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nLab > Latest Changes: Fundamental theorem of calculus

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    • CommentRowNumber1.
    • CommentAuthorColin Tan
    • CommentTimeJan 20th 2014
    After receiving Spitters and Toby's answer to my question on the mean value theorem, I begun reading the article "Uniform Calculus and the Law of Bounded Change" by Bridger and Stolzenberg. I started filling up the stub on the fundamental theorem of calculus.

    I have a feeling that the two parts of the fundamental theorem of calculus are different in foundational strength. In particular, I believe that the part of the fundamental theorem of calculus which is Stokes's theorem for the interval [0, 1] is equivalent to the law of bounded change. I am aware that the previous assertion is trivial at present, since of these statements are manifestly true (constructively).
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2014

    just for the record, here is the link to the entry that this is about: fundamental theorem of calculus

    • CommentRowNumber3.
    • CommentAuthorColin Tan
    • CommentTimeJan 21st 2014
    I articulated my feeling in a theorem under the section "in synthetic differential geometry". I will check this more thoroughly for errors. Is there a name for Axiom 1 of Kock's book "Synthetic differential geometry"? With the definition of integration in terms of the antiderivative, part of the fundamental theorem of calculus becomes redundant.
    • CommentRowNumber4.
    • CommentAuthorColin Tan
    • CommentTimeJan 21st 2014
    My feeling is wrong. I have edited out those errors I found from the nlab entry.
    • CommentRowNumber5.
    • CommentAuthorColin Tan
    • CommentTimeJan 21st 2014
    In synthetic differential geometry, does the monotonicity of the definite integral follow from the integration axiom (plus any other required axiom)?
    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeJan 26th 2014

    I put in the FTC as I understand it, but I would like to restore some of your proofs … the parts without errors, that is.

    • CommentRowNumber7.
    • CommentAuthorColin Tan
    • CommentTimeFeb 2nd 2014
    Referring to the section on the integral formula for the antiderivative, does this mean that, in the FTC, existence implies the first part?

    Referring to the introduction, what is the difference between the infinitesimal calculus and the differential calculus?
    • CommentRowNumber8.
    • CommentAuthorColin Tan
    • CommentTimeFeb 2nd 2014
    Toby, thanks for improving my writeup.

    It would be satisfying to work together to include the requisite proofs. In particular, the classical versions should follow from the constructive versions.
    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 3rd 2014
    • (edited Feb 3rd 2014)

    does this mean that, in the FTC, existence implies the first part?

    Existence, together with the second part, implies the first part. This explains why you could write the article to say that the first part (numbered 2 by you by the end of your edits) was only existence (well, and uniqueness).

    There is a dual to this. Uniqueness, together with the first part, implies the second part. This is implicit in the proof outline that I wrote up, but not spelt out.

    what is the difference between the infinitesimal calculus and the differential calculus

    This sort of thing is kind of vague, but ‘infinitesimal calculus’ (a fairly old term, now not much used) should refer to any method of calculating with infinitely small quantities, while ‘differential calculus’ would just be limited to calculating with differentials (and derivatives, of course, which are essentially the same thing). So in particular, to not include integration as such.

    On the other hand, antidifferentiation goes naturally with differentiation, and even direct integrals involve adding up infinitesimal quantities. Still, you can make a distinction between taking differentials/derivatives and taking (direct) integrals. The FTC links these, by saying that the direct integral of a differential/derivative and taking the differential/derivative of a direct integral amount to simple substitution and subtraction; so it's not just about one or the other.

    I generally use ‘infinitesimal calculus’ in place of the layperson's simple ‘calculus’; but this is not a perfect replacement, since the layperson's Calculus also includes a section on infinite series (and perhaps some analytic geometry).

    • CommentRowNumber10.
    • CommentAuthorColin Tan
    • CommentTimeFeb 1st 2016

    Toby, thanks for the edits. Given that you have explained how to rephrase the second part of the FTC as ∫ a bdF(t)=F(b)−F(a)\int_{a}^b \, \mathrm{d} F(t) = F(b) - F(a), it seems that the last section “Integral formula of the antiderivative” is now redundant.

    Should we edit this last section?

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