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assignment_ind Small White Board Question

10 min.

Curvilinear Coordinates Introduction

Cylindrical coordinates spherical coordinates curvilinear coordinates

Curvilinear Coordinate Sequence

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.

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.

assignment Homework

Distance Formula in Curvilinear Coordinates

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.

  1. Find the distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between the point \(\vec r\) and the point \(\vec r'\) in rectangular coordinates.
  2. 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}
  3. 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}
  4. Now assume that \(\vec r\,{}'\) and \(\vec r\) are in the \(x\)-\(y\) plane. Simplify the previous two formulas.

group Small Group Activity

120 min.

Box Sliding Down Frictionless Wedge

Lagrangian Mechanics Generalized Coordinates Special Cases

Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.

accessibility_new Kinesthetic

30 min.

The Distance Formula (Star Trek)

distance formula coordinate systems dot product vector addition

Ring Cycle Sequence

A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. \[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]

accessibility_new Kinesthetic

10 min.

Curvilinear Basis Vectors

symmetry curvilinear coordinate systems basis vectors

Curvilinear Coordinate Sequence

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 Small Group Activity

30 min.

Vector Surface and Volume Elements

Integration Sequence

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 more sophisticated vector approach to find surface, and volume elements.

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