On page 112 of Tenenbaum and Pollard's [i]Ordinary Differential Equations[/i] I am confronted with the a diagram like this:
The value Δ controls the accuracy of the displayed approximations ds, A and dA. Tenenbaum goes to the heart of the matter. dA is the area enclosed by OP, OQ, and the arc [i]ds[/i]. I will use a physical definition of equality: "Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal." (Newton) As the evanescent quantity vanishes, quantities whose ultimate ratios are those of equality can be used interchangeably. [b]The first obstacle is that, for (a) dA/dθ = r²/2 to hold, dθ must be indistinguishable from sin(dθ).[/b] Next: _______________________ To measure the ratio of areas, I would like very much to have, as [i]dθ[/i] vanishes, the ultimate ratios *rdθ/u = 1, *ds/h = 1 *h sinϕ/dr = 1 Then I could write interchangeably, in the limit, u² + dr² = h² = (rdθ)² + dr² = ds² I find the geometric details more instructive than the abstract answers which I know and am happy to take for granted.