Students compute vector line integrals and explore their properties.
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?
Ask the students if their level curves are equally spaced.
(They shouldn't be.)