Students predict from graphs of simple 2-d vector fields whether the curl is positive, negative, or zero in various regions of the domain using the definition of the curl of a vector field at a point as the maximum circulation per unit area through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
For each of the vector fields below, decide whether the \(z\)-component of the curl is postive, negative, or zero in each quadrant. Be prepared to defend your answers.
We precede this activity with a derivation of the rectangular coordinate expression for curl: (the magnitude of a particular component of) the curl is the circulation per unit area around an appropriately chosen planar loop. Our derivation follows the one in Div, grad, curl and all that, Schey, 2nd edition, Norton, 1973, p. 74.
Then the students are presented with a number of examples of 2-d vector fields in dry eraseable plastic sleeves. Most vector fields are shown as a cross-section of the field and it is assumed the the vector field is independent of the third (unshown) dimension. Students are asked to use the definition of curl to predict the sign and relative magnitude of the curl in each quadrant. Optionally, a prepared Mathematica worksheet can be used to calculate the curl, so students can check their predictions.
A quick whole class discussion of the items listed in Student Conversations.