Calculus

11
What is a good contemporary book (notation-wise) on calculus of variations?

I'm re-reading Gelfand and Fomin, which is a great classical treatise on the subject, but is there anything contemporary that is recommendable?

Thank you! It sure has a welcoming size for a primer on the subject.

4

I plan to take a formal course(s); however, given the reputation, I would love to get a jump start on learning as soon as possible!

This is wonderful! Thank you, Kwara Nantomah and Artur Sergyeyev! I will look into each of these resources.

9
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.

Dear Octav Olteanu

I resolve the integral abs(x)^(-p) dx, x from -1 to 1, this is my mistake, but the result is the same, if p<1. Exuse me, thank You. Since p is a parameter, there is no difference whether you decide the integral of (abs(x))^p  or ithe integral of (abs(x))^(-p)., but the  question concerns the definite integral of (abs(x))^p,You are right. Sincerely Yours, Anna Tomova.

I

41
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 ?

Characterizations of zero-derivative points have some interesting applications. Illustration: Consider a C1 Lipschitz function f and an interior point x* in a compact interval K of its domain. Denote by I(x*,x) the integral of f from x* to x in K and F(x*,x) =  I(x*,x) - f(x*) (x-x*). Then F'(x*,x*) = f(x*) - f(x*) = 0. The quadratic envelope characterization says that abs F(x*,x) is overestimated by 1/2 max abs f'(x) on K times square of (x - x*) for every x in K. This "third part" of the fundamental theorem of calculus compares integration with differentiation. (The first two parts of the theorem say that these processes are "inverse". )Thanks.

4
Can someone help on an exercise on the consistency of the SKI calculus?

The exercise I'm dealing with asks me to show that by adding S = K to the usual reduction rules for the SKI-calculus, one obtains an inconsistent equivalence. This must be done without using Böhm's theorem.

Now, I've found two terms (not combinators) M and N with the following properties:

M = x
N = y
M can be obtained from N by replacing one or more occurrences of S with K

From the rule S = K we thereby get that M = N, and hence that x = y. Which means that any term can be proved equal to any other.

Do you think such an answer could work? Actually, Böhm's theorem (as shown in the book I am studying) establishes that, for distinct combinators G and H, there is a combinator D such that DxyG = x and DxyH=y. So, I feel it would have been more appropriate to find a combinator D such that Dx1x2S = x_i and Dx1x2K = x_j with i , j = 1 , 2 and i ≠ j. But I have HUGE problems in finding combinators, so I have only found terms respectively reducing to one of the variables, and obtainable one from another by replacing S with K or vice versa.

Yeah I see now there's a much simpler solution. But I wondered if I had "well done" the exercise. Actually, it seems to me it works. I found two terms, M and N, the first one reducing to an arbitrary x and the second one to an arbitrary y. Moreover, M can be obtained from N by replacing an occurrence of S with K and therefore, as soon as one admits S = K as a rule, it follows that M = N. Since M = x and N = y, by transitivity one should get x = y for every x and y.

Actually, M and N, as I built them, contain other terms which are in the end not relevant to respective reducibility to x and y. But that's should not be a problem. It should be like to have two terms with variables (x, y, z, w), possibly with z = w, respectively reducing to x and y.

That's how I did the exercise...

PS: inconsistent in the sense that one can trivialize the equality relation.

5
What is new about the global injectivity in terms of the leading minors of the Jacobian ?

In communication with Mehmet Cankaya, we consider the subject.
There is known result of D. Gale and H. Nikaido, The Jacobian matrix and the global univalence of mappings,
Mathematischen Annalen 159 (1965) 81-93.

For simplicity we will consider $2$- dimensional case. Let $D=(a,b) \times (c,d)$ and suppose that $F=(f_1,f_2)$ is $C^1$ on $D$.

Suppose that (i) $D_1f_1= f_{11}\neq 0$ and $det F' \neq 0$ on $D$.

We outline a proof that (i) implies (I) $F$ is injective on $D$.

Let $c\in F(D)$ and let $F(a_1,b_1)=c$. Find solutions of equation $F(x,y)=c=(c_1,c_2)$ on $D$.

Consider (2) $f_1(x,y)=c_1$. We will  show that the set $f_1^{-1}(c_1)$ is graph of a function. Since $f_{11}=D_1f_1 \neq 0$ on $D=(a,b) \times (c,d)$ we can suppose that $f_{11}> 0$. Then (a1) the function $f_1(x,y)$ is increasing in $x$ for every fixed $y$; then there exists an open set $U$ containing $a_1$, an open set $V$ containing $b_1$, and a unique continuously differentiable function $g: V \rightarrow U$ such that $f_1(g(y),y)=c_1$, $y \in V$. Using continuity we can extend $g$ on $(c,d)$ such that $g \in C^1(c,d)$.
Thus there is a function $x=g(y)$, in a
$y \in (c,d)$ such that
$f_1(g(y),y)=c_1$, $y \in (c,d)$. By (a1), $\Gamma_g= f_1^{-1}(c_1)$, where $\Gamma_g$ is the graph of $g$ over $(c,d)$.

Hence (3) $f_{11}(g(y),y) g'(y) + f_{12}(g(y),y) =0$, $y \in (c,d)$.

Let us show that equation $h_2(y)=c_2$ has a unique solution on $(c,d)$. Contrary suppose that $h_2(y_1)=h_2(y_2)$ for $y_1\neq y_2$. Then there is
$y_0$ such that $h_2'(y_0)=0$. Hence (4) $f_{21}(g(y_0),y_0) g'(y_0) + f_{22}(g(y_0),y_0) =0$. Since $det F' \neq 0$, from (3) and (4) it follows that $g'(y_0)=0$ and
therefore $f_{12}(g(y_0),y_0) = f_{22}(g(y_0),y_0) =0$, which is a contradiction.

It seems that using a modification of the above proof we can get a corresponding version  of this  result   for convex sets in n-dimensional space.

Dear Miodrag Mateljevic, may I ask you to prepare your answer as an attached file, by using mathematical notations? Your first answer is very difficult to be followed..

1
Can you explain to me how you actually control for variables (confounding) in matrix calculus?

I would like to know in which part of the matrix calculation the control or adjustment is done.

Thanks

Please disclose the matrix or contact  H.J. Korsch or convert the matrix in to physical problem ,then it will be easy to control the variable.

B.Rath

15
Close to a Tauberian theorem?

Let a>0 and f:(0, \infty)\rightarrow (0, \infty) be fixed. The following statements are equivalent:

(i) f(x)/{x^{a+\varepsilon}} approaches 0 and f(x)/{x^{a-\varepsilon}} approaches infinity as x approaches infinity, for any \varepsilon >0.

(ii) ln f(x)/ln x approaches a as x approaches infinity.

In analogy with regularly varying functions, is it possible to find Potter bounds for the class of functions in my exercise?

11
Is the limit equal to 0?

Let f be a real function defined and differentiable on (0, \infty). If the limit of f(x), as x approaches infinity, exists and is finite, then there exists a sequence x_n , n\geq 1, such that the limit of x_n f'(x_n), as n approaches infinity, equals 0. Find a function f for which the limit of x f'(x), as x approaches infinity, is not equal to 0.

Let f(x)=cos(1/x),then x f'(x) = x(1/x^2)sin(1/x) =(1/x)sin(1/x) goes to 0, as x goes to infinity, while f(x) goes to 1 as x goes to infinity.

4
Do you know sources about problems posed using quadratic function across the ages?

I'm looking for problems about quadratic function across the ages. For example, in the  Babylonian civilization, there are problems which are related with quadratic equation. Besides that, the concept of function was developed through relation between numbers. On the other hand, in Greek Culture the problems were focusing in geometrical interpretation for solving quadratic equation. Al-Khwarizmi, in Ithe Islamic culture, was the most important scholar because he posed a formula to solve quadratic equation. I've been reading some Al-Khwarizmi's problems and his solution , which could be solved through algebraic ideas.

My principal references are

Swetz, F. J. (2012). Mathematical expeditions: Exploring word problems across the ages. JHU Press.

Kline, M. (1990). Mathematical thought from ancient to modern times (Vol.1, 2 y 3). Oxford University Press.

Boyer, C. B., & Merzbach, U. C. (2011). A history of mathematics. John Wiley & Sons.

Irving, R. (2013). Beyond the quadratic formula. Washington, D.C.: Mathematical Association of America.

Bashmakova, I. G., & Smirnova, G. S. (2000). The beginnings and evolution of algebra (No. 23). Cambridge University Press.

My focus is to study problems posed about quadratic function in some scenes of history of mathematics.

Please, could you help me in this endeavour?

Hello Carlos,

You could be interested in

BEDNARZ, N., KIERAN, C., LEE, L. (1996) Approaches to algebra: Perspectives for Research and Teaching. Dordrecht: Kluwer, 364 p.

This is a collection of individual contributions.

Louis Charbonneau and Luis Radford have some interesting points of view.

11
What theorem helps solving the following exercise?

A function f satisfies f(0)=0 and, for any x\geq 0, f''(x) exists and is finite. Prove that, for any x \geq 0, there exists c\in (0, x) such that xf'(x) -f(x) = x2 f''(c)/2.

The most general statement (I think) in this direction is as follows:

If f  has an n+1 derivative (n>0) with f(0)=0, then for each nonzero x there is some real c between 0 and x such that

(-1)nf(x)+(-1)n-1xf'(x)+...+(-1)n-kxkf(k)(x)/k!+...+xnf(n)(x)/n!=xn+1f(n+1)(c)/(n+1)!

This is proved exactly as in my previous post, using Roll's theorem.

23
What differential equation has a solution of the form F(x)=(1+1/Kx)^-2?

I'm aware that the solution to xd/dxF(x)=F(x)(1-F(x)) is (1+1/Kx)-1, where K is an arbitrary constant.  I'm looking for a similar equation with solution (1+1/Kx)-2.  It's been a long time and I'm a bit rusty with substitutions and so forth, and I thought some mathematician on RG might have the answer off the top of his or her head.  Thanks in advance for the help.

OK, thanks all.  Moving on, I have posted a phase 2 question.  See link below.

99+
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.

Michael, I say sincere thanks for your valuable advice. Actually, I have been working on finite geometries for more than forty years. It is my fault that my results have drawn so little attention.  One of my first papers appeared in 1983, in Italian: 'Una generalizzazione del concetto di piano proiettivo' co-authored with the late Ferenc Kárteszi. However, my studies have been based on the axiomatic system mentioned in my previous post. Akin  in  many respects to classical projective planes, my 'projective spheres' show many different characteristics as well. For example, associative elements correspond to altitudes of a triangle, but the operation itself is not associative. Or, there are two non-isomorphic finite spheres with 21 elements, in contrast with only one existing in projective planes (and I found four non-isomorphic spheres with 91 elements): You are perfectly right: Finite models are of great help in studying many interesting features, including extension of calculus into a different axiomatic environment. Thank you for your advice anyway.

2
How would it be possible to combine real figures, complex figures and rational figures in order to build a better structural model?
Utilising real figures with a view to linearise equations for production, complex figures in order to explain production, and re-production instead, and at least rational figures to have a transversal view with a possibility to make statistical tests whenever needed.

Yes that is really what I mean

With kindest wishes

the answer to my question will be : a linear approach to the economy before to get one's own capital access should mean real figures , then a matricial ( say marxian etc. ) approach would mean complex resolution , and at the end when we include conventions we get to rational numbers , and a new linear approach through the mask of capital

I am not really sure it could investigate in the same time evolutionary games with references to biology mathematics ( for instance adaptation + evolution games , but it is possible ;-)

14
Does anyone know how to solve this integral ?

the integral is attached here

It must be noted that Ei is the exponential integral

regards

Do you talk about series representation of the exponential integral ?

38
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?

Ooops.  Just read some of the earlier postings in this thread and Andrew has already made these points, even quoting me.   But to repeat (unnecessarily) I have always hated to use the phrase "improper integral" that somehow implies that any decent self-respecting integral would only treat of bounded functions and that unbounded functions represent some kind of pathology.  (Sorry-not used to the way this site works.)

17
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?

With modern limit theory and skill, can we really change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant?

This is a “strict proven modern version of ancient Zeno’s Paradox by the limits”. The more we try to explain the more doubts be aroused and we feel more helpless.

So, why don't we teach limits in calculus?!

Regards, Geng

6
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 ?

Thanks Prof. Stoica.

7
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 .

6
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?

A nice paper

i will  go through it

9
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.

39
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.

37
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.

2
What is the best reason for Conformable fractional derivative is not related to standard fractional derivatives like Riemann-Liouville or Caputo?

.

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

5
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?

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.

8
Does any one know how to compute definite integrals involving zero oreder first kind modified bessel function ?

i want to compute integral of f(x) which is defined as

f(x)=e(-ax)* I_0(b*(x^ 0.5))

where

I_0 is zero order first kind modified bessel function.

and, the integral is done from x=0 to x=+\infinity

i know the answer in the case of zero order first kind bessel finction (J_0), but im not sure to use the transformation from unmodified to modified. i would be really grateful to have your comments.

The general method of "analytic continuation" lets you take the function J_0(by) into I_0(by) . In effect you replace b by ib in the answer.

26
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?

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.

88
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?

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) ?