Figure 22.1 A runner works along an oriented course through a wind. Over the whole race,
did the wind help or hinder? The course behind them is tinted by which it was
doing at each moment, and beneath, the same story is told against distance
travelled. The two arrows at the runner are all there is to work with: the wind
⃗𝐹 in ink, and the direction of travel ⃗𝑇 in teal.Figure 22.2 Only the part of the wind pointing along the course counts. Drag the point: the
wind never changes, but as the course turns beneath it the along-part swings
from helping to hindering and back. The faint arrow across the course is the
part thrown away — a pure crosswind neither helps nor hinders, however strong.
22.2Vector Line Integrals
Figure 22.3 The runner goes out, comes back, and goes out again. Reversing the course
reverses every ⃗𝑇, so it reverses every ⃗𝐹⋅⃗𝑇, so the whole
profile turns over about its axis and the total comes back the same size with
the opposite sign.Figure 22.4Every piece has to be parameterized the same way round as the walk, and if
it isn't, the pieces are not the pieces of a path at all.Figure 22.5 Two very different laps, both totalling nothing. In ⟨𝑥,𝑦⟩ the
wind is square to the course at every point and never contributes anything at
all — the profile lies flat on the axis. In a steady wind the runner is pushed
hard for half the lap and pushed back just as hard for the other half, and the
zero is a cancellation of two large numbers. Totals alone cannot tell these
apart; the profile can.Figure 22.6 Nothing about this construction needed the plane. Here is a helix climbing
through space, and the same picture as before: the path above, the walker on it
with ⃗𝐹 and ⃗𝑇, and ⃗𝐹⋅⃗𝑇 accumulating beneath. The
dot product now runs over three terms instead of two, and that is the only
change.Figure 22.7 When the curve closes and the field is a flow, we call the integral the
circulation and put a ring on the sign: ∮𝐶⃗𝐹⋅𝑑⃗𝑟. Neither
the ring nor the word is a new construction.