24.1Conservative Fields and the Fundamental Theorem
Figure 24.1 Two routes leave 𝑝 and arrive at 𝑞 through the same wind. Is the work the
same? Beneath, each route's total is accumulated as it is walked, so the
question is whether the two traces land on the same dot at the right-hand end.Figure 24.2 Where the number comes from, when the field is a gradient. The orange arrows on
the floor are ∇𝑓, and above them stands the graph of 𝑓 itself; the
route on the floor lifts to a walk on that graph, and the work accumulated so
far is exactly the height gained so far.Figure 24.3 Now close the route. The walk on the graph closes with it, and a closed walk
ends at the height it started, so the lap comes to nothing.
24.2Finding Potentials
24.3When Potentials Do—and Do Not—Exist
Figure 24.4 Try to build a potential for ⟨−𝑦,𝑥⟩ by the method of the last
section and it comes apart: integrating 𝑃=−𝑦 gives the candidate −𝑥𝑦, whose
𝑦-slope is −𝑥 where the field wants +𝑥, and the 2𝑥 left over is a
function of 𝑥 where the only thing left to adjust is a function of 𝑦. Here
is that failure as a picture rather than an algebraic obstruction.Figure 24.5 Zero curl is necessary. It is not sufficient — and exactly one missing point is
enough to break it.