\) where \(a={1\over10}\,{\hbox{ft}\over\hbox{mi}^2}\).

Your location corresponds to \(x=y=1~\hbox{mi}\). Your goal is to reach the road located at \(y=0\).
  • Choose one of the following paths, and sketch it on your map.
    I: \(x^2+y^2=2\)       II: \(y=x\)       III: \(y=x^2\)       IV: \((y-1)=3(x-1)\)       V: \(x=1\)
  • Determine \(\boldsymbol{\vec\nabla} h\) at your location.
  • Calculate \(\oint_{C}\boldsymbol{\vec\nabla} h\cdot d\boldsymbol{\vec{r}}\) along your path.
  • Compute \(\int_C dh\) along your path.
  • Compare your answers to these two integrals. What do your answers represent?
    Is there an easier way to get the same answer?

Learning Outcomes