Choose a vector field \(\boldsymbol{\vec{F}}\) from the first column below. Choose a small loop \(C\) (that is, a simple, closed, positively-oriented curve) which does not go around the origin.
Is \(\oint\boldsymbol{\vec{F}}\cdot d\boldsymbol{\hat{r}}\) positive, negative, or zero?
Will a paddlewheel spin if placed inside your loop, and, if so, which way?
Do you think \(\nabla\times\boldsymbol{\vec{F}}\) is zero or nonzero inside your loop?
Explain.
Compute \(\nabla\times\boldsymbol{\vec{F}}\). Did you guess right?
Explain.
Is \(\oint\boldsymbol{\vec{F}}\cdot\boldsymbol{\hat{n}}\,ds\) positive, negative, or zero?
(\(\boldsymbol{\hat{n}}\) is the outward pointing normal vector to \(C\).)
Is the net flow outwards across your loop positive, negative, or zero?
Do you think \(\nabla\cdot\boldsymbol{\vec{F}}\) is zero or nonzero inside your loop?
Explain.
Compute \(\nabla\cdot\boldsymbol{\vec{F}}\). Did you guess right?
Explain.
Repeat the above steps for vector fields \(\boldsymbol{\vec{G}}\) and \(\boldsymbol{\vec{H}}\) chosen from the
second and third columns.
Geometry of divergence and curl, either through a geometric definition or
through Stokes' Theorem and the Divergence Theorem.
Warmup
Students may need to be reminded what circulation is.
Students may not have seen flux in 2 dimensions.
Students may only have seen \(\boldsymbol{\hat{n}}\) for surfaces, not curves. Some students
will set \(\boldsymbol{\hat{n}}=\boldsymbol{\hat{z}}\)! Emphasize that \(\boldsymbol{\hat{n}}\) is horizontal (and that \(ds\ne\boldsymbol{d\vec{S}}\)).
Props
whiteboards and pens
formula sheet for div and curl in spherical and cylindrical coordinates
(Each group may need its own copy.)
divergence and curl transparency
blank transparencies and pens
Wrapup
Discuss the effect of choosing loops of different shapes, especially those
adapted to the given vector field.
Talk about the geometry of sinks and sources (for divergence) and paddlewheels
(for curl).
Details
In the Classroom
While students are working on this activity, draw the vector fields on the
board to use during the wrapup. Alternatively, bring an overhead transparency
showing the vector fields (and blank transparencies for students to write on).
Students like this lab; it should flow smoothly and quickly.
Students may need to be reminded what \(\oint\) means, and that the positive
orientation in the plane is counterclockwise.
Yes, two pairs of questions are really the same.
Make sure the paths do not go around the origin.
Encourage each group to work on at least two vector fields, which are in
different rows and columns. Include one vector field from the third column if
time permits.
Encourage each group to consider, for a single vector field, moving their loop
to another location. This is especially effective (and in fact essential) for
the two vector fields in the third column.
See the discussion of using transparencies for Group Activity The Hill.
Students may eventually realize that the vector fields in the middle column
are linear combinations of the vector fields in the first column, which are in
turn “pure curl” and “pure divergence”, respectively.
Subsidiary ideas
Divergence and curl are not just about the behavior near the origin.
Derivatives are about change --- the difference between
nearby vectors.
Homework
(MHG refers to McCallum, Hughes Hallett, Gleason, et al.
MHG 19.1:20
MHG 20.2:16
MHG 20.3:10,12,20
MHG 20.4:22
Essay questions
Which operation, curl or divergence is easier to understand?
Which is more useful?
Do you prefer to gauge curl from a plot or from a calculation? What about divergence?
Enrichment
Emphasize the importance of divergence and curl in applications.
Ask students how to determine which vector fields are conservative!
(A single closed path with nonzero circulation suffices to show that a vector
field is not conservative. The best geometric way we know to show
that a vector field is conservative is to try to draw the level
curves for which the given vector field would be the gradient.)
Discuss the fact that \(\boldsymbol{\hat{r}}\over r\) and \(\boldsymbol{\hat{\phi}}\over r\) are both
curl-free and divergence-free; this is counterintuitive, but crucial for
electromagnetism. (These are, respectively, the electric/magnetic field of a
charged/current-carrying wire along the \(z\)-axis.)
Discuss the behavior of \(\boldsymbol{\hat{r}}\over r^n\) and \(\boldsymbol{\hat{\phi}}\over r^n\), emphasizing
that both the divergence and curl vanish when \(n=1\).
Relate these examples to the magnetic field of a wire (\(\boldsymbol{\vec{B}}={\boldsymbol{\hat{\phi}}\over r}\))
and the electric field of a point charge (\(\boldsymbol{\vec{E}}={\boldsymbol{\hat{r}}\over r^2}\); this is the
spherical \(r\)).
Show students how to compute divergence and curl of these vector fields in
cylindrical coordinates.
Trying to estimate divergence and curl from a single plot of a vector field
confronts students with the need to zoom in. Technology can be useful here.
Point students to our paper on Electromagnetic Conic Sections, which
appeared in Am. J. Phys. 70, 1129--1135 (2002), and which is also
available on the Bridge Project website.
Most physical applications of the divergence are 3-dimensional, rather than
2-dimensional. Each vector field in this activity could be regarded as a
horizontal 3-dimensional vector field by assuming that there is no
\(z\)-dependence, in which case the flux can be computed through a
3-dimensional box whose cross-section is the loop, and whose horizontal
top and bottom do not contribute.
Consider the vector field \(\vec F=(x+2)\hat{x} +(z+2)\hat{z}\).
(2pts)
Calculate the divergence of \(\vec F\).
(2pts)
In which direction does the vector field \(\vec F\) point on the plane
\(z=x\)? What is the value of \(\vec F\cdot \hat n\) on this plane
where \(\hat n\) is the unit normal to the plane?
(4pts)
Verify the divergence theorem for this vector field where the volume
involved is drawn below. (“Verify” means calculate both sides of the divergence theorem, separately, for this example and show that they are the same.)
Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)