Activity: The Hill

Vector Calculus II 23 (8 years)
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.
What students learn
  • The gradient is perpendicular to the level curves.
  • The gradient is a local quantity, i.e. it only depends on the values of the function at infinitesimally nearby points.
  • Although students learn to chant that "the gradient points uphill," the gradient does not point to the top of the hill.
  • The gradient path is not the shortest path between two points.

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.

Learning Outcomes