3 ms·

I'm a math idiot, too, but I know that he's confusing indefinite and definite integration. Definite integration returns a number; indefinite integration return

by bkovitz 17y ago

I'm a math idiot, too, but I know that he's confusing indefinite and definite integration. Definite integration returns a number; indefinite integration returns a function. Differentiation does undo indefinite integration.

likpok 17y ago
Sort of, for well-defined functions (for some definition of well-defined). I remember seeing an example in analysis which showed that it was possible to get some nonintuitive behavior in functions.
- madars 17y ago
  You might enjoy "Counterexamples in analysis" [1], it shows, for example, a two argument function for which the order of integration matters, while the geometric interpretation would suggest otherwise. A must have! [1] www.amazon.com/dp/0486428753
- bkovitz 17y ago
  Yeah, that brings up another of those peculiar things about math, and especially, dealing with math culture. It's hard to talk about a main idea without getting mired in weird exceptions. There are indeed precise conditions that guarantee that a function can (a) be integrated and (b) the integral can be differentiated, and there are subtly different definitions of "integral" that lead to different sets of extremely weird functions being integrable or not. I can't remember any of the details despite having come across them many times. In software culture, it seems much easier to talk in broad strokes when appropriate and in precise details when appropriate. You might get an argument, but it's usually about something relevant.
  - bkovitz 17y ago
    Duh. It just hit me. Math is mostly edge cases. Consequently, the urge to be have perfect logical validity (same as the urge to write a bugless program) requires you to spend most of your attention on the edge cases. That urge is stronger in mathematicians than in software engineers.
ggchappell 17y ago
> ... I know that he's confusing indefinite and definite integration. Yes he is. The real solution is to banish the term "indefinite integral" from the language, and substitute "antiderivative". We should also not use the integral sign for antiderivatives, but unfortunately there is no well known alternative. :-( Using the same terminology and symbols for antiderivatives as for (definite) integrals is confusing, as here. It also makes the Fundamental Theorem of Calculus -- a very profound fact if there ever was one -- appear to be something trivial about getting rid of limits on an integral sign. But consider: one can compute a definite integral using an antiderivative ... whodathunkit?
- bkovitz 17y ago
  Isn't it weird how awful math terminology is? Even the word "derivative" is an awful choice. The special and wonderful thing about a derivative is that it's the rate at which another function changes, not that it's derived somehow.
  - deleted 17y ago
    [deleted]
- madcaptenor 17y ago
  And why is the verb for "take the derivative" "differentiate"? Both words make sense individually to describe things -- df/dx is derived from f by taking differences -- but it doesn't make sense that we use both words. I had an abstract algebra professor, in my first year of grad school in math, who referred to operators D that satisfy the product rule D(fg) = D(f) g + f D(g) , which are not necessarily defined in terms of a limit, as "derivations". This is a natural generalization of the derivative, and apparently "derivation" is the correct technical term, but for a week or so I thought he was just not speaking English properly. I'm not an algebraist now.

by bkovitz 17y ago

likpok 17y ago

Sort of, for well-defined functions (for some definition of well-defined). I remember seeing an example in analysis which showed that it was possible to get some nonintuitive behavior in functions.

madars 17y ago
You might enjoy "Counterexamples in analysis" [1], it shows, for example, a two argument function for which the order of integration matters, while the geometric interpretation would suggest otherwise. A must have! [1] www.amazon.com/dp/0486428753
bkovitz 17y ago
Yeah, that brings up another of those peculiar things about math, and especially, dealing with math culture. It's hard to talk about a main idea without getting mired in weird exceptions. There are indeed precise conditions that guarantee that a function can (a) be integrated and (b) the integral can be differentiated, and there are subtly different definitions of "integral" that lead to different sets of extremely weird functions being integrable or not. I can't remember any of the details despite having come across them many times. In software culture, it seems much easier to talk in broad strokes when appropriate and in precise details when appropriate. You might get an argument, but it's usually about something relevant.
- bkovitz 17y ago
  Duh. It just hit me. Math is mostly edge cases. Consequently, the urge to be have perfect logical validity (same as the urge to write a bugless program) requires you to spend most of your attention on the edge cases. That urge is stronger in mathematicians than in software engineers.

ggchappell 17y ago

> ... I know that he's confusing indefinite and definite integration. Yes he is. The real solution is to banish the term "indefinite integral" from the language, and substitute "antiderivative". We should also not use the integral sign for antiderivatives, but unfortunately there is no well known alternative. :-( Using the same terminology and symbols for antiderivatives as for (definite) integrals is confusing, as here. It also makes the Fundamental Theorem of Calculus -- a very profound fact if there ever was one -- appear to be something trivial about getting rid of limits on an integral sign. But consider: one can compute a definite integral using an antiderivative ... whodathunkit?

bkovitz 17y ago
Isn't it weird how awful math terminology is? Even the word "derivative" is an awful choice. The special and wonderful thing about a derivative is that it's the rate at which another function changes, not that it's derived somehow.
- deleted 17y ago
  [deleted]
madcaptenor 17y ago
And why is the verb for "take the derivative" "differentiate"? Both words make sense individually to describe things -- df/dx is derived from f by taking differences -- but it doesn't make sense that we use both words. I had an abstract algebra professor, in my first year of grad school in math, who referred to operators D that satisfy the product rule D(fg) = D(f) g + f D(g) , which are not necessarily defined in terms of a limit, as "derivations". This is a natural generalization of the derivative, and apparently "derivation" is the correct technical term, but for a week or so I thought he was just not speaking English properly. I'm not an algebraist now.