Student handout: The Hillside

Vector Calculus I 2022
Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
What students learn
  • The gradient is a geometric object.
  • At a given point, the gradient points in the direction in which the function is changing most rapidly.
  • At a given point, the magnitude of the gradient represents the rate of change of the function (in this case, the slope of the hill).

Warm-up: Imagine you are standing on the side of a tall hill. List three things you would want to know about your position.

On your Mark: Place your surface on the grid. Label the \(x\) and \(y\) directions on the grid and surface. Measure the slope in the direction of greatest increase of the surface at the blue dot. Include units.

Slope in steepest direction: \(\underline{\hspace{2in}}\)

Get Set: The surface's height \(h\) is a function of \(x\) and \(y\), giving \(h = h(x,y)\). At the blue dot, measure both \(\frac{\partial h}{\partial x}\) and \(\frac{\partial h}{\partial y}\). Then form the vector \(\frac{\partial h}{\partial x}\,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y}\,\boldsymbol{\hat{y}}\). Include units.

\(\frac{\partial h}{\partial x} = \underline{\hspace{.75in}} \hspace{.5in} \frac{\partial h}{\partial y} = \underline{\hspace{.75in}} \hspace{.5in} \frac{\partial h}{\partial x} \,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y} \,\boldsymbol{\hat{y}} = \underline{\hspace{.75in}} \,\boldsymbol{\hat{x}} + \underline{\hspace{.75in}} \,\boldsymbol{\hat{y}}\)

Go: At the blue dot, which way does your vector \(\frac{\partial h}{\partial x}\,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y}\,\boldsymbol{\hat{y}}\) point on the surface?

  1. What is your vector's magnitude?
  2. How does your vector relate to the level curve through the blue dot?

Challenge: Rotate the surface \(30^\circ\) on the grid and redraw the \(x\) and \(y\) directions on your surface. Which of your answers to On your Mark, Get Set, and Go remain the same?

Copyright 2014 by The Raising Calculus Group

Author Information
Surfaces team
Learning Outcomes