# Calculus

5
What's the most accurate way to estimate derivatives from a table?

Suppose we have a scalar function of one variable f(t) and a series of points (ti,xi=f(ti)), where ti - ti-1 = \Delta t = Const. What is the most accurate way of estimating f'(ti) at any given ti?

The most intuitive way would be f'(ti)=(xi+1-xi)/(\Delta t), while f'(ti)=(xi+1-xi-1)/(2 * \Delta t) is more accurate as it eliminates first-order error. Is there any better idea? And which textbook are similar topics covered in?

Assuming your tabular values are sufficiently accurate, an obvious idea is to interpolate all the data in your table, e.q. by a polynomial p(t) or some other interpolating function i(t). Then, the derivative f'(t) can be approximated by p'(t) or i'(t), respectively. For noisy data, a least square approximation a(t) of the (smoothed) data by a lower order polynomial may be preferable, and the derivative then is approximated by a'(t). The more you know about the function f or the data points in the table, the better one may be able to determine a good class of interpolating or approximating  functions. Thus, it may be prefererable to use rational interpolants for functions with poles near to the interval of arguments in the table, or some spline interpolant.  A general answer which approach is "best" is very difficult to find.

I recommend to read  the classic book

Applied Analysis
by Cornelius Lanczos

15
How can I use the fundamental theorem of calculus to describe roots of functions?

Consider a C2 function f of the single variable and an interior point x* in a compact interval of its domain. How to use the fundamental theorem of calculus to describe f(x*)?

Here is a complete formulation of the quadratic envelope property (QEP). For a function of the single variable x, QEP is an inequality in two variables x and x*. It recovers some of the old and introduces new properties of f.

Theorem Consider a C1 function f of the single variable x  and a point x* in the interior  of a compact interval I in its domain. For every x in I the following holds

(QEP)    abs [F(x,x*) - f(x*)(x - x*)] is overestimated by c (x - x*)^2.

Here F(x,x*) = integral of f from x* to x and constant  c = half of max [abs f"(t) : t in I].

Proof: It follows from the fundamental theorem of calculus and the characterization of  zero-derivative point.

Let us call a relation between x and x*, that follows from (QEP), "augmentor" of f on I.

Illustrations: (a) Trivial augmentor. For f(x) = x  and any compact interval I, (QEP) yields, after multiplication by 2,  abs (x^2 - x*^2 - 2x* (x - x*) <= (x - x*)^2 which is     (x - x*)^2 <- (x - x*)^2.

(b) For f(x) = cos x, an augmentor on I = [- pi, pi] says that                                  abs [sin x - sin x* - cos x* (x - x*)] is overestimated by half (x - x*)^2 for every  x, x*.

In particular, at x* = 0, it says that abs (sin x - x) is overestimated by half x^2 for every x. etc.

11
What is Monotonus function?

When I study the properties of autonomous differential equation, they said that the solutions to the autonomous equations are monotonus functions.

I can't undestand what is monotonus function?

what is the difference between monotonic function and monotonus function?

Thank you for all your suggestions.....

53
“Potential infinitesimal” or “actual infinitesimal” ?

There are “potential infinite” and “actual infinite” in our present infinite related classical science theory system at least since Zeno’s time, so we can not avoid (logically have) “potential infinitesimal” or “actual infinitesimal” in mathematics. But the confused “infinite”, confused “potential infinite--actual infinite” have being creating a lot troubles (such as infinite related paradoxes) when we are treating (cognizing) some “infinite things” in mathematics. The Harmonic Series Paradox is a typical example:

1, no one cares what y---->0 (Un---->0)in Harmonic Series means-------“potential infinitesimal” or “actual infinitisimal”and where do they locate in number system,

2, no one cares what differences between y---->0 (Un---->0) in Harmonic Series and y---->0  (dx ---->0) in calculus,

3, no one cares whether or not we should have different ways when treating “potential infinitesimal” or “actual infinitesimal”.

4, no one cares whether or not we can really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000000000000,… from the Un--->0 Harmonic Series and no one cares the exact (how many) items being consumed for just the first 10000000000000000000 from Harmonic Series.

The suspended infinite related paradoxes have been disclosing the fundamental defects of “infinite theory, limit theory, number theory” in present infinite related science theory system.

Dear Dr. Panchatcharam Mariappan, thank you.

4
Are there mathematics self-efficacy scales developed in university level (various sub-topics)?

The sub-topcis of mathematics, I mean self efficacy scale in diffeential equations or self efficacy scale in rings.

I am sorry, I don't have yet. May be next time. Thank
40
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?

If we consider brain (meant as memory device) as an elaborator of what you call <<learned schemas>>, the domain of these <<mental actions>> should be a topographic space, being that a schema is not simply a linear list but resembles a graph, with plural conditions to satisfy for parallel nodes.
As you say, senses animate a cascade of associations, so I find logic that, for each step, external stimuli and stored memories (input signals) are linked/applied to a schema in order to test their "meaning of association": initial input signal joins with some aspect (node) of some schema, then the output joins with "neighboring" nodes [and schemas - I'm inspired by non-synaptic transmission between neighboring neurons], whose outputs join with respectively near nodes and schemas, and so on - each step inhibits some routes; finally, after enough iterations, some schema proves to be stable (enough internal conditions - nodes' trigger point - are satisfied and there is no signal dispersion with close schemas) - it might be precisely what is <<loaded into our conscious work space>>. Maybe, brain as memory device is a simplifier: it trains itself to keep track exclusively between stable schemas - what we may call "architecture of memory".
Furthermore, what if I call these schemas... memes?

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

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.

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 ;-)

39
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?

One of the most important improper integrals are those involving Gamma and Beta functions. The relationship between these two functions, as well as the basic properties of Gamma function further lead to the well known important integrals on R from [x^(2n)*exp(-x^2)], and integral on [0, infinity) from [(x^p)*exp(-x^2)), p>=0. Passing to several variables, the most simple improper integrals are those on Cartesian products of unbounded intervals, involving the moments of the functions exp(-Sum x_j, j =1,...,n), exp( - Sum (x_j)*2). Here Fubini's theorem is the ingredient. Going back to the one variable case, the properties of the Gamma and related functions are basic in probability theory, statistics, physics. In the case of several variables, some functions mentioned above serve as weights or basic elements in various concrete problems, such as polynomial approximation on unbounded subsets, moment problems, etc.

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.

40
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?

Definitely, first we introduce the limit concept of a function. A function may have have one or more limits. Next verify continuity of a function and then we shall teach derivative of a function. Moreover limit is related to neighborhood concept.

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.