group Small Group Activity

30 min.

Vector Surface and Volume Elements

Students use known algebraic expressions for vector line elements \(d\boldsymbol{\vec{r}}\) to determine all simple vector area \(d\boldsymbol{\vec{A}}\) 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.

  • Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop course(s) Found in: Integration Sequence sequence(s)

group Small Group Activity

30 min.

Scalar Surface and Volume Elements

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.

  • Found in: Static Fields, AIMS Maxwell course(s) Found in: Integration Sequence sequence(s)

None

Paramagnet (multiple solutions)
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?

  • Found in: Energy and Entropy course(s)

group Small Group Activity

30 min.

Paramagnet (multiple solutions)
  • Students evaluate two given partial derivatives from a system of equations.
  • Students learn/review generalized Leibniz notation.
  • Students may find it helpful to use a chain rule diagram.

None

Differentials of Two Variables
Find the total differential of the following functions:
  1. \(y=3u^2 + 4\cos 3v\)
  2. \(y=3uv\)
  3. \(y=3u^2\cos wv\)
  4. \(y=u\cos(3v^2-2)\)
  • Found in: Static Fields, AIMS Maxwell course(s)

None

Differentials of One Variable
Find the total differential of the following functions:
  1. \(y=3x^2 + 4\cos 2x\)
  2. \(y=3x^2\cos kx\) (where \(k\) is a constant)
  3. \(y=\frac{\cos 7x}{x^2}\)
  4. \(y=\cos(3x^2-2)\)
  • Found in: Static Fields, AIMS Maxwell course(s)

group Small Group Activity

30 min.

Vector Differential--Curvilinear

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 Small Group Activity

5 min.

Acting Out Flux
Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.

assignment_ind Small White Board Question

10 min.

Vector Differential--Rectangular

In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

group Small Group Activity

30 min.

Flux through a Cone
Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .
  • Found in: Static Fields, AIMS Maxwell course(s) Found in: Integration Sequence sequence(s)

group Small Group Activity

30 min.

Finding \(d\boldsymbol{\vec{r}}\)
  • Found in: Vector Calculus II course(s)

None

Cone Surface

  • Find \(dA\) on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
  • Using integration, find the surface area of an (open) cone with height \(H\) and radius \(R\). Do this problem in both cylindrical and spherical coordinates.

  • Found in: Static Fields, AIMS Maxwell course(s)

group Small Group Activity

30 min.

Total Charge
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.

group Small Group Activity

30 min.

The Cone
Students set up and compute a scalar surface integral.
  • Found in: Vector Calculus II, Surfaces/Bridge Workshop course(s)

None

Electric Field and Charge
Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}
  1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
  2. Find a formula for the charge density that creates this electric field.
  3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
  • Found in: AIMS Maxwell, Static Fields course(s)