Polar vs. Spherical Coordinates
- accessibility_new Curvilinear Basis Vectors
accessibility_new Kinesthetic
10 min.
Curvilinear Basis Vectors
Static Fields 2023 (9 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 Activity
30 min.
Vector Surface and Volume Elements
Static Fields 2023 (4 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 Question
10 min.
Curvilinear Coordinates Introduction
Static Fields 2023 (10 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 2023 (6 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.
-
Find the distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between
the point \(\vec r\) and the point \(\vec
r'\) in rectangular coordinates.
- keyboard Electrostatic potential of spherical shell
keyboard Computational Activity
120 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. - computer Visualizing Combinations of Spherical Harmonics
computer Mathematica Activity
30 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 Vector Differential--Curvilinear
group Small Group Activity
30 min.
Vector Differential--Curvilinear
Vector Calculus II 23 (9 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.
- group Scalar Surface and Volume Elements
group Small Group Activity
30 min.
Scalar Surface and Volume Elements
Static Fields 2023 (7 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 Activity
30 min.
Total Charge
Static Fields 2023 (6 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.