Differentials of Two Variables

    • 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 Flux through a Plane

      assignment Homework

      Flux through a Plane
      Static Fields 2023 (4 years) Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).
    • assignment Differentials of One Variable

      assignment Homework

      Differentials of One Variable
      Static Fields 2023 (6 years) 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 Fourier Transform of a Shifted Function

      group Small Group Activity

      5 min.

      Fourier Transform of a Shifted Function
      Periodic Systems 2022

      Fourier Transforms and Wave Packets

    • assignment Rubber Sheet

      assignment Homework

      Rubber Sheet
      Energy and Entropy 2021 (2 years)

      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 The puddle

      assignment Homework

      The puddle
      differentials Static Fields 2023 (5 years) 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 Find Area/Volume from $d\vec{r}$

      assignment Homework

      Find Area/Volume from \(d\vec{r}\)
      Static Fields 2023 (5 years)

      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}

    • format_list_numbered Integration Sequence

      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 Cube Charge

      assignment Homework

      Cube Charge
      charge density

      Integration Sequence

      Static Fields 2023 (6 years)
      1. Charge is distributed throughout the volume of a dielectric cube with charge density \(\rho=\beta z^2\), where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
      2. On a different cube: Charge is distributed on the surface of a cube with charge density \(\sigma=\alpha z\) where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge on the cube? Don't forget about the top and bottom of the cube.
    • group Scalar Surface and Volume Elements

      group Small Group Activity

      30 min.

      Scalar Surface and Volume Elements
      Static Fields 2023 (7 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.

  • Static Fields 2023 (8 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. \(y=u\cos(3v^2-2)\)