Kinesthetic: Acting Out the Gradient

Static Fields 2021
Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".
What students learn
  • Geometric understanding of the directionality of the gradient vector.
  • Clarify misconceptions about the phrase “the gradient always points up hill”
  • Media
    • 2255/hill.png

There is a hill in this classroom. The top of the hill is in the middle of the room at the ceiling. This topo map below describes the hill in the room. You are standing at some point on the topo map. Use your right arm to point in the direction of the gradient.

Instructor's Guide


For this activity, the class is asked to stand from their seats. The students are told that they are all standing on an elliptical hill, represeted by the topo map, and one location of the classroom is selected as the top of a hill, typically in the center of the room. (If you are in a tiered lecture hall, then make use of the actual hill in the room, istead!)

The students are asked to close their eyse and point in the direction of the gradient.

Student Conversations

  1. Many students will incorrectly point towards the top of the hill, rather than perpendicular to the level curves. The gradient is not always the direction of the top of the hill. despite the gradient only lying in the \(x\), \(y\)-plane.
  2. Many students will incorrectly point upward. For a function of two variables, the gradient does not have a third, vertical component. The gradient lives in the topo map, not in 3-d space.


Ask students to generalize the concepts in this activity to functions of three dimensions. Emphasize the understanding that the gradient is always perpendicular to the level curves (for two dimensions) or level surfaces (for three dimensions).

Author Information
Corinne Manogue, Tevian Dray
gradient vector fields electrostatics
Learning Outcomes