Reading: Griffiths, Chapter 1

You may work together and get help from other students. Your solutions must be written in your own words, without looking at someone else’s solutions while you write them.

Don’t forget the 9 points that we are looking for in your solutions (see Moodle).

In order to make sure you get your context and meaning/make sense points, next to your answers, please put a “c” with a circle around it for context, and an “m” with a circle around it for meaning.

  1. Griffiths, Chapter 1, Problem 1.12.
    Main point: This problem helped me think less abstractly about the gradient by applying it to topography. Looking for the top of the hill amounts to finding the location of zero slope. That actually completely makes sense to me when thinking about topography.

  2. Griffiths 1.20 (Can you find an interesting vector function of which the curl and the divergence are both zero? It’ll be fun to see how varied your group’s examples end up being.)

  3. Griffiths, Chapter 1, Problem 1.33
    Main point: This problem helped me think about the mechanics of the divergence theorem, how to think about dotting the vector into the area, and integrating it over the volume, etc. It also reminded me that it’s really true for any vector, i.e. it’s pure math, not physics.

  4. Suppose a fluid rotates about the axis such that velocity of flow at any point is given by (in cylindrical coordinates)

    \[v=v(s)\hat{\phi}\]

    where the speed \(v(s)\) is a function of \(s\) only. A model that approximates vortices in viscous fluids is the Rankine vortex for which

    \[\nabla \times \vec{v} = \begin{cases} \omega \hat{z}, & \text{if } s < R\\ 0, & \text{if } s \geq R \\ \end{cases}\]

    where \(\omega\) is constant. (a) What is the radial velocity profile, \(v(s)\), for a Rankine vortex? Note: Assume that \(v(s)\) is continuous at \(s = R\). (b) Evaluate the “circulation”,\(\oint \vec{v}\cdot \vec{dl}\) , for this flow as a function of \(s\).
    Main point: This one reminded me how to deal with the curl in cylindrical coordinates (it’s in the back of your book by the way.) I also got practice apply Stokes’ theorem.

  5. Griffiths 1.44 (Practicing integrating over delta functions.)

  6. Griffiths 1.50 (More practice on divergence and curl, but also confronting scalar and vector potentials)