3 ms·
> this foundation (called "real analysis", largely developed in the 19th century) is quite different from the intuitive ideas presented in this book. ... (in pa
by sundarurfriend 10y ago
> this foundation (called "real analysis", largely developed in the 19th century) is quite different from the intuitive ideas presented in this book. ... (in particular the idea that dx² is negligible)
Thank you for posting this! Such things always bothered me in high school, seemed like approximations that ought to bite you in the behind at least in some corner cases. Another example from TLA:
> dy = 2cos(θ + 1/2 dθ) · sin 1/2 dθ
> But if we regard dθ as indefinitely small, then in the limit we may
neglect 1/2 dθ by comparison with θ, and may also take sin 1/2 dθ as being the same as 1/2 dθ. The equation then becomes:
> dy = 2cosθ × 1/2 dθ
This again seems like very sloppy and careless kind of approximation that ought to bite you in the back - but knowing there are just (supposed-to-be) intuitive non-rigorous methods, and that these have actual rigorous backing, somehow soothes me.
- thanatropism 10y agoBut this is true of every level of mathematics. See humorous proofs that 2=3 amounting to manipulations like 0a = 0b, cut the zeros, a=b. This is no fault of "nonrigorous elementary algebra", it's a matter of remembering all notations are abbreviations and you have to know how to manipulate them. You can get away with dy and dx if you just think "I'm not writing integral signs because they're annoying" and remember the rules of working with integrals. This is how stochastic differential equations work -- they're not even differential equations because Brownian motion is not differentiable, they're just notation.