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 definition of the divergence of a vector field at a point: The divergence of a vector field at a point is flux per unit volume through an infinitesimal box surrounding that point. Optionally, students can use a Mathematica notebook to verify their predictions.

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral
\(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative.
They do a similar activity for the vector field \(\vec{G}\) which is not conservative.

Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.

Use the cross product to find the components of the unit vector
\(\mathbf{\boldsymbol{\hat n}}\) perpendicular to the plane shown in the figure below, i.e.
the plane joining the points \(\{(1,0,0),(0,1,0),(0,0,1)\}\).

Consider the vector field \(\vec F=(x+2)\hat{x} +(z+2)\hat{z}\).

Calculate the divergence of \(\vec F\).

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?

Verify the divergence theorem for this vector field where the volume
involved is drawn below.

Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).