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Results: curvilinear coordinates

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.

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.

Scalar Surface and Volume Elements

Integration Sequence

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.

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.

format_list_numbered Sequence

Integration Sequence
Students learn/review how to do integrals in a multivariable context, using the vector differential \(d\vec{r}=dx\, \hat{x}+dy\, \hat{y}+dz\, \hat{z}\) and its curvilinear coordinate analogues as a unifying strategy. This strategy is common among physicists, but is NOT typically taught in vector calculus courses and will be new to most students.

assignment Homework

Divergence Practice including Curvilinear Coordinates

Calculate the divergence of each of the following vector fields. You may look up the formulas for divergence in curvilinear coordinates.

  1. \begin{equation} \hat{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z} \end{equation}
  2. \begin{equation} \hat{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z} \end{equation}
  3. \begin{equation} \hat{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z} \end{equation}
  4. \begin{equation} \hat{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z} \end{equation}
  5. \begin{equation} \hat{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z} \end{equation}
  6. \begin{equation} \hat{K} = s^2\,\hat{s} \end{equation}
  7. \begin{equation} \hat{L} = r^3\,\hat{\phi} \end{equation}

assignment Homework

Curl Practice including Curvilinear Coordinates

Calculate the curl of each of the following vector fields. You may look up the formulas for curl in curvilinear coordinates.

  1. \begin{equation} \vec{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z} \end{equation}
  2. \begin{equation} \vec{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z} \end{equation}
  3. \begin{equation} \vec{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z} \end{equation}
  4. \begin{equation} \vec{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z} \end{equation}
  5. \begin{equation} \vec{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z} \end{equation}
  6. \begin{equation} \vec{K} = s^2\,\hat{s} \end{equation}
  7. \begin{equation} \vec{L} = r^3\,\hat{\phi} \end{equation}

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.
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