Estimated Time:

• Tabletop Whiteboard with markers
• Pineapples for cylindrical coordinates
• Pumpkins for spherical coordinates
• Pumpkin carving kit
• A handout for each student

Typically this activity has been done as a whole class activity where students answer a series of small whiteboard questions as the instructor makes cuts in a pumpkin to construct a volume element in spherical coordinates. However, this activity can be done as a small group activity while doing Scalar Infinitesimal Line, Area, and Volume Elements.

If done as an instructor led whole class activity, the following provides a structure for the activity done in spherical coordinates using a pumpkin.

• Let's define a set of coordinates $\hat{x}$ and $\hat{y}$ ($\hat{z}$ is through the pumpkin stem). I'm going to start by drawing a line around the equator to help my drawing.
  • SWBQ: What is the circumference of this circle? Answer: $l=2\pi r$
• Now I'm going to pick a vector $\vec{r}$, which we will examine. When drawing figures (or pumpkins), I recommend that you never pick an angle that is close to $90^{\circ}$ or $45^{\circ}$, since that makes it way easier to distinguish between $\theta$ and $\frac{\pi}{2}-\theta$.
• Draw a longitude line.
  • SWBQ: What is the arc length from $\vec{r}$ to the stem? Answer: $l=r\theta$
  • SWBQ: What is the distance along the equator from $\hat{x}$? Answer: $r\phi$
• Now let's consider an infinitesimal volume of pumpkin. Pro tip: you always want to draw infinitesimal quantities as medium-sized, otherwise your drawing will be illegible, and won't actually help you. In this case, I'll pick a $d\theta$ and a $d\phi$.
  • SWBQ: What is this small distance along the equator? Answer: $dl=rd\phi$
  • SWBQ: What is this small distance along the longitude? Answer: $dl=rd\theta$
• Draw a latitude circle at $\vec{r}$.
  • SWBQ: What is the circumference of this circle? Answer: $l=2\pi s = 2\pi r\sin{\theta}$
• Cut out the pumpkin chunk.
  • SWBQ: What is the width of my pumpkin chunk? Answer: $dl=sd\phi=r\sin{\theta}d\phi$
• So when we put these distances together (including the thickness of the chunk dr), we find that the volume of our pumpkin chunk is $d\tau=dr(rd\theta)(r\sin{\theta}d\phi)=r^2\sin{\theta}drd\phi d\theta$

If done as a small group activity, this should be done while doing Scalar Infinitesimal Line, Area, and Volume Elements so that students have some guidance for making cuts in the pumpkin and the geometry of spherical volume elements is emphasized.

• Size of the Volume Element: When done as part of a small group activity, students may try to make their volumes as small as possible because they are thinking of an infinitesimally small volume element used for integration. Because this activity addresses the geometry of volume elements, encourage students to make large enough cuts in their pumpkins.

This is the initial activity within a sequence of activities addressing Scalar Integration in Curvilinear Coordinates. The following activities are included within this sequence:

• Preceding activities:
  • Internal Energy of Derivative Machine: This small group activity uses a modified Partial Derivative Machine to measure the internal energy of a nonlinear, one dimensional system while emphasizing integration as an experimentally measurable quantity.
  • Curvilinear Coordinates: This lecture introduces students to curvilinear coordinates and highlights the notation difference of $\theta$ and $\phi$ in physics and mathematics.
  • Scalar Distance, Area, and Volume Elements: In this small group activity students derive expressions for infinitesimal distances in order to find area and volume elements in cylindrical and spherical coordinates and can be done with Pineapples and Pumpkins to give students a three dimensional object to explore the geometry and construction of a volume element.
• Follow-up activities:
  • Acting Out Charge Densities: This kinesthetic activity provides students with an embodied understanding of charge density and total charge by using their bodies to represent charges and 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: In this small group activity, students calculate the total charge within spherically or cylindrically symmetric volumes by using multivariable integration in various coordinate systems in order to find the total charge contained within the volume due to a specific charge density.