Static Fields: Fall-2025
HW 01 Practice: Due W1 D3: Math Bits

  1. Cube Charge
    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.
  2. Curvilinear coordinate drawings
    1. Using the conventions from class for cylindrical and spherical coordinates, draw each of the surfaces below by hand:
      1. \(\phi=\frac\pi3\)
      2. \(\theta=\frac{3\pi}4\)
    2. Using the conventions from class for cylindrical and spherical coordinates, draw each of the following vectors by hand at the indicated point.
      1. \(\boldsymbol{\hat\phi}\) at the point where \(s=2\), \(\phi=\frac\pi3\), and \(z=0\).
      2. \(\boldsymbol{\hat\theta}\) at the point where \(r=1\), \(\theta=\frac{3\pi}4\), and \(\phi=\frac\pi2\).
  3. Exponential and Logarithm Identities Make sure that you have memorized the following identities and can use them in simple algebra problems: \begin{align*} e^{u+v}&=e^u \, e^v\\ \ln{uv}&=\ln{u}+\ln{v}\\ u^v&=e^{v\ln{u}} \end{align*}
  4. Derivative Rules

    Make sure that you have memorized or can quickly find the derivative of all of the common transcendental functions: powers, trigonometric functions (especially sine and cosine), exponential, logarithms. Make sure that you can use these rules, even when the argument has parameters in it, e.g. \(\sin{kx}\). Also, make sure you can use the chain rule.

  5. Basic Calculus: Practice Exercises Determine the following derivatives and evaluate the following integrals, all by hand. You should also learn how to check these answers on Wolfram Alpha.
    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. \(\displaystyle\int v\cos v\,dv\)