• Reinforces both the Master Formula and differentials.
• Sets the stage for path-independence.

Estimated Time: 30–45 minutes

Students work in groups to evaluate vector integrals along multiple paths in a variety of vector fields.

• Some familiarity with differentials;
• $d\rr$ in rectangular (and ideally also polar) coordinates;
• Brief exposure to/discussion of vector line integrals (Valley activity is sufficient).

• Dry-Erasable sleeves and markers;
• A handout for each student;
• A selection of vector field maps (available here).

This activity encourages students to interpret vector line integrals geometrically, and to think about whether the choice of path matters.

Initially assign each group one or two the vector fields; groups that finish quickly can try again with a different vector field.

Depending on time constraints, either assign different vector fields to different groups, or give a conservative and a nonconservative vector field to each group.

Students may initially assume their path should follow the vector field. This would be a good time to remind students of the differences between the “where” and the “what” of integration.

Students may then choose a complicated path, making it difficult to evaluate the integral. Remind the students that not all integrals can be evaluated!

Most students will still be quite unfamiliar with vector line integrals, and may need guidance. What is $d\rr$? What is changing? What are the limits? Use what you know!

Students may have difficulty choosing appropriate paths for the polar examples; a radial line may be a good choice.

WARNING: The last question about optimization is only relevant for nonconservative vector fields. Students can be allowed to discover this for themselves, or this question can be moved to the wrapup.

Call someone from each group to the board to draw both their path(s) and $d\rr$ and ask how they found $d\rr$. Discuss the different methods used by different groups.

Ask each group whether their answer depended on the path, or only on the endpoints. Start with conservative vector fields, then nonconservative (without using those names). Conclude that the path matters for some vector fields, but not for others.

This activity pairs well with the Valley and Wire activities.