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Scalar Distance, Area, and Volume Elements

Keywords: Small Group Activity, Vector Calculus, Coordinate Systems, Symmetry, Curvilinear Coordinates, Differential Elements

Highlights of the activity

  1. This small group activity is designed to help students construct scalar differential elements ($ds$, $dA$, $d\tau$) for integration in curvilinear coordinates.
  2. Students calculate infinitesimal line elements in curvilinear coordinates and use these to construct area and volume elements.
  3. The whole class discussion focuses on infinitesimal scalar elements used in integration in curvilinear coordinates.

Reasons to spend class time on the activity

Integration in curvilinear coordinates is typically introduced to students in calculus courses prior to upper-division electricity and magnetism, however, some students do not recall or are not comfortable with integrating over lines, areas, and volumes in curvilinear coordinates. This maybe a result of memorizing various aspects of curvilinear coordinates. Therefore, this activity serves as a review of integration by emphasizing the geometry of differential line, area, and volume elements which are commonly used upper-division electricity and magnetism.


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