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If you think of the variable being derived/integrated over as time, then conceptually the derivative at a time only represents information from that one time pl
by GrantMoyer 2y ago
If you think of the variable being derived/integrated over as time, then conceptually the derivative at a time only represents information from that one time plus or minus some tiny delta, while an antiderivative repesents information from all of time.
Formally, I'm not sure how to show that's related to the relative difficilties of the two operarions, or if it actually is related in the first place, but intuitively the explanation is appealing.
- almostgotcaught 2y agoIt's not that simple https://en.m.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) https://en.m.wikipedia.org/wiki/Liouville%27s_theorem_(diffe... So it's important to keep in mind the difference between integral and anti-derivative. And note that as soon you say "within an epsilon" you're already on the other side of "not a lot of info" vs "infinite info". So global, local doesn't matter, especially since eg an analytic function is completely determined by it's derivatives at a single point (good ol Taylor's theorem).
- 6gvONxR4sf7o 2y ago> Formally, I'm not sure how to show that's related to the relative difficilties of the two operarions, or if it actually is related in the first place, but intuitively the explanation is appealing. I wonder if you could relate it to the difference between local and global sections on bundles and connect “global sections don’t always exist” to “global data is harder.”