In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

Infinitesimal reasoning in cylindrical and spherical coordinates.

Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and we demonstrate the answers with coins under a doc cam.

• How to find area, and volume elements in curvilinear coordinates using geometric methods.

Students use \(d\boldsymbol{\vec{A} }= d\boldsymbol{\vec{r}}_1 \times d\boldsymbol{\vec{r}}_2\) and \(d\tau=(d\boldsymbol{\vec{r}}_1\times d\boldsymbol{\vec{r}}_2)\cdot d\boldsymbol{\vec{r}}_3\) to find differential surface and volume elements for cylinders and spheres.

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Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .

Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.

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