Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the geometric definition of the divergence of a vector field at a point as flux per unit volume (here: 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 divergence 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 divergence: the divergence is the flux per unit volume through an appropriately chosen closed surface. Our derivation follows the one in div grad curl and all that, Schey, 2nd edition, Norton, 1973, p. 36 or see our versions. One can also use clicker questions or SWBQs about divergence to help get them started (see Visualization of Divergence).
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 divergence as the flux per unit volume through an infinitesimal box to predict the sign and relative magnitude of the divergence in each quadrant. Optionally, a prepared Mathematica worksheet can be used to calculate the divergence, so students can check their predictions.
A quick whole class discussion of the items listed in Student Conversations.