*accessibility_new*Curvilinear Basis Vectors*accessibility_new*Kinesthetic10 min.

##### Curvilinear Basis Vectors

Static Fields 2022 (8 years) Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).*group*Vector Surface and Volume Elements*group*Small Group Activity30 min.

##### Vector Surface and Volume Elements

Static Fields 2022 (3 years)Students use known algebraic expressions for vector line elements \(d\vec{r}\) to determine all simple vector area \(d\vec{A}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

*assignment_ind*Curvilinear Coordinates Introduction*assignment_ind*Small White Board Question10 min.

##### Curvilinear Coordinates Introduction

Static Fields 2022 (9 years)Cylindrical coordinates spherical coordinates curvilinear coordinates

First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles \(\theta\) and \(\phi\). Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.*assignment*Distance Formula in Curvilinear Coordinates*assignment*Homework##### Distance Formula in Curvilinear Coordinates

Static Fields 2022 (5 years)The distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between the point \(\vec r\) and the point \(\vec r'\) is a coordinate-independent, physical and geometric quantity. But, in practice, you will need to know how to express this quantity in different coordinate systems.

- Find the distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between the point \(\vec r\) and the point \(\vec r'\) in rectangular coordinates.
- Show that this same distance written in cylindrical coordinates is: \begin{equation} \left|\vec r -\vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi-\phi\,{}') +(z-z\,{}')^2} \end{equation}
- Show that this same distance written in spherical coordinates is: \begin{equation} \left\vert\vec r -\vec r\,{}'\right\vert =\sqrt{r'^2+r\,{}^2-2rr\,{}' \left[\sin\theta\sin\theta\,{}'\cos(\phi-\phi\,{}') +\cos\theta\cos\theta\,{}'\right]} \end{equation}
- Now assume that \(\vec r\,{}'\) and \(\vec r\) are in the \(x\)-\(y\) plane. Simplify the previous two formulas.

*keyboard*Electrostatic potential of spherical shell*keyboard*Computational Activity120 min.

##### Electrostatic potential of spherical shell

Computational Physics Lab II 2022 Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize.*group*Vector Differential--Curvilinear*group*Small Group Activity30 min.

##### Vector Differential--Curvilinear

Vector Calculus II 2022 (8 years)vector calculus coordinate systems curvilinear coordinates

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

*computer*Visualizing Combinations of Spherical Harmonics*computer*Mathematica Activity30 min.

##### Visualizing Combinations of Spherical Harmonics

Central Forces 2023 (3 years) Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.*group*Scalar Surface and Volume Elements*group*Small Group Activity30 min.

##### Scalar Surface and Volume Elements

Static Fields 2022 (6 years)Students use known algebraic expressions for length elements \(d\ell\) to determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

*format_list_numbered*Curvilinear Coordinate Sequence*format_list_numbered*Sequence##### Curvilinear Coordinate Sequence

The curvilinear coordinate sequence introduces cylindrical and spherical coordinates (including inconsistencies between physicists' and mathematicians' notational conventions) and the basis vectors adapted to these coordinate systems.*group*Total Charge*group*Small Group Activity30 min.

##### Total Charge

Static Fields 2022 (5 years)charge charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates

In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.-
Central Forces 2023 (3 years)
Show that the plane polar coordinates we have chosen are equivalent to spherical coordinates if we make the choices:

- The direction of \(z\) in spherical coordinates is the same as the direction of \(\vec L\).
- The \(\theta\) of spherical coordinates is chosen to be \(\pi/2\), so that the orbit is in the equatorial plane of spherical coordinates.