## Geometry of Vector Fields

After students have spent time working with both the algebra and geometry of scalar fields, they move on to vector fields in the context of electric fields. Vector fields can be considered more complicated than scalar fields because at each point there is a vector with both a magnitude and direction while scalar fields have one value at each point. Not only does this increase the difficulty in representing the field, but there are also additional properties of vector fields–divergence and curl. Building off the previous activities on the electrostatic potential, a scalar field, students begin to wrestle with the more complicated geometry of vector fields.

These activities can be used independently of the others in order to address a particular property of vector fields, or they can be used as a sequence to address important aspects of vector fields in physics. Curvilinear Basis Vectors, Visualizing Gradient, Visualizing Divergence, and Visualizing Curl describe vector fields in general and do not specifically address vector fields in a physical context. Visualizing Divergence and Visualizing Curl can be paired together to address the geometric meanings of divergence and curl using similar methods with Mathematica.

### Activities

• Curvilinear Basis Vectors (Estimated time: 15 minutes): This kinesthetic activity students are asked to point in $\hat{r}$, $\hat{\theta}$, $\hat{\phi}$, and $\hat{z}$ directions in reference to an origin within the classroom. A class discussion ensues about the directions of curvilinear basis vectors and how the direction changes at different points in space. This is in contrast to rectangular unit vectors, $\hat{x}$, $\hat{y}$, and $\hat{z}$, which have fixed directions at each point in space. Many mathematics courses do not cover curvilinear basis vectors, so it is expected that students will not be familiar with these basis vectors. This activity serves as an introduction to the notations which physicists use to represent vector fields in various coordinate systems.
• Visualizing Gradient (Estimated time: 10-15 minutes): This activity utilizes Mathematica to show the gradient of several scalar fields. Students can see that the gradient is always perpendicular to level curves, is of the same dimension as the number of variables, and points in the direction of greatest change locally. The gradient relates the scalar electrostatic potential field to the electric field by the expression $\vec{E} = - \vec{\nabla}V$, however, many middle-division students do not understand the geometry of this relationship. This computer visualization provides scalar fields without explicit physical meaning so students can begin to understand the geometry which relates the electrostatic potential and electric field.
• Drawing Electric Field Vectors (Estimated time: 20-30 minutes): In direct analogy to Drawing Equipotential Surfaces, this small group activity has students sketch the electric field due to a quadrupole. This activity requires one of two approaches: use of vector addition of electric field from each point charge or the use of the gradient on the known potential field. Students can compare this activity with Drawing Equipotential Surfaces to observe how geometry leads to a physical relationship between the electric field and electrostatic potential.
• Visualizing Electric Flux (Estimated time: 20 minutes): This computer visualization activity uses Mathematica to explore the effects of placing a point charge inside, outside, and on a cubical Gaussian surface. Electric field vectors are a useful representation when students are learning about flux which is essential for using Gauss's law. Students have familiarity with Gauss's law from introductory courses and may have preconceived notion of flux integrals. Often students confuse the flux of an electric field with the “flow” of an electric field. This activity allows students to visualize the electric flux of a point charge through a Gaussian surface in different locations with respect to the point charge.
• Visualizing Divergence (Estimated time: 20 minutes): This computer visualization activity allows students to predict the sign of divergence at various points in many vector fields generated by a Mathematica notebook. Many students are familiar with the algebraic expression of the divergence of a vector field, however, many have difficulty in the geometric interpretation of divergence–the flux per unit volume at a point in a vector field. Using this computer visualization allows students to expand their preexisting algebraic knowledge of divergence to include a geometric interpretation.
• Visualizing Curl (Estimated time: 20 minutes): Similar to Visualizing Divergence, this activity uses a Mathematica notebook of various vector fields to assist students in the geometric interpretation of the curl of a vector field–the circulation per unit area at a local point in the vector field. Students predict the value of the curl at various locations within many vector fields. Add quantum activities, add a verbal description of this sequence including problems visualizing scalar field in 3 dimensions–use of color.

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