A math trick I like a lot is the approach to taking derivatives using hyperreal numbers. Thread:
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thanks, I was about to ask about the relationship to similar tricks via dual numbers, https://blog.demofox.org/2014/12/30/dual-numbers-automatic-differentiation/ … etc... are they both equally practical, or suited to certain uses?
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Mathematically the "problem" with this approach is that what you get is not an ordered field, since ε is not invertible and ε^2 can't be bigger than zero even though ε should be. This leads to Synthetic Differential Geometry instead of NSA.
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If you're just using them as a bookkeeping device for a program they are fine of course. This kind of ε is restricted to the first derivative (or up to an nth derivative fixed at the beginning), which is good for space but maybe less flexible?
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