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_ind Small White Board Question

10 min.

Vector Differential--Rectangular
Vector Calculus II Fall 2021 AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Vector Calculus II Summer 21 Static Fields Winter 2021

vector differential rectangular coordinates math

Integration Sequence

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..

assignment Homework

Differentials of Two Variables
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022 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. \(y=u\cos(3v^2-2)\)

assignment Homework

Zapping With d 1
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

Find the differential of each of the following expressions; zap each of the following with \(d\):

  1. \[f=3x-5z^2+2xy\]

  2. \[g=\frac{c^{1/2}b}{a^2}\]

  3. \[h=\sin^2(\omega t)\]

  4. \[j=a^x\]

  5. \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]

assignment Homework

Differential Form of Gauss's Law
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.

assignment Homework

Coffees and Bagels and Net Worth
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

In economics, the term utility is roughly related to overall happiness. Many things affect your happiness, including the amount of money you have and the amount of coffee you drink. We cannot directly measure your happiness, but we can measure how much money you are willing to give up in order to obtain coffee or bagels. If we assume you choose wisely, we can thus determine that your happiness increases when you decrease your amount of money by that amount in exchange for increasing your coffee consumption. Thus money is a (poor) measure of happiness or utility.

Money is also a nice quantity because it is conserved---just like energy! You may gain or lose money, but you always do so by a transaction. (There are some exceptions to the conservation of money, but they involve either the Fed, counterfeiters, or destruction of cash money, and we will ignore those issues.)

In this problem, we will assume that you have bought all the coffee and bagels you want (and no more), so that your happiness has been maximized. Thus you are in equilibrium with the coffee shop. We will assume further that you remain in equilibrium with the coffee shop at all times, and that you can sell coffee and bagels back to the coffee shop at cost.*

Thus your savings \(S\) can be considered to be a function of your bagels \(B\) and coffee \(C\). In this problem we will also discuss the prices \(P_B\) and \(P_C\), which you may not assume are independent of \(B\) and \(C\). It may help to imagine that you could possibly buy out the local supply of coffee, and have to import it at higher costs.

  1. The prices of bagels and coffee \(P_B\) and \(P_C\) have derivative relationships between your savings and the quantity of coffee and bagels that you have. What are the units of these prices? What is the mathematical definition of \(P_C\) and \(P_B\)?

  2. Write down the total differential of your savings, in terms of \(B\), \(C\), \(P_B\) and \(P_C\).

  3. Solve for the total differential of your net worth. Your net worth \(W\) is the sum of your total savings plus the value of the coffee and bagels that you own. From the total differential, relate your amount of coffee and bagels to partial derivatives of your net worth.

assignment Homework

Differentials of One Variable
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022 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. \(y=\cos(3x^2-2)\)

group Small Group Activity

5 min.

Leibniz Notation
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 This short small group activity introduces students to the Leibniz notation used for partial derivatives in thermodynamics; unlike standard Leibniz notation, this notation explicitly specifies constant variables. Students are guided in linking the variables from a contextless Leibniz-notation partial derivative to their proper variable categories.

group Small Group Activity

30 min.

Vector Surface and Volume Elements
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021

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.

assignment_ind Small White Board Question

10 min.

Partial Derivatives from a Contour Map
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?

assignment Homework

The puddle
differentials AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 The depth of a puddle in millimeters is given by \[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\] Your path through the puddle is given by \[x=3t \qquad y=4t\] and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.
  1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
  2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
  3. FOOD FOR THOUGHT (optional)
    There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.

assignment Homework

Rubber Sheet
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

Consider a hanging rectangular rubber sheet. We will consider there to be two ways to get energy into or out of this sheet: you can either stretch it vertically or horizontally. The distance of vertical stretch we will call \(y\), and the distance of horizontal stretch we will call \(x\).

If I pull the bottom down by a small distance \(\Delta y\), with no horizontal force, what is the resulting change in width \(\Delta x\)? Express your answer in terms of partial derivatives of the potential energy \(U(x,y)\).

assignment Homework

Cone Surface
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

Using integration, find the surface area of a cone with height \(H\) and radius \(R\). Do this problem in both cylindrical and spherical coordinates.

assignment Homework

Paramagnet (multiple solutions)
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 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?

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.

assignment Homework

Find Force Law
Central Forces Spring 2021

Find the force law for a central-force field that allows a particle to move in a spiral orbit given by \(r=k\phi^2\), where \(k\) is a constant.

assignment Homework

Adiabatic Compression
ideal gas internal energy engine Energy and Entropy Fall 2020

A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.

In this problem, you may treat air as an ideal gas, which satisfies the equation \(pV = Nk_BT\). You may also use the property of an ideal gas that the internal energy depends only on the temperature \(T\), i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by \(C_V=\frac52Nk_B\), which means the internal energy is given by \(U=\frac52Nk_BT\).

Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is not considered a solution at this level.

  1. If the air is initially at room temperature (taken as \(20^{o}C\)) and is then compressed adiabatically to \(\frac1{15}\) of the original volume, what final temperature is attained (before fuel injection)?

  2. By what factor does the pressure increase?

group Small Group Activity

30 min.

Scalar Surface and Volume Elements
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022 Static Fields Winter 2022

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.

Total Charge
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

charge charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates

Integration Sequence

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.

Flux through a Cone
AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

Integration Sequence

Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .