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## Homework Statement

just looking for opinions regarding the best text for introductory calculus as in refresher--I used Taylor eons ago, which i recall as being pretty descent.

- Thread starter denverdoc
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just looking for opinions regarding the best text for introductory calculus as in refresher--I used Taylor eons ago, which i recall as being pretty descent.

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I like Stewart...

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egad! Stay away from Stewarts. His definition of integration is not well defined. And I do mean in the mathematical sense.

I guess it's not bad for an intro text.

I guess it's not bad for an intro text.

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stewart seems to be the most controversial--on amazon it seems to get 5 starts or 1-2.

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your just looking for a refesher right?

get a 2nd ed stewart book

it is going to be pretty cheap.

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There is no better introduction calculus text then calculus by spivak.

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Courant's text is one of the deepest out there, but it is time consuming.

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I'm afraid I do not see what you mean. A very important result in calculus is that the Riemann sum of any repartition of an interval [a, b] as the largest interval goes to zero will always yield the same value. This is due to the fact that any function defined as integrable is also uniformly continuous (in a closed interval, if we chose an appropriate h, then f(x + h) - f(x) < k, where k is any value we wish and x is any value in the in the interval). Your argument with irrational numbers does not hold. Irrational numbers are given all the arithmetic properties of rationals, hence making Riemann summation consistent. Also, the integral of the identity function over [0, 1] is 1/2, not 1. This said, Stewart is certainly a masterful mathematician, and I deem his calculus textbook beyond reproach.

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morphism

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The indicator function of the rationals is not Riemann integrable on any interval of positive length, precisely because the limit as the mesh fineness goes to zero of the Riemann sums doesn't exist. I don't actually have a copy of Stewart's book (it was the assigned text for my first year calculus course, but I never bothered to open it and I can't find it now!), but my second-year vector calculus text (and every other text I've seen) defines the Riemann integral in a way that is certainly equivalent (unless he does something very silly!).

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But in this case the integral cannot be defined since the function g(x, h) = f(x+h) - f(x) would have an infinite number of jump discontinuity on any closed interval - the concept of a limit does not apply to the Riemann sum of this function because the Riemann sum has, how would you say, a "bad behavior".indicator-not identity- function (of the rationals), that is the function which is 1 on the rationals and 0 otherwise. Although I don't really follow his argument. It seems to me that in both cases you'd get 0.

Edit: Oops, just saw Data's post. That makes mine just a repetition of what has already been said.

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I just want to reacquaint myself with the tools of the trade, I have a good diffy text (kreyzig) but some of the simple stuff i haven't used for 20+ years. Just looking for a good cookbook with enuf theory to understand whats being assumed. I know the forum has to be carful re endorsement, else there should be some stickies. Weakest area on PF was book reviews, which is where Neutrino suggested I look.So I'm back til we get moved.

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mathwonk

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My point was that this is how stewarts defines his integrals, as far as I remember. As the mesh goes to zero and it comes out to some number, then that is the definite integral. But that is not a well-defined definition, as my indicator function shows. If, however, you're integrating a continuous function and you get some number, then it is Riemann integrable with that limit.

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