In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
1. << Electric Field of a Line Source From Potential | Gradient Sequence |
Suppose you are standing on a hill. You have a topographic map, which uses rectangular coordinates \((x,y)\) measured in miles. Your global positioning system says your present location is at one of the following points (pick one):
A: \((1,4)\qquad\) B: \((4,-9)\qquad\) C: \((-4,9)\qquad\) D: \((1,-4)\qquad\) E: \((2,0)\qquad\) F: \((0,3)\)Your guidebook tells you that the height \(h\) of the hill in feet above sea level is given by \[h = a - b x^2 - c y^2\] where \(a=5000\hbox{ft}\), \(b=30\,{\hbox{ft}\over\hbox{mi}^2}\), and \(c=10\,{\hbox{ft}\over\hbox{mi}^2}\).
Starting at your present location, in what map direction (2-d unit vector) do you need to go in order to climb the hill as steeply as possible?
Draw this vector on your topographic map.
How steep is the hill if you start at your present location and go in this compass direction?
Draw a picture which shows the slope of the hill at your present location.
In what direction in space (3-d vector) would you actually be moving if you started at your present location and walked in the map direction you found above?
To simplify the computation, your answer does not need to be a unit vector.
It's tempting to use a hill as a nice geometric example of a function of two variables. However, doing so opens a can of worms. In examples like this, when the function has dimensions of length, students are confused as to whether the gradient is 2-dimensional or 3-dimensional. In most applications, involving physical quantities such as temperature, this confusion does not arise. If you want to use hills as an important example, then it's best to confront this confusions head-on; this lab is a good way to do so, although this requires fairly sophisticated geometric reasoning. If you choose to restrict to other applications, you may prefer to skip this lab.
Consider a valley whose height \(h\) in meters is given by \(h={~x^2\over10}+{~y^2\over10}\), with \(x\) and \(y\) (and 10!) in meters. Suppose you are hiking through this valley on a trail given by \(x=3t\), \(y=2t^2\), with \(t\) in seconds. How fast are you climbing per meter along the trail when \(t=1\)? How fast are you climbing per second when \(t=1\).
Discuss which way you should go to get to the top of the hill the fastest. What does this mean? The shortest path (geodesic)? The one with the largest average steepness? The smallest average steepness?
As one student put it, the answer depends not only on the shape of the hill, but also on the shape of the hiker!
Regard the hill as a level surface of the function of three variables \(z-h(x,y)\). What is the gradient of this function? What does it mean geometrically? How is it related to the gradient of \(h\) and to the 3-dimensional vector found in the last problem?
(This is a good problem for honors students.)