Figure 27.1 One picture, three times. The interior is filled by a derivative, the boundary
is lit by the quantity itself, and the two numbers underneath agree. Step
through the tabs: only the dimension changes.Figure 27.2 The one-variable theorem, recognised: it was internal-boundary cancellation all
along. This is Chapter 25's figure with the region reduced to an interval — same
holding apart, same red and blue, same refinement, same number that never moves.Figure 27.3 Four conventions get memorised separately — the minus sign in 𝑓(𝑏)−𝑓(𝑎),
counterclockwise, the right-hand rule, outward normals — and they look unrelated
because their outputs have different type. They are one rule: point out of
the object, and everything else is forced.
27.2A Glimpse of Differential Forms
Figure 27.4 The chapter's device applied twice in a row. A square's boundary is four
oriented edges; take the boundary of those — a plus at each head, a minus at
each tail — and at every corner one edge's head lands on the next one's tail.
They annihilate in pairs and nothing is left at all: