Activity: Visualising the Gradient

Static Fields 2022 (6 years)
Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
What students learn
  • Model the gradient using a computer algebra system.
  • Visualize the geometric relationship between the gradient and the contours plot of a scalar field
  • Clarify how gradients look in two and three dimensions.

Use the Sage code at Using Technology to Visualize the Gradient to plot scalar functions of two variables, predict what a plot of the gradient looks like, and then check it.

Instructor's Guide

Student Task

This activity is a great follow up to Acting Out the Gradient. The students should know that the gradient is perpendicular to level curves (surfaces) and that its magnitude is the maximum slope. They use this information to predict the gradient vector field from looking at level curves for scalar fields. Then check their predictions with the Mathematica worksheet. The worksheet is open-ended and students can plot some scalar fields of their own.

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  • assignment Contours

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    Shown below is a contour plot of a scalar field, \(\mu(x,y)\). Assume that \(x\) and \(y\) are measured in meters and that \(\mu\) is measured in kilograms. Four points are indicated on the plot.

    1. Determine \(\frac{\partial\mu}{\partial x}\) and \(\frac{\partial\mu}{\partial y}\) at each of the four points.
    2. On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the gradient of \(\mu(x,y)\) using your answers to part a above. How did you choose a scale for your vectors? Describe how the direction of the gradient vector is related to the contours on the plot and what property of the contour map is related to the magnitude of the gradient vector.
    3. Evaluate the gradient of \(h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3\) at the point \((x,y)=(3,-2)\).

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Author Information
Corinne Manogue, Tevian Dray
Learning Outcomes