23 · Flux Across Curves and Surfaces
Chapter 23

Flux Across Curves and Surfaces

23.1Flux Across Curves

Figure 23.1 How much of the flow gets through the fence? This is the last chapter's opening picture with one word changed — there we asked how much the wind helped a runner along a course, and here how much water crosses a barrier standing in it. Everything else is deliberately the same, because the construction is the same.
Figure 23.2 Only the part of the flow crossing the barrier gets through it. This is the previous chapter's decomposition with the two roles exchanged: it is now the part along the fence that is thrown away, drawn faintly, and the part across it that is kept. Water running parallel to a fence pours past forever and never crosses it.
Figure 23.3 Here is what the crossing actually is. Take a short stretch of fence, ⃗𝑟′Δ𝑡, and the flow carrying material across it. Everything inside the parallelogram those two span is what goes through in that moment, so the amount crossing is its signed area:
Figure 23.4 The normal is not a separate decision: it is a quarter turn clockwise from whichever way you chose to walk. Go counterclockwise round the loop and every one of those quarter turns points out of it; go the other way and every one points in, and the flux changes sign. "Outward" is a property of the curve and a direction of travel, never of the curve alone.
Figure 23.5 The loop from the last chapter, measured the other way — how much crosses out of it rather than how much carries you round it. Drag it about: in a general flow the answer swings hard.

23.2Flux Across Surfaces

Figure 23.6 First, what a flow in space even looks like. The arrow lattice that served so well in the plane does not survive the extra dimension — arrows at the back hide behind those at the front, every one is foreshortened by an unknown amount, and the picture becomes a thicket. So a field in space is drawn as streamlines: the paths specks of dust would take through it, with arrowheads to say which way along.
Figure 23.7 The fence becomes a net, 𝑑𝑠 becomes 𝑑𝑆, and the sum becomes a double one. Nothing else changes. The net hangs in the same streamlines as before, so you can watch the flow arrive at it, pass through, and leave on the far side. At a sparse lattice on the surface are the same two arrows the chapter has carried throughout — the flow in ink, the normal in violet — and the same two colours, red where the flow leaves through the surface and blue where it enters.
Figure 23.8 A surface has no positive side until you parameterize it, and the trap is sharper here than it was for curves. A curve at least tells you which way you walked it; a surface has no direction of travel to take a hint from, so the orientation comes from nothing but the order you wrote the two parameters in. ⃗𝑟𝑢 ×⃗𝑟𝑣 points one way and ⃗𝑟𝑣 ×⃗𝑟𝑢 points the other, and the flux comes out with opposite signs. The surface itself never moves.
Figure 23.9 Close the bowl with a disc and you have a surface with an inside. Now "outward" is a demand, and the two pieces answer it in opposite directions: outward on the bowl is up and away, outward on the cap is straight down. Writing "outward" secures neither — each parameterization must be checked, and the cap's has to be reversed to comply.