Activity: The Hill (editing)

Prerequisite concepts
Multivariable Calculus
Derivatives can be found while holding one or more variables constant

Derivatives of multivariable functions can be found by holding one or more variables constant (subject to physical limitations).
There is a partial derivative in every direction at any point

For a smooth, two dimensional function, a partial derivative can be found at any point and in any direction.
The value of a partial derivative depends on the value(s) of what is held constant

FIXME: Is this really a statement about paths? I don't think we have path-related concepts.
The magnitude of the gradient is proportional to the density of contour lines
Contour Maps

As adjacent contour lines represent equal changes in the function, the denser the contour lines, the quicker the function is changing. Therefore, the magnitude of the gradient is greater where the contour lines are denser.
The Hot Plate
The magnitude of the gradient is proportional to the density of contour lines
Contour Maps

As adjacent contour lines represent equal changes in the function, the denser the contour lines, the quicker the function is changing. Therefore, the magnitude of the gradient is greater where the contour lines are denser.
The magnitude of the gradient is proportional to the density of contour lines
Contour Maps

As adjacent contour lines represent equal changes in the function, the denser the contour lines, the quicker the function is changing. Therefore, the magnitude of the gradient is greater where the contour lines are denser.
The magnitude of the gradient is proportional to the density of contour lines
Contour Maps

As adjacent contour lines represent equal changes in the function, the denser the contour lines, the quicker the function is changing. Therefore, the magnitude of the gradient is greater where the contour lines are denser.
There is a partial derivative in every direction at any point
There is a tangent line in every direction at every point
The value of a partial derivative depends on the value(s) of what is held constant
Instructor guide vchill

The Hill

This small group activity is designed to reinforce the geometric definition of the gradient. Students work in small groups to construct the gradient vector at different points on a hill. Then, students compare and contrast their findings to reinforce that the gradient is a vector field, but also that it is a local quantity. The whole class wrap-up discussion emphasizes that the gradient lives in the domain, not on the graph.

Representations used

Contour Maps
$\vec \nabla f$

Concepts taught

The magnitude of the gradient is proportional to the density of contour lines
Contour Maps

As adjacent contour lines represent equal changes in the function, the denser the contour lines, the quicker the function is changing. Therefore, the magnitude of the gradient is greater where the contour lines are denser.
The direction of the gradient is perpendicular to contour lines
Contour Maps

As the gradient lies along the direction of greatest change for a function, and contour lines lie along the direction of no change, these two objects (the gradient and contour lines) are perpendicular.
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 gradient lives in the domain and has the same number of spatial dimensions as the original function

The gradient of an $n$ dimensional function is $n$ dimensional. This is not to be confused with the gradient having the same physical dimensions (e.g., Newtons) as the original function.