• Gradient

Estimated Time: 30 minutes

Students should be familiar with vectors and with differential calculus.

• Tabletop Whiteboard with markers
• Computers with Maple (optional)
• A handout for each student

We preface this activity with a mini-lecture about the gradient. Students should be familiar with how to calculate a gradient:

$$\Vec{\nabla} f = \frac{\partial f}{\partial x}\;\hat{\imath}+ \frac{\partial f}{\partial y}\;\hat{\jmath}+\frac{\partial f}{\partial z}\;\hat{k}$$

and the geometric property that the gradient points in the direction of greatest increase in the function.

• Students very quickly figure out parts a) the location of the top of the hill and b) the height of the hill.
• Topo Maps We find that students are eager to skip over part c) and must be encouraged to complete this step. We allow students to use Maple to generate a topo map if they want (we want to encourage students to use Maple to help them visualize physics problems).
• Where does the gradient live? (relevant for parts (c) and (d)). Students will not realize that because the height is a function of two variables in this problem, the gradient of the height function is a 2-D that lives in the topo map. Have the students stand up and imagine the room is a hill, with the top of the hill in one of the corners of the room. Have the students close their eyes and use their rights arms to point in the direction of the gradient. Students should be alerted to the fact that their arms should be parallel to the floor.
• Compass direction versus slope The gradient tells the students the direction of steepest ascent, and it also contains information about how quickly the height function is changing. $$df = \Vec{\nabla}f\cdot d\Vec{r}$$ Therefore, the magnitude of the gradient is the slope. To find the 3-D vector direction of travel, students need to find a unit vector in the direction of gradient as well as the change in height. Most students will forget that they need a normalized vector in the x-y plane to give the 3-D vector pointing along the steepest direction at their point.

It is useful to have a couple groups present their reasoning and their answers.

This activity is part of a sequence of activities which address the Geometry of the Gradient. The following activities are included within this sequence.

• Preceding activity:
  • Acting Out the Gradient: This kinesthetic activity introduces students to the geometric concept of the gradient through an imaginary elliptic hill in the classroom where students use their arms to represent the gradient at their local point within the classroom.

• Follow-up activity:
  • Visualizing Gradient: This activity uses a Mathematica (or Maple) worksheet to visualize the relationship between scalar fields and the gradient.