Table of Contents

Curl

Prerequisites

In-class Content

  • Curl as circulation (lec - 10 min) We follow “div, grad, curl and all that”, by Schey
  • Visualizing Curl (SGA - 30 min)
  • Curl in curvilinear coordinates (lec - 10 min)

Homework for Static Fields

  1. (CurlPracticeMMM) Calculate the curl for several made-up functions using rectangular and curvilinear coordinates. Part (d) can be included if you know how to find a potential from a vector field.

    Calculate the curl of each of the following vector fields. You may look up the formulas for curl in curvilinear coordinates.

    1. $\FF=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z}$

    2. $\GG = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z}$

    3. $\HH = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z}$

    4. $\II = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z}$

    5. $\JJ = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z}$

    6. Compare the curl to the divergence for each field (see Homework 2 Practice).

    7. For each vector field in the preceding problems which have zero curl, find the corresponding potential function.

  2. (CurlPractice2)

    Choose some simple vector fields of your own and find the curl of them both by hand and using Mathematica or Maple. Choose some that are written in terms of rectangular coordinates and others in cylindrical and/or spherical.

  3. (CurlVisualizePractice)

    If you need more practice visualizing curl, go through the Mathematica Notebook on the course website.


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