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 Apr 27 comment Where does the gap come from? @StevenGregory: No, I'm just looking at the overall shape, not the individual shapes. I'm saying the difference of the two shapes' areas seems to be way less than that of the white rectangle. Apr 27 comment Where does the gap come from? Yes, actually I probably would... to my eyes that hole looks way more massive than the added area! Apr 18 comment Why does gradient descent make sense? @CharlieParker: I was analyzing two separate cases. Apr 17 comment Function with no roots @MorganRodgers: I wouldn't, but I might include it in $\mathbb{R}$... Apr 15 comment Function with no roots If you include $-\infty$ then $\exp$ does have a root... I like the absolute value one better. Apr 13 comment Why does this way of solving inequalities work? What your teacher meant was, "We have [...] which is always true. Hence, since these statements are equivalent, what was assumed originally is always true." Apr 8 awarded Good Question Apr 7 comment Understanding Euclid's proof that the number of primes is infinite. +1 This is definitely more clever than the usual contradiction formulation... Apr 4 comment Derivative over variable vs. partial derivative over variable I would use the "evaluated at" bar notation instead, since this was confusing to me when I read your answer. Mar 31 comment Is there a math function to find an element in a vector? +1 this is the proper answer. Mar 26 comment Is there an “intrinsic” difference between a plane and a cylinder? One of them flies, the other doesn't. Mar 24 awarded Popular Question Mar 20 comment When are differential operators not equivalent to variables? I mean if you take what I said too literally (re: sloppiness etc.) then you kind of miss my actual point. I'm sure you see what I mean regarding the confusion, right? And why the explanations aren't satisfactory? Try explaining all these to a high-schooler learning calculus. You can't just tell them it's a shorthand and $dy$ has a dozen different meanings in differential geometry and then expect them to be satisfied with that answer. They won't even know what differential geometry is, but they really do have the capacity to know (and deserve to know) exactly what is going on here. Mar 20 comment When are differential operators not equivalent to variables? If it's just a shorthand (which I take to be an informal thing etc.) then why is it then that when I look at Wikipedia it clearly very formally says that $dy$ is a thing in itself, called the "total differential"? Nowhere does it suggest that it's just some kind of sloppy notation or shorthand... it seems to be a very formally well-defined thing in itself that I should be able to treat separately from $dx$. Mar 20 comment When are differential operators not equivalent to variables? But is their meaning actually different? Or is merely the situations in which they are used different? If you're really saying their meanings are different then I'd love to hear what you see as the difference. (i.e. what would be each of their definitions? This should be irrespective of their use case.) Mar 20 comment When are differential operators not equivalent to variables? The problem I've always had with people saying you can't treat $dx$ and $dy$ as separate entities is that I see them violate their own decree and write nonsense like $dy = (\partial y/\partial x_1)dx_1 + (\partial y/\partial x_2)dx_2$... excuse me, if you said you can't treat them separately then what the heck is this supposed to mean? So can you treat them as separate variables or no? Why don't mathematicians follow their own advice? I think this really deserves a good explanation somewhere... Mar 19 comment Is linear algebra more “fully understood” than other maths disciplines? Is it really that linear algebra is more "well understood"? I feel that it's inherently "easier"? Like, I feel like it's possible to understand things that are difficult very well, and it's also possible to not understand things that are easy... no? Mar 16 awarded Popular Question Mar 13 accepted Conversion of explicit to implicit ODEs (uniqueness & algorithm) Mar 13 comment Conversion of explicit to implicit ODEs (uniqueness & algorithm) Oooh, totally missed that somehow since I missed that $x$ would have to be just a function of $t$ and not of other variables. Thanks!