Basic Calculus: Practice Exercises

    • assignment Mass of a Slab

      assignment Homework

      Mass of a Slab
      Static Fields 2022 (5 years)

      Determine the total mass of each of the slabs below.

      1. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho=A\pi\sin(\pi z/h). \end{equation}
      2. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big) \end{equation}
      3. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose surface density is given by \(\sigma=2Ah\).
      4. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose mass density is given by \(\rho=2Ah\,\delta(z)\).
      5. What are the dimensions of \(A\)?
      6. Write several sentences comparing your answers to the different cases above.

    • assignment Differentials of Two Variables

      assignment Homework

      Differentials of Two Variables
      Static Fields 2022 (6 years) 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 Pressure of thermal radiation

      assignment Homework

      Pressure of thermal radiation
      Thermal radiation Pressure Thermal and Statistical Physics 2020

      (modified from K&K 4.6) We discussed in class that \begin{align} p &= -\left(\frac{\partial F}{\partial V}\right)_T \end{align} Use this relationship to show that

      1. \begin{align} p &= -\sum_j \langle n_j\rangle\hbar \left(\frac{d\omega_j}{dV}\right), \end{align} where \(\langle n_j\rangle\) is the number of photons in the mode \(j\);

      2. Solve for the relationship between pressure and internal energy.

    • assignment Icecream Mass

      assignment Homework

      Icecream Mass
      Static Fields 2022 (5 years)

      Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).

    • assignment Coffees and Bagels and Net Worth

      assignment Homework

      Coffees and Bagels and Net Worth
      Energy and Entropy 2021 (2 years)

      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 Free Expansion

      assignment Homework

      Free Expansion
      Energy and Entropy 2021 (2 years)

      The internal energy is of any ideal gas can be written as \begin{align} U &= U(T,N) \end{align} meaning that the internal energy depends only on the number of particles and the temperature, but not the volume.*

      The ideal gas law \begin{align} pV &= Nk_BT \end{align} defines the relationship between \(p\), \(V\) and \(T\). You may take the number of molecules \(N\) to be constant. Consider the free adiabatic expansion of an ideal gas to twice its volume. “Free expansion” means that no work is done, but also that the process is also neither quasistatic nor reversible.
      1. What is the change in entropy of the gas? How do you know this?

      2. What is the change in temperature of the gas?

    • assignment Energy, Entropy, and Probabilities

      assignment Homework

      Energy, Entropy, and Probabilities
      Energy Entropy Probabilities Thermodynamic identity

      The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier \(\beta\), we can prove that \(\beta=\frac1{kT}\) based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.

      The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align}: We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier \(\beta\) as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate \(U\) and \(S\) to \(\beta\). Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that \(\beta = \frac1{kT}\).

    • assignment Energy, Entropy, and Probabilities

      assignment Homework

      Energy, Entropy, and Probabilities
      Thermal and Statistical Physics 2020

      The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier \(\beta\), we can prove that \(\beta=\frac1{kT}\) based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.

      The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align} We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier \(\beta\) as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate \(U\) and \(S\) to \(\beta\). Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that \(\beta = \frac1{kT}\).

    • assignment Bottle in a Bottle 2

      assignment Homework

      Bottle in a Bottle 2
      heat entropy ideal gas Energy and Entropy 2021 (2 years)

      Consider the bottle in a bottle problem in a previous problem set, summarized here.

      A small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated.

      The volume of the small bottle is 0.001 m23 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).

      1. How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

      2. Compute the integral \(\int \frac{{\mathit{\unicode{273}}} Q}{T}\) and the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).

      3. Discuss your results.

    • assignment Cone Surface

      assignment Homework

      Cone Surface
      Static Fields 2022 (5 years)

      • 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.

  • Static Fields 2022 (3 years) Determine the following derivatives and evaluate the following integrals.
    1. \(\frac{d}{du}\left(u^2\sin u\right)\)
    2. \(\frac{d}{dz}\left(\ln(z^2+1)\right)\)
    3. \(\displaystyle\int v\cos(v^2)\,dv\)
    4. \(\displaystyle\int v\cos v\,dv\)