Figure 21.1 Follow one particle and all you learn is where the flow carried it. Drop a
whole blob of dye and you learn how the vectors near it differ from one
another — which is the only thing a derivative was ever about. Four things
happen at once: the blob is carried along, it swells, it turns, and it shears
out of round. The rest of the chapter isolates the second and the third. Drag
the drop point.
21.1Vector Fields
Figure 21.2 One field, two pictures. Arrows draw the rule — here is the vector at this
point, and this one, and this one. Flow lines draw the consequence — here is
where the rule would carry you if you let it. Flip between them: it is the same
field either way, and the drifting specks are the same specks.Figure 21.3 The four ways a field can behave, in the order the section builds them: the
same vector everywhere; the same direction at every point but a length that
changes; the same length everywhere but a direction that turns; and finally
both at once. The particles are there because a vector field is a thing that
pushes, not a pattern of arrows on a page.Figure 21.4 Galileo's constant downward gravity and Newton's inverse-square law are not two
competing models. The window closes in on one spot on the surface and opens out
again: near the ground the arrows are parallel and all one length, which is
⃗𝑎=⟨0,−𝑔⟩; pull back and those same arrows are aiming at
the centre and getting shorter. The dashed square in the wide shot is the close
view, so you can see that nothing about the field changed — only how much of it
fits on screen.
21.2Divergence
Figure 21.5 Where the number comes from. Walk the boundary of a small square and keep only
the part of the field pointing through each side — red for what leaves, blue for
what enters. Add them for the net outflow, divide by the area, and let the
square shrink.Figure 21.6 At every point these two fields aim the same way — straight out from the
origin. The only difference between them is how fast the arrows shorten: on
the left they never shorten at all, on the right they fall off like 1/𝑟. Drop
the same cloud of particles in each and carry it for the same time. On the left
the particles drift apart and the cloud swells. On the right they stay exactly
as densely packed as they started, however far the cloud is stretched along the
way. Arrow length alone is doing the work.Figure 21.7 Divergence in its plain form. The same cloud dropped in three fields: it swells
in the first, collapses in the second, and in the third it is carried right
round without changing size at all. Try predicting the sign from the arrows
before reading it off — the next figure is about what happens when that guess
goes wrong.
21.3Curl
Figure 21.8 The same square, the same clock, the same two panes, the same colours — and one
thing changed. On each side we now keep the part of the field pointing along
the boundary instead of through it: red where it carries you counterclockwise,
blue where it pushes back. Divide the total by the area, shrink the square, and
the limit is the curl.Figure 21.9 Curl is what a small paddle wheel does, and the painted spoke is how you tell.
On the left the wheel rides the field. On the right the camera is bolted to the
wheel: the dashed window shows what it is looking at, and inside it are the
field's own vectors at the points beside the wheel, streaming past as it
travels.Figure 21.10 The same trick as the divergence counterexample, one section later. Both fields
run round the same circles, and both wheels are carried round the same path.
Only the arrow lengths differ — and on the right the wheel completes the whole
circuit without turning once. Going around is not the same as spinning.