Section 2.6 Circulation and Flow
When a vector field is a velocity field, a natural phenomenon we can measure is the Flow. This accumulates the tendency of the vector field to be tangential to the curve. If we imagine our field is a velocity field, the flow is measuring the degree to which a curve just naturally flows along with that velocity field. As a line integral, this is identical to our formula for work, it is just interpreted differently physically. When the path is a closed curve - i.e. starts and ends at the same point - then the flow integral is called the circulation of the vector field along a path.
Learning Objectives:
State the definition for the Flow or Circulation
Geometrically interpret Flow or Circulation.
Compute Flow or Circulation for a specific parametrization
Post-Video Activities Post-Video Activities
1.
One sort of annoying thing about this section is there are a LOT of different looking formulas to compute the exact same thing. Five, to be precise as listed
in Thomas' Calculus 1 . It's partly because we have a definition in terms of ds, but we compute it with a parametrization, and then sometimes easier to remember using the the line integrals with respect to x or y or z. Anyways, just take note of the differences and why you might prefer one or the other.
2.
Imagine you compute the circulation along some curve from point A to point B. Now imagine it is the same curve, but oriented exactly backwards, starting at point B and going to point A. How does this affect the circulation? Can you both guess the answer AND prove it using the formulas?
media.pearsoncmg.com/cmg/pmmg_mml_shared/mathstats_html_ebooks/ThomasCalcET14e/page_982.html
