Activity: The Valley

Prerequisite concepts

Multivariable Calculus The components of the gradient are partial derivatives

FIXME

Static Fields The gradient is a local quantity

The magnitude and direction of the gradient depends on the behavior of the function infinitesimally near the point of interest. One implication of this is that the gradient does not point in the direction of maxima a finite distance away (except sometimes accidentally), but rather in the direction of steepest slope at the point of interest.

The Hill

The gradient is a local quantity

The magnitude of the gradient is proportional to the density of contour lines The direction of the gradient is perpendicular to contour lines The gradient lives in the domain and has the same number of spatial dimensions as the original function

Instructor guide vcvalley

The Valley

This small group activity is designed to reinforce the geometry of line integrals, and to set the stage for path independence, and builds on the Hill activity. Students work in small groups to evaluate the elevation change along different paths. The whole class wrap-up discussion reinforces the importance of a “Use what you know!” strategy to evaluate line integrals, and emphasizes that different paths with the same endpoints must have the same elevation gain.

Representations used

Contour Maps

Concepts taught

The gradient can tell you a small change in a function in any direction (differentials edition)

$df=\vec \nabla f \cdot d \vec r$