Activity: 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.

What students learn Students use \(d\vec{A} = d\vec{r}_1 \times d\vec{r}_2\) and \(d\tau=(d\vec{r}_1\times d\vec{r}_2)\cdot d\vec{r}_3\) to find differential surface and volume elements for cylinders and spheres.

Instructor's Guide

Notes:

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

This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins.

Introduction

In a previous activity, Vector Differential--Curvilinear, students are asked to find the vector line element, \(d\vec{r}\), along each side of an “infinitesimal box” in cylindrical and spherical coordinates. Using the \(d\vec{r}\), they are now asked to construct the area (\(d\vec{A}\)) and volume (\(dV\)) elements in each coordinate system. This prepares students to integrate vector- and scalar-valued functions in curvilinear coordinates.

Begin with a brief lecture which “derives” the formula \[d\vec{A}=d\vec{r}_1\times d\vec{r}_2\] by drawing a differential area element on an arbitrary surface and appealing to the geometric definition of the cross product as a directed area. Label the sides of the surface element with vectors \(d\vec{r}_1\) and \(d\vec{r}_2\) with both vectors' tails at the same point.

Similarly, derive the formula \[d \tau=(d\vec{r}_1\times d\vec{r}_2)\cdot d\vec{r}_3\] from a picture of an arbitrary differential volume element and the geometric definition of the scalar triple product as the volume of a parallelopiped. Label the sides of the volume element with vectors \(d\vec{r}_1\), \(d\vec{r}_2\), and \(d\vec{r}_3\) with all the vectors' tails at the same point. Make sure to choose a right-handed orientation.

Next, ask the students to use these formulas to find the surface and volume elements for a plane, for a finite cylinder (including the top and bottom), and for a sphere.

Student Conversations

  • Students who do not have much experience with the “Use what you know” strategy have trouble getting from the generic expression for \(d\vec{r}\) in cylindrical and spherical coordinates to specific ones for the vectors that they care about. Encourage the students to draw pictures.
  • A remarkable number of students have trouble finding the cross product. Emphasize that the curvilinear basis vectors \((\hat{s}, \hat{\phi}, \hat{z})\) or \((\hat{r}, \hat{\theta}, \hat{\phi})\) are orthonormal, just like their rectangular conterparts: \((\hat{x}, \hat{y}, \hat{z})\). You can do dot and cross products in curvilinear coordinates as long as the two vectors have their tails at the same point.
  • A few students will want to jump to formulas that they know without actually doing the computation and/or they will want to simply multiply the lengths of the two \(d\vec{r}\)'s together. There is nothing wrong with this. You can save some time by going directly to these strategies. The only downside is that the students may not get any practice with the computational strategies that work in the (rarely needed) generic cases.
Find the formulas for the differential surface and volume elements for a plane, for a finite cylinder (including the top and bottom), and for a sphere. Make sure to draw an appropriate figure.
  • group Scalar Surface and Volume Elements

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    Scalar Surface and Volume Elements
    Static Fields 2022 (6 years)

    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.

  • assignment Find Area/Volume from $d\vec{r}$

    assignment Homework

    Find Area/Volume from \(d\vec{r}\)
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    Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.

    1. Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
    2. Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
    3. Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

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  • groups Pineapples and Pumpkins

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    Pineapples and Pumpkins
    Static Fields 2022 (5 years)

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    There are two versions of this activity:

    As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

    As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distribued to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

  • assignment Gravitational Field and Mass

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    Gravitational Field and Mass
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    The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: \begin{equation} \vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases} \end{equation}

    This problem explores the consequences of the divergence theorem for this shell.

    1. Using the given description of the gravitational field, find the divergence of the gravitational field everywhere in space. You will need to divide this question up into three parts: \(r<a\), \(a<r<b\), and \(r>b\).
    2. Briefly discuss the physical meaning of the divergence in this particular example.
    3. For this gravitational field, verify the divergence theorem on a sphere, concentric with the shell, with radius \(Q\), where \(a<Q<b\). ("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
    4. Briefly discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

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  • group Vector Differential--Curvilinear

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    Vector Differential--Curvilinear
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    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.

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  • assignment Paramagnet (multiple solutions)

    assignment Homework

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    Energy and Entropy 2021 (2 years) We have the following equations of state for the total magnetization \(M\), and the entropy \(S\) of a paramagnetic system: \begin{align} M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\} \end{align}
    1. List variables in their proper positions in the middle columns of the charts below.

    2. Solve for the magnetic susceptibility, which is defined as: \[\chi_B=\left(\frac{\partial M}{\partial B}\right)_T \]

    3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

      \[\left(\frac{\partial M}{\partial B}\right)_S \]

    4. Evaluate your chain rule. Sense-making: Why does this come out to zero?

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    • Students may find it helpful to use a chain rule diagram.

Learning Outcomes