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==== Vector Line Integrals ====

=== Essentials ===

== Main ideas ==

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

==== Students' Task ====
//Estimated Time: 30--45 minutes//

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

==== Prerequisite Knowledge ====

  * Some familiarity with differentials;
  * $d\rr$ in rectangular (and ideally also polar) coordinates;
  * Brief exposure to/discussion of vector line integrals ([[courses:activities:vcact:vcvalley|Valley]] activity is sufficient).

==== Props/Equipment ====

  * [[:Props:start#contour maps & vector maps|Dry-Erasable sleeves]] and markers;
  * A handout for each student;
  * A selection of vector field maps (available {{vfields.pdf|here}}).

==== Activity: Introduction ====

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.

==== Activity: Student Conversations ====

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.

==== Activity: Wrap-up ====

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.

==== Extensions ====

This activity pairs well with the [[courses:activities:vcact:vcvalley|Valley]]
and [[courses:activities:vcact:vcwire|Wire]] activities.