Integration in Curvilinear Coordinates to Determine Total Charge

Integration using curvilinear coordinates has been introduced to most students in a multivariable calculus course prior to entering into upper-division electricity and magnetism courses, however, some students may not carry those skills immediately into physics. By reintroducing curvilinear coordinates, with the physics convention for $\theta$ and $\phi$, in a physical context, students can determine the total charge of a line, surface, or volume which are prevalent in electrostatic problems and essential to certain techniques such as Gauss's law.

Additionally, this sequence is intended to expand student understanding of integration to include integrals as measurable experimentally. Integrals can be measured by the accumulation of small pieces or small changes in measurable quantities. By including an experimental activity of integration, students can begin to more fluently use different representations of integration which likely already include FIXME (symbolic manipulation, area under a curve, and …).


  • Internal Energy of Derivative Machine (Estimated time: ): This small group activity uses a modified Partial Derivative Machine to measure the internal energy of a nonlinear, one dimensional system. This activity emphasizes integration as an experimentally measurable quantity as students must add together small changes they measure in order to calculate the internal energy of the system. Students begin to think of integration as accumulating small pieces which is important in many instances in electricity and magnetism.
  • Curvilinear Coordinates (Estimated time: ): This lecture introduces students to curvilinear coordinates. The notation difference in physics and mathematics should be highlighted: that $\theta$ and $\phi$ are switched in each discipline. This is likely to be review for students who have been previously introduced to curvilinear coordinates in math courses.
  • Scalar Distance, Area, and Volume Elements (Estimated time: ): In this small group activity students derive expressions for infinitesimal distances in order to find area and volume elements in cylindrical and spherical coordinates. This activity can be done with Pineapples and Pumpkins to give students a three dimensional object to explore the geometry and construction of a volume element.
  • Pineapples and Pumpkins (Estimated time: ): This activity can be done in small groups or as an instructor led whole class activity. A pineapple (for cylindrical) and/or pumpkin (for spherical) can be cut to demonstrate the geometry of an infinitesimal volume element used in integration. If done as a small group activity, it can be combined with Scalar Infinitesimal Distance, Area, and Volume Elements. If done as a whole class activity, the instructor cuts a pumpkin and/or pineapple prompted by students answering a series of small whiteboard questions. This emphasizes the construction of infinitesimal volume elements using a three dimensional representation.
  • Acting Out Charge Densities (Estimated time: 10 minutes): This kinesthetic activity provides students with an embodied understanding of charge density and total charge by using their bodies to represent charges. Students move around the room to act out linear, surface, and volume charge densities which prompts a whole class discussion on the meaning of constant charge density, the geometric differences between linear, surface, and volume charge densities, and what is “linear” about linear charge density.
  • Total Charge (Estimated time: 30 minutes): In this small group activity, students calculate the total charge within spherically or cylindrically symmetric volumes. Students use multivariable integration in various coordinate systems in order to find the total charge contained within the volume due to a specific charge density.

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