# Calculus

How should I explain and justify a jacobians matrix for which the eigenvalues are equal to zero?Is my system of equations stiff or not?

in my equations system ..i write jacobians matrix and i  calculate the eigenvalue of my jacobian matrix.. but the two eigenvalues from three eigenvalues are zero and my stiifness ratio is infinity .

Note that the stiffness ratio is just one of the measures for describing stiffness; and it  works inefficiently in a case like yours. In your case the ration is infinity while the stiff. index (max of Re(\lambda)) could be moderate.

You can take a look at this

http://www.iact.ugr-csic.es/personal/julyan_cartwright/papers/rkpaper/node6.html

i.e. you simply need to look at other measure, like stiffness index, c.f.

http://www.scholarpedia.org/article/Stiff_systems

Though there are some replies regarding stability, there is no relation between stability analysis and stiffness, here we assume our system is stable and are just concerned with its extreme time behavior.

i hope this helps.

Is there any use in constructing/defining integration over (some subset of) the rationals?

I was working on 2 papers on statistics when I recalled a study I’d read some time ago: “On ‘Rethinking Rigor in Calculus...,’ or Why We Don't Do Calculus on the Rational Numbers’”. The answer is obviously trivial, and the paper was really in response to another suggesting that we eliminate certain theorems and their proofs from elementary collegiate calculus courses. But I started to wonder (initially just as a thought exercise) whether one could “do calculus” on the rationals and if so could the benefits outweigh the restrictions? Measure theory already allows us to construct countably infinite sample spaces. However, many researchers who regularly use statistics haven’t even taken undergraduate probability courses, let alone courses on or that include rigorous probability. Also, even students like engineers who take several calculus courses frequently don’t really understand the real number line because they’ve never taken a course in real analysis.

The rationals are the only set we learn about early on that have so many of the properties the reals do, and in particular that of infinite density. So, for example, textbook examples of why integration isn’t appropriate for pdfs of countably infinite sets typically use examples like the binomial or Bernoulli distributions, but such examples are clearly discrete. Other objections to defining the rationals to be continuous include:
1) The irrational numbers were discovered over 2,000 years ago and the attempts to make calculus rigorous since have (almost) always taken as desirable the inclusion of numbers like pi or sqrt(2). Yet we know from measure theory that the line between distinct and continuous can be fuzzy and that we can construct abstract probability spaces that handle both countable and uncountable sets.
2) We already have a perfectly good way to deal with countably infinite sets using measure theory (not to mention both discrete calculus and discretized calculus). But the majority of those who regularly use statistics and therefore probability aren’t familiar with measure theory.

The third and most important reason is actually the question I’m asking: nobody has bothered to rigorously define the rationals to be continuous to allow a more limited application of differential and integral calculi because there are so many applications which require the reals and (as noted) we already have superior ways for dealing with any arbitrary set.

Yet most of the reasons we can’t e.g., integrate over the rationals in the interval [0,1] have to do with the intuitive notion that it contains “gaps” where we know irrational numbers exist even though the rationals are infinitely dense. It is, in fact, possible to construct functions that are continuous on the rationals and discontinuous on the reals. Moreover, we frequently use statistical methods that assume continuity even though the outcomes can’t ever be irrational-valued. Further, the Riemann integral is defined in elementary calculus and often elsewhere as an integer-valued and thus a countable set of summed "terms" (i.e., a function that is Riemann integrable over the interval [a,b]  is integrated by a summation from i=1 to infinity of f(x*I)Δx, but whatever values the function may take, by definition the terms/partitions are ordered by integer multiples of i). As for the gaps, work since Cantor in particular (e.g., the Cantor set) have demonstrated how the rationals “fill” the entire unit interval such that one can e.g., recursively remove infinitely many thirds from it equal to 1 yet be left with infinitely many remaining numbers. In addition to objections mostly from philosophers that even the reals are continuous, we know the real number line has "gaps" in some sense anyway; how many "gaps" depends on whether or not one thinks that in addition to sqrt(-1) the number line should include hyperreals or other extensions of R1. Finally, in practice (or at least application) we never deal with real numbers anyway (we can only approximate their values).
Another potential use is educational: students who take calculus (including multivariable calculus and differential equations) never gain an appreciable understanding of the reals because they never take courses in which these are constructed. Initial use of derivatives and integrals defined on the rationals and then the reals would at least make clear that there are extremely nuanced, conceptually difficult properties of the reals even if these were never elucidated.

However, I’ve been sick recently and my head has been in a perpetual fog from cold medicines, so the time I have available to answer my own question is temporarily too short. I start thinking about e.g., the relevance of the differences between uncountable and countable sets, compact spaces and topological considerations, or that were we to assume there are no “gaps” where real numbers would be we'd encounter issues with e.g., least upper bounds, but I can't think clearly and I get nowhere: the medication induced fog won't clear. So I am trying to take the lazy, cowardly way out and ask somebody else to do my thinking for me rather than wait until I am not taking cough suppressants and similar meds.

David Gilat · Tel Aviv University

To Miodrag Mateljevic:

The whole point of my question 1 was to argue the impossibility of coverage from first principles, without the use of Lebesgue's theory. The argument may go as follows:

WLOG assume that the intervals are open (otherwise, each interval can be slightly extended to an open interval so that the lengths-sum of the extended intervals remains strictly less than 1). If the union of the original intervals cover I, thenclearly so does the union of the extended open intervals. If so, using compactness of (the Heine-Borel theorem), extract a finite sub-cover and prove (non trivial, but easy) that the union of a finite sequence of intervals with lengths-sum strictly less than 1 cannot cover (nor Q).

This argument illustrates the enormous difference between finite and countable unions of intervals, or if you will - between the algebra of finite unions of intervals and the sigma-algebra (of Borel sets) generated by intervals. These ideas are at the root of the difference between Jordan content and Lebesgue measure, or if you will - between Riemann and Lebesgue integration.

Is there a book in English where one can find characterizations of zero-derivative (stationary) points ?

In non-English literature two  such characterizations for C2 functions of the single variable can be found in the text Neralic, Sego: Matematika (second edition), Element, Zagreb, 2013 (ISBN 978-953-197-644-2) but they do not seem to be widely known. They appear to be important in analysis, calculus, optimization and other areas.  Where can one find such results in functional analysis ?

Miodrag Mateljević · University of Belgrade

Hello Sanjo and followers,

Regarding the possible Morse property (MP) characterization of zero-derivative points. ("Around zero-derivative points and only around such points the functions have Morse property.") We  have to assume here that we work with non-degenerate critical points of f.

However, It seems that some characterizations of degenerate critical points are possible (see item 2. below).

1. If $f$ is $C^2$ function of
a single variable in a neighborhood $V$ of $0$ and $f(0)=0$, $f'(0)=f''(0)=0$ and there is $s=f'''(0)\neq 0$, then there is a function $g$ (diffeomrphism, g'(0)=1) such that $f(g(x))=k x^3$
in a neighborhood $W$ of $0$.

It seems that  some characterizations   of degenerate critical points  are possible.

2. One can describe when f is of form $f(g(x))=s x^n$
in a neighborhood $W$ of $0$ .

If $f$ is $C^{n-1}$ function of
a single variable in a neighborhood $V$ of 0 and if there is $s=f^(n)(0)\neq 0$ and the derivatives up to order n-1 are 0 at 0, then there is a function $g$ (diffeomrphism, g'(0)=1) such that $f(g(x))=k x^n$
in a neighborhood $W$ of $0$.

How can I develop $(p,q)$-analogue of Gamma and Bea functions?

Those who are working in the field of approximation by positive linear operators know how to use $q$-calculus to develop $q$-operators. And we have very good operators like $q$-Benrstein, $q$-Bernstein-Stancu operators etc. We introduce a new generalization of the above discussed operators based on $(p,q)$-integers. Can anybody suggest me how to develop $(p,q)$-analogue of Gamma and Beta functions with the help of $q$-Gamma and $q$-Beta functions?

Khursheed J. Ansari · Aligarh Muslim University

A nice paper

i will  go through it

Is there a general method to convert Sn to Tn in Sequence and Series?

I am just wondering. For example, for series 1+2+3+4+5+6+.... the Tn is k, while the Sn is (n)(n+1)/2. I do know there are rules to reach the summation for each case (for example, for series k2, the summation is n(n+1)(2n+1)/2, etc), so is there a more general way to convert the summation Sn to Tn just like the the case of derivation and integration in Calculus?

For each k you can find a polynomial of order k+1. You can prove it by induction.

Why don't we teach limits in calculus?

Limits underlie everything in calculus and analysis. To see this, simply look at some old textbooks that use infinitesimals (I do not mean the infinitesimals from non-standard analysis that require hyperreals) or spend some time programming functions for numerical integration or calculus operations for some CAS in general (or look at the content of a real analysis textbook). You can’t actually understand anything much in calculus without understanding limits. Yet your standard textbook introduces them in chapter two, reintroduces them somewhat when defining Riemann integrals (and then again with improper integrals), and of course with sequences and series. But apart from that 2nd chapter introduction, limits are never covered in any kind of comprehensive, detailed way; rather, they are mostly introduced to move on to differentiation and then used when needed to introduce various other components of single or multivariable calculus.

Is there a good reason for this? Is there a way to teach calculus at least somewhat via limits as the foundation they are (as in e.g., classic textbooks such as Courant’s)? Is there a good reason not to?

Geng Ouyang · MinNan Normal University

Dear Ahmed，I agree with you that limit theory and skill is an important tool in present infinite related area in mathematics. But I am very sorry to say that we sometimes have to stuff our students some other mysterious things.

Just see following divergent proof of Harmonic Series, very elementary and important, which can be found in many current higher mathematical books written in all kinds of languages:

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                  （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．   （２）
>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．            （3）
=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                          （4）

We teach our students that we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Un--->0 items in Harmonic Series by “brackets-placing rule" with modern limit theory and skill to change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity.

The more we try to explain the more doubts be aroused and we feel more helpless.

I think it is miserable.

Regards, Geng

Why should we teach improper integrals?

I've taught using maybe a dozen calculus textbooks (not through my own choice) and examined many more, and the only ones that do not include improper integrals are those that I've never heard of anybody using. Yet

1) Improper integrals do not generalize. That is, once we leave the real number line and start working in Rn, improper integrals do not work (indeed, as defined they make no sense).

2) The only pedagogical function they might serve has to do with convergence, but textbooks invariably include the so-called "integral test" anyway and anything of value for understanding convergence/divergence we obtain from first learning of improper integrals could be introduced merely with the "integral test".

3) By the time most students get to improper integrals, integration has come to be understood (at least almost) entirely in terms of finding antiderivatives and using the "fundamental theorem" of calculus. Thus improper integrals represent a conceptual break from previous work with integration. Also, as whatever failings the Riemann integral may have in terms of what functions it can't deal with, Lebesgue integrals (and others) will do all that improper integrals can and more.

4) Improper integrals are deceptively (even for advanced students) tricky if not absolutely convergent.

So what's the justification for using them to teach elementary calculus?

Geng Ouyang · MinNan Normal University

Sometimes we complain that students are not taught “correct ‘finite—infinite’ related mathematics”, but now I fully understand that at least since Zeno’s time, everyone in present traditional “finite--infinite” related science theory system have to teach our students all the theories in our text book. After all, we have no choices and so do our students. Sometimes we are helpless to teach our students something we are not agree in the bottom of our heart but sometimes we teach our students “wrong mathematical things” unconsciously because we ourselves are within the defected present traditional “finite--infinite” related science theory system------ some fallacious cases （the typical things are paradoxes）are produced logically by many reasons.

It is a long being suspended syndrome of infinite related fundamental defects (confusions): what are infinite, infinitesimal, 0, limit theory, infinite related numbers,…? Anyone working in present traditional infinite related science branches (mathematics, physics, …) is sure to be confused with infinite, infinitesimal, 0, limit theory, infinite related numbers,… And some fallacious cases many be produced naturally by this syndrome.

I think the thing really worries us scientists is: only a few people care how to avoid “purposely or unconsciously teaching our students wrong mathematical things” or how to solve the defects in present traditional “finite--infinite” related science theory system disclosed by the growing family members of “finite--infinite” related paradoxes.

“Applying mathematics” is only small branches of “theoretical mathematics (trunk)”, both “applying mathematics” and “theoretical mathematics” is needed in our science.

Something should be done sooner or later to get rid of those “finite--infinite” related fundamental defects in our science------ it is a huge project

Should we teach limit or derivative first?
Should we teach limit as a tool for derivation or should we teach in depth as a subject before derivative concept?

We should teach the concepts of limits first. It is the basic idea that provide the foundation to the studies in Calculus.  Derivatives can be taught using the concepts of limits.

How do we, as instructors, explain our students that one needs other types of integrals besides Riemann and Lebesgue?

Not only Stieltjes, but also Henstock-Kurzweil or Denjoy-Perron, etc.

Excellent idea, dear Guy! This is how I have been trained as a student, and this is how I proceed as a teacher.

What is the best reason for Conformable fractional derivative is not related to standard fractional derivatives like Riemann-Liouville or Caputo?

.

Boddu Muralee Bala Krushna · Maharaj Vijayaram Gajapati Raj College of Engineering

Thank you for answer. But  some conditions failed like arbitrary real order derivative of some functions and existence of non local.

Is the order of coordinates important in the calculus of tensors?

For example, if I use spherical coordinates, r is X1, theta is X2 and phi is X3. But could I use, for example, phi as X1, theta as X2 and r as X3 in the calculus of tensors?

Antonio Marmo · Instituto Tecnológico de Aeronáutica (ITA)

I also agree with Prof. Hady.

In fact, we must use oriented base vectors , because we can not always have coordinates (as in non-holonomic). The order is important because it determines an orientation of the axes and the signal of the Jacobian.

How to characterize differentiability of multifunctions by continuity?

A (single variable) function is differentiable iff some other function is continuous. Can we get similar characterizations for (different kinds of) multifunctions? How to express: a multifunction is X-differentiable iff some other (multi)function is continuous?

Shahram Rezapour · Azerbaijan Shahid Madani University

Hello dear friends

You know we can put such type questions in pure classification field and there is no reason for  thinking about any applications (may be it could be find many applications in future). But, it is not bad we could get some sights about the question and we could get some imagines. Anyway, I interest more answers from members of this group. Finally, may be it will be interest for somebody that some researchers are investigating some systems of fractional differential inclusions.

What if calculus had never been discovered?
We all know how the discovery of calculus has changed the course of mathematics. There is hardly any branch of science that has not been effected by calculus. But what would have happened had calculus not been there? How would the world of science have proceeded in this kind of situation?
Marković G. Đoko · University of Montenegro

In the event that Isaac Newton 1666. was not discovered the magic formula - the connection between the differential and integral calculus would probably be someone else found the Archimedes method before Dane Heiberg (1909), so that history went back similar lines.

This is also an opportunity to respond to my question I recently asked.

Does anyone know where is the papyrus (pergament) Archimedes method (Archimedes' letter to Dositej) ?

Any suggestion to represent "thinking" in mathematical language or logic?

How could we describe the act of "thinking" with mathematical tools? Which paradigm is best suited for? What does "thought" mathematically mean? Is there any alternative to the procedural (linear) conception of neural calculus?

Arturo Tozzi · University of North Texas

See also:

http://www.ncbi.nlm.nih.gov/pubmed/20068583

Riemann-Liouville fractional integral operator I_(a+ ) is defind for x>a but what happens if x=a?

Riemann-Liouville fractional integral operator Ia+(x)  is defined for x>a but what happens if x=a?

Olaniyi Samuel Iyiola · King Fahd University of Petroleum and Minerals, KFUPM

The question is interesting. I have also been working on fractional derivatives for some time now. You can find answers to some of thess issues if you have access to this book:

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and Applications of
Fractional Differential Equations, volume 204 of Mathematics Studies. Elsevier,
Amsterdam, 2006.

I use it alot together with

I. Podlubny. Fractional Differential Equations, volume 198 of Mathematics in
Science and Engineering. Acad. Press, 1999.

They contain some of the conditions under which thess kind of limits exist.

Hope it would be useful. By the way, good responses from Mohamed and Jukka.

Can anyone tell me how to check the convergence and divergence of the function like integral(1/{Sqrt(x^4-1)}), with lower limit 2 and upper limit infinity?
I also need to check for which real of p the integrals converge or diverge
Integral(|x|^p), with lower limit -1 and upper limit 1.
Anna Valkova Tomova · ....

My answer for the first definite integral and for the second indefinite integral are done by http://www.wolframalpha.com/ on the following figures. Your sincerely: Anna Tomova.

Is a histopathologic or cytologic confirmation of malignancy really needed for suspicious biliary stricture before surgery?
A biliary stricture (common bile duct or confluence) without history of calculi is often proposed for exploratory surgery due to low sensivity of bioptic procedures and the need to avoid further bilirubin rise.
Alexandre Costa · Centro Hospitalar de Vila Nova de Gaia/Esphino

Yes, pre-operative cytologic or, preferably, histologic confirmation is the ideal, and new biopsy devices may help - but still are a little too far from being reliable. Unfortunately, not always we can obtain that information and, if those procedures results are negative for malignancy, we have to rely on the clinical presentation and, to some extent, abdominal MRI-MRI cholangiopancreatography. Nevertheless, when in doubt, I think it is in the best interest of the patient to propose a surgical approach so an opportunity for a cure not be missed (albeit all the negativity of cytology or biopsy).

DO you have any scalar- and vector field modeling software recomendations?

I would like to know what standard softwares are normally used to model and visualize scalar and vector fields on a PC.

Is there any geometrical relation between fractional calculus and fractal ?
Of course there are some concepts of fractional calculus that are co-related to fractal. But how are they geometrically related ?

An emerging branch of stochastic analysis is related to the fractal nature of the Brownian motion, of its increments, and of related processes, especially Levy-type. An Ito's formula has been obtained for such processes, which helps solve the associated stochastic differential equations driven by such processes.

Is there any set-valued map which is not satisfy any properties of subdifferential calculus?

I'm looking for optimizing multivalued  vector valued function.

Sanjeev Gupta · Indian Institute of Technology Kanpur

Dear Sir my problem is associated with the nonsmooth convex vector optimization problem. I want to give an concrete example of set-valued which is not the subdifferential map.

Thanks for your kind anticipation. But my problem is not solve yet now. Kindly give some suggestion on this question.

Is there a definition of the Mittag-leffler function for variable-order fractional calculus?
E_q(t) with q a bounded function.
Hammouch Zakia · FSTE Université Moulay Ismail

Dear Professor  R.Heremann,

Thank you !

What does it mean to the fractional calculus if the constant is equal to zero in some definitions and not equal to zero in another's definition?
Riemann–Liouville fractional derivative & Caputo fractional derivative & Modified Riemann–Liouville
Is it possible to apply the Leibniz product rule for variable-order fractional derivation?
D_q(t) (f.g) = D_q(t) f .g + f.D_q(t)g
What is integral of (secx)^3 wrt x ?
If limits considered from 0 to (pi/4) and without limits also.
Anna Valkova Tomova · ....

My answer from wolfralpha.com is in the file attachment. Your sincerely; Anna Tomova

What do you think about the definition of *-differentiation which is given via "Generalized Runge-Kutta method with respect to non-Newtonian calculus?

The non-Newtonian calculi were created in the period from 1967 to 1970 by Michael Grossman and Robert Katz. In July of 1967, they created an infinite family of calculi that includes the classical calculus, the geometric calculus, the harmonic calculus, and the quadratic calculus. In August of 1970, they created infinitely-many other calculi, including the bigeometric calculus, the biharmonic calculus, and the biquadratic calculus. All of the calculi can be described simultaneously within the framework of a general theory. They decided to use the adjective "non-Newtonian" to indicate any of the calculi other than the classical calculus.

M. Grossman sent a letter to R. Katz on 21 July 2014; he said that

"As you well know, for many years lots of people, especially various pure mathematicians, claimed that our work was useless. But, despite their discouraging and sometimes arrogant comments, we always knew that NNC has considerable potential for application in science, engineering, and mathematics. And we were right!! ".

I agree strongly with you Professor.

Anatolij K. Prykarpatski · AGH University of Science and Technology in Kraków

Dear Ugur, thanks - yet I guess you are slightly mislead concerning my  in no sense critical comments (being strongly math backgrounds educational, in particular concerning   things, which are related to general of analytical nature   math problems - as you well, I am sure, understand too - that any  such a math problem is not solvable on the whole, if it does not allow some  either local  or global linearization. - Even more, the math analysis as science would not exist,  if there were no possibility to linearize a mapping in some  vicinity of a domain point! )  - As you can also easily observe,  my  clearly formulated  intentions  were completely aimed at  the pure mathematical essence of these "calculi" activity subject to this fundamental linearity property,  and concerned in no sense their , eventually,  possible applications! Why not? - -nobody knows!

All the best and regards!

What is your opinion that students find difficult in learning introductory calculus?
Michael Livshits · mathfoolery

@Andrew Messing, it looks easy until you try, and when you try you will understand how difficult it is, especially in our real world, not in the ideal world that you may be imagining while discussing this or that "improved" approach. And if you tell me how "dumping the Riemann integral" or "rigorous formulation of infinitesimals" would make introductory calculus more understandable, I will be all ears.

What is difference between greatest value and maximum value of a real valued function? When do they coincide?
Real valued functions

Dear Suruchi,

Max is the largest number within a set, whereas sup bounds the set from above. Sup may or may not be part of the set itself. If sup is part of the set, it is also the max. Examples are straightforward.

Sincerely,

George

When is the power series of an even function alternating?

This is motivated by a calculation using Fourier transforms of filters / mollifiers. Since it is such a classical question, I suspect there is a classical answer out there and would appreciate help satisfying my curiosity.

Roman Sznajder · Bowie State University

Dear Bill:

If I understand it correctly, you are expanding a function f in Taylor series at a=0, which is natural, as the function is even. As such, all odd-order derivatives of f at 0 must be zero. Thus, we deal only with f^(2k)(0) and this sequence should alternate. That's it. It seems not to exist any reasonable classification of functions with this property.

All the best,

Roman