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## Representations of Two-Dimensional Scalar Fields

This sequence can be used following Representations of Ordinary Derivatives sequence at the beginning of middle-division courses to quickly introduce students to using multiple representations. These two sequences do not require knowledge beyond what is introduced in introductory physics and calculus courses, however, they can be used to discuss previous information in more detail.

### Activities

**Representations of Temperature in Two Dimensions***(Estimated time: 5 minutes)*: This small whiteboard questions prompts students to think about ways in which to represent temperature in two dimensions which is an example of a two dimensional scalar field.

**Isoheights on Surfaces***(Estimated time: 5 minutes)*: This small group activity first has students draw lines of “isoheight” on the surfaces. Next, students are given topographic maps which correspond to the surface and asked to match the surface to the contours using their “isoheights”. This requires students to become familiar with two representations of a two-dimensional scalar field and work between the two representations.

**Scalar Field Examples***(Estimated time: 5 minutes)*: This small whiteboard question is prompted by a definition of a scalar field in terms of the temperature. Students are asked to give examples of scalar fields and responses may include sound intensity, mass density, charge density, and electrostatic potential.

**Representations of Two-Dimensional Scalar Fields Discussion***(Estimated time: 10 minutes)*: This whole class discussion involves student ideas about ways in which to represent scalar fields. The strengths and weaknesses of many different representations of two-dimensional scalar fields are discussed among students and the instructor.