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## Curvilinear Coordinates

This sequence introduces cylindrical and spherical coordinate systems. While middle-division students may have been introduced to the cylindrical and spherical coordinates themselves, in a multivariable calculus course, they have almost certainly NOT been introduced to the basis vectors such as $\hat{r}$, $\hat{\theta}$, etc. that are adapted to them.

BEWARE: The conventions for the names of angles in spherical coordinates are opposite for American physicists and American mathematicians, see Tevian Dray and Corinne A. Manogue, *Spherical Coordinates*, College Mathematics Journal **34**, 168–169 (2003). (MAA listing)

### Activities

**Definition of Curvilinear Coordinates***(Estimated time: 5 minutes)*: This lecture serves as an introduction to the notations which physicists use to represent vector fields in various coordinate systems.

**Drawing Surfaces in Cylindrical and Spherical Coordinates***(Estimated time: 5 minutes)*: In this sequence of small whiteboard questions, the students are asked to draw surfaces of equal values of coordinates in cylindrical ($s$, $\theta$, and $\phi$) and spherical coordinates ($r$, $\theta$, and $\phi$). This can lead into a whole class discussion on the range of values allowed for each coordinate in cylindrical and spherical coordinate systems.

**Curvilinear Basis Vectors***(Estimated time: 15 minutes)*: In this kinesthetic activity students are asked to point in $\hat{r}$, $\hat{\theta}$, $\hat{\phi}$, $\hat{s}$, and $\hat{z}$ directions in reference to an origin within the classroom. A class discussion ensues about the directions of curvilinear basis vectors and how the direction changes at different points in space. This is in contrast to rectangular unit vectors, $\hat{x}$, $\hat{y}$, and $\hat{z}$, which have fixed directions at each point in space. Many mathematics courses do not cover curvilinear basis vectors, so it is expected that students will not be familiar with these basis vectors.

**Introducing $d\vec{r}$***(Estimated time: 5 minutes)*: This mini-lecture introduces the $d\vec{r}$ in terms of rectangular coordinates. This lecture can be used to introduce the dr in Cylindrical and Spherical Coordinates activity which finds $d\vec{r}$ in those coordinates.

Rationalize the following four activities:

**Curvilinear Coordinates (scalar version)***(Estimated time: 30 minutes)*:

**Curvilinear Coordinates (vector version)***(Estimated time: 30 minutes)*:

**Pumpkins and Pineapples and dr in Cylindrical and Spherical Coordinates***(Estimated time: 30 minutes)*: These small group activities have students find $d\vec{r}$ in curvilinear coordinates by using pumpkins and pineapple slices to construct volume elements while determining how to construct $d\vec{r}$ in curvilinear coordinates. This activity is intended to introduce students to small changes in each coordinate direction. (finish)