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Activities & Homework Problems
The Valley
(PDF)
Student handout
(PDF)
Student handout: The Valley
Surfaces/Bridge Workshop 2023
Students compute vector line integrals and explore their properties.
group
Small Group Activity
schedule
60 min.
description
Student handout
(PDF)
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?
Keywords
Learning Outcomes