A forum set up for physics questions from students in the courses PHYS1121, 1131, 1221 and 1331 at the University of New South Wales. It is intended for questions that cannot readily be answered in class,
either because they fall outside the main syllabus and therefore would be distraction (however interesting) or for other reasons.

Here, I want to present another view and my almost complete proof of the paradox which I have found (further in this thread I post the proof with proving h = delta x), in my opinion. The proof consists of 3 pages: in this post - the first page; and in two next posts - the 2nd and 3rd page appropriately. In my opinion, all the algebraic operations are implemented in the right way?
By the way, please, notice that my proof is based on Algebra - the derivative is represented as a fraction in a term of the series. Calculus is based on Algebra as well.

P.S. I strongly recommend to read this thread starting at the end!

P.S.S. I should say that I am not a mathematician, and I am not planning to become a mathematician.

Thank you.

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Last edited by Boris Lagutin on Thu Oct 20, 2016 1:57 am, edited 14 times in total.

Here is the second page of my proof of the paradox of integral calculus (attachment). Please, notice that F(x) = x^3/3 in the example given in the proof.

Thank you.

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Last edited by Boris Lagutin on Sun Oct 16, 2016 10:13 am, edited 6 times in total.

Boris Lagutin wrote:Here is the second page of my proof of the paradox of integral calculus (attachment). Please, notice that F(x) = x^3/3 in the example given in the proof.

Thank you.

Well for one thing, you've claimed here that f(x) is equal to some expression involving Δx, but this is not the case; rather, f(x) is that expression limited as Δx → 0, which means that f(x) is actually just x².

a.s.h, I think you "forgot" that a derivative is a slope. The latter is defined as: a difference of a function's outputs / a difference of a function's inputs which is an algebraic expression called "fraction". Therefore, F(x+h) and F(x) are just outputs, and (x+h) and x are just inputs.
If the expression with h was not equal to the expression with delta x, then it would violate Fundamental Theorem of Calculus and the limit of Riemann's sum as Integral.

The derivative is the instantaneous slope at a point. The expressions involving h and Δx are slopes of secant lines and are known as difference quotients. The derivative at a point is the limit of this difference quotient (slope of a secant line) as the width of the interval of the secant goes to 0. Note that h is just a dummy variable used in the limit (you could replace h with any other letter. It doesn't make sense to talk about h being equal to Δx because it's just a dummy variable without any value.).

Both h and delta x in the limits in the proof express some very small values which are equal to each other. By the way, it would be interesting to learn history of creating Calculus in details. I seem that the originators of Calculus could not invent a mathematical mechanism (formula etc.) to evaluate limit expressions with very small values like h and simply ignored (removed) terms of algebraic expressions containing h . However, when it comes to infinite series like Riemann's sum, ignoring such very small values leads to the paradox, in my opinion. Maybe it is possible to create such a mathematical mechanism which would calculate very small values like h. Possibly, I will think of it in some day.

P.S. I like Algebra more...

Have a nice day

Last edited by Boris Lagutin on Wed Sep 14, 2016 3:54 am, edited 1 time in total.

To sum up my previous comment, I think that this paradox can be solved only in one way - creating a mathematical mechanism (formula etc.) able to calculate limit expressions containing very small values like h.

Have a nice day

Last edited by Boris Lagutin on Wed Sep 14, 2016 3:54 am, edited 1 time in total.

I believe when Calculus was first being invented/discovered, they used things like "infinitesimals", which are what you seem to be describing. In standard calculus today though, these h and Δx things aren't actually infinitisemals, because we (in standard calculus) use limits in calculus now rather than infinitesimals. So the h and Δx don't have actual values.

And by liking Algebra more, were you mainly referring to Linear Algebra? Or just Algebra in general (including Abstract Algebra etc.)? (These are also things that have many applications in Physics of course.)

In any case, a derivative represents a slope which is just dividing a difference of function's outputs by a difference of function's inputs. They all are some values. We cannot divide nothing by nothing and get some slope.
I should have added that it is necessary to highlight that the paradox appears only when the definition of the derivative (limit) is applied to the limit of Riemann's sum (integral) which is an infinite series. In other cases (finite series etc.), I think, the definition of the derivative (limit) can be applied nicely (without the paradox).
I mentioned Algebra on the whole, as a base for Calculus and other stuff.

I would like to add the last thing: x_i in lim F(x_i + h) - F(x_i) / h (shown in the proof above) must correspond x_i in F(x_i + delta x) - F(x_i) / delta x (shown in the proof above). If this limit with h was not equal to this expression with delta x, it would violate Fundamental Theorem of Calculus and the limit of Riemann's sum as Integral. I have already written about it in one of my comments above. It is one more proof that h is equal to delta x !

But in your post of Mon Sep 12, 2016 6:22 am,
h is dummy variable, inside a limit statement, while delta x is a quantity that multiplies the limits. These are defined very differently.
Joe

Thank you, Joe. F'(x_i) = f(x_i), and the limit of Riemann's sum (Integral) is an infinite sum each term of which is f(x_i) * delta x. It means that F(x_1 + h) - F(x_1) / h = f(x_1) and x_2 = (x_1+h) so on and so forth, hence f(x_1) * delta x is the first term of Riemann's sum, f(x_2) * delta x is the second term of Riemann's sum so on and so forth. It causes h to be equal to delta x. Please, pay attention to a width of Riemann's sum partition (small rectangle) and its height which is a slope of F(x). There is only the same slope for each x_i for both F(x_i) and Riemann's sum consequent term f(x_i) * delta x which causes h to be equal to delta x ! Otherwise, we do not get a continuous curve of a function f(x)!

P.S. Moreover, we cannot discern h which goes to zero from delta x = (b-a) / n where n goes to infinity.