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".

• The gradient is perpendicular to the level curves.
• The gradient is a local quantity, i.e. it only depends on the values of the function at infinitesimally nearby points.
• Although students learn to chant that "the gradient points uphill," the gradient does not point to the top of the hill.
• The gradient path is not the shortest path between two points.

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.

Students are asked to "find the derivative" of a plastic surface that represents a function of two variables. This ambiguous question is designed to help them generalize their concept of functions of one variable to functions of two variables. The definition of the gradient as the slope and direction of the "steepest derivative" is introduced geometrically.

Find the gradient of each of the following functions:

1. \begin{equation} f(x,y,z)=e^{(x+y)}+x^2 y^3 \ln \frac{x}{z} \end{equation}
2. \begin{equation} \sigma(\theta,\phi)=\cos\theta \sin^2\phi \end{equation}
3. \begin{equation} \rho(s,\phi,z)=(s+3z)^2\cos\phi \end{equation}

Consider the fields at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\).

1. Write down an expression for the electrostatic potential \(V(\vec{r})\) at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to calculate here.)
2. Write down an expression for the electric field \(\vec{E}(\vec{r})\) at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to calculate here.)
3. Working in rectangular coordinates, compute the gradient of \(V\).
4. Write several sentences comparing your answers to the last two questions.

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.

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).

In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).