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Sequences

This sequence starts with an introduction to partial derivatives and continues through gradient. While some of the activities/problems are pure math, a number of other activities/problems are situated in the context of electrostatics. This sequence is intended to be used intermittently across multiple days or even weeks of a course or even multiple courses.

Activities

Kinesthetic

10 min.

Acting Out the Gradient
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".
  • gradient vector fields electrostatics
    Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence, Gradient Sequence sequence(s)

Small Group Activity

30 min.

The Hill
  • 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.
  • Gradient
    Found in: Vector Calculus II, Vector Calculus I, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Gradient Sequence sequence(s)

Mathematica Activity

30 min.

Visualising the Gradient
Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
  • Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Visualizing Scalar Fields, Geometry of Vector Fields Sequence, Gradient Sequence sequence(s)

Small White Board Question

5 min.

Find the Derivative
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}

  • Found in: AIMS Maxwell, Static Fields, Problem-Solving course(s) Found in: Gradient Sequence sequence(s)

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.

  • Found in: Gradient Sequence sequence(s)

Small Group Activity

30 min.

Directional Derivatives
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.

Small Group Activity

30 min.

The Hillside
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).
  • Found in: Vector Calculus I course(s) Found in: Gradient Sequence, Workshop Presentations 2023 sequence(s)

Small Group Activity

30 min.

DELETE Navigating a Hill
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.
  • Found in: Static Fields, AIMS Maxwell course(s)

Small Group Activity

30 min.

The Hillside (Updated)
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).
  • Found in: Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Workshop Presentations 2023 sequence(s)