25 · Fundamental Theorems in Two Dimensions: Green's Theorem
Chapter 25
Fundamental Theorems in Two Dimensions: Green's Theorem
25.1Circulation and Curl
Figure 25.1 The whole chapter is in this picture. Two cells, each walked counterclockwise,
held apart so that both copies of the edge they share can be seen at once. The
left cell walks that edge upward and the right cell walks it downward, so their
two contributions are not merely opposite in colour but opposite numbers —
reversing a curve reverses every ⃗𝐹⋅⃗𝑇 along it, which is the one
fact from Chapter 22 this chapter spends.Figure 25.2 The same thing at scale. Two cells become four, then sixteen, then a hundred and
twenty-eight, and each stage tells the identical story in the identical place:
every cell walked counterclockwise and held apart, then closed up so that every
interior pair cancels, leaving the boundary and nothing else.Figure 25.3Figure 25.4 Rectangles were for the proof. They tile a region exactly, which is what let the
last three figures cancel interior edges without a word about approximation —
but the theorem itself never mentions them. Here the boundary is a loop that
never stops changing shape, and the equality does not so much as flicker.Figure 25.5 What the hypothesis is for. The field is the one Chapter 24 ended on: its curl
is zero everywhere it is defined, so the mosaic has nothing to show. Every cell
is blank and the interior integral is flatly zero wherever the rectangle is put.
25.2Flux and Divergence
Figure 25.6 The same cancellation with one substitution: across each edge instead of along
it. Same region, same tiling, same stages, same place on the page — everything
you learned about reading the last figure still applies, which is the point — what
carries over is the argument, not the measurement. Circulation and flux are
two different readings of a boundary, and each gets the same cancellation for
the same reason.Figure 25.7