Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrates several different ways of plotting the potential.

Students use known algebraic expressions for length elements \(d\ell\) to
determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

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.

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 are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.

Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical
coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.