# Calculus

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
Kai Diethelm · Technische Universität Braunschweig
No, you can't do that. Actually you cannot even do that if q(t) is a constant function, i.e. in the more standard case of a fractional derivative with non-variable order. Just try an example: f(t) = t, g(t) = t, q(t) = q = constant with 0 < q <= 1 (using 0 as the starting point of the fractional operator). In this example it doesn't matter if you use Riemann-Liouville or Caputo operators. In either case the left-hand side of your equation is D^q (t^2) = \Gamma(3) t^{2-q} / \Gamma(3-q) and the right-hand side is 2 t D^q (t^1) = 2 t \Gamma(2) t^{1-q} / \Gamma(2-q) = \Gamma(3) t^{2-q} / \Gamma(2-q) so you can see that the fractions have the same numerator but different denominators (except, of course, for the non-fractional case q = 1 when \Gamma(3-q) = \Gamma(2) = 1 = \Gamma(1) = \Gamma(2-q)). The correct formula is given, e.g., in my book "The analysis of fractional differential equations", Springer, Berlin, 2010. Theorem 2.18 gives the result that you need if your differential operators are of Riemann-Liouville type; Theorem 3.17 is the corresponding result if you want to use operators of Caputo type.
What is your opinion that students find difficult in learning introductory calculus?
Christopher Landauer · The Aerospace Corporation
It my experience as a student and teacher, it was usually the teacher's inability to understand how many different ways there are for understanding calculus - if you can learn what style of learning the student uses, there is almost certain to be a corresponding style of explanation of calculus
Could anyone tell me what the mathematical definition/expression for a polyhedral convex set and polyhedral cone in general Euclidean space is?
I have searched through the internet and found "Convex Polyhedron", "Convex Polytope". Are they the same thing? Plus it would be great if there is some elaboration on the properties of such object.
Aria Tsam · Aristotle University of Thessaloniki
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?
Qefsere Doko Gjonbalaj · University of Prishtina
The idea of a limit underlies the various branches of calculus. It is therefore appropriate to begin the study of calculus by investigating limits and their properties. The special type of limit that is used to find tangents and velocities gives rise to the central idea in differential calculus, the derivative. Therefor we should teach limit first then differentiation.
Can anyone help with this problem?
If B(x,1/k) is open ball in a Banach space M and there exist open set G_k in M such that G_k \subset B(x,1/k) for all natural number k. What can we say about the family {G_k} when k \to \infty? is G_k stay open ? Can we say the family converges to x?
Milen Ivanov · Sofia University "St. Kliment Ohridski"
And what is then $\limsup_k G_k$? If it is the set of all cluster points of all sequences $(x_k)$ s.t. $x_k \in G_k$, then $$\limsup_k G_k = \{x\}$$ simply because your condition implies $\|x-x_k\| \le 1/k$ for any such sequence.
• Hammouch Zakia asked a question:
Is there a definition of the Mittag-leffler function for variable-order fractional calculus?
E_q(t) with q a bounded function.
Can we define the Cartesian product of two functions in a different way from the classical one that is {(fxg)(a,b)=(fa,gb)}?
The properties of their product should not depend totally on the properties of the two functions
Jean-Pierre Magnot · Académie de Clermont-Ferrand
Basically, if $f : X \rightarrow Y$ and $g : X \rightarrow Z$, $f \times g$ is a map $X \rightarrow Y \times Z$ defined by $f \times g (x) = (f(x); g(x))$. The cartesian product of two functions is defined with the cartesian product of the image sets. The property that you mention is a consequence in the following sense: $f$ and $g$ are here acting on two domains $X$ and $X'$ and the link with my definition got saying $f(a,b) = f(a)$ and $g(a,b) = g(b)$ (constant maps projectionwise). Conversely, if you have your definition with $X = X'$, tatke the diagonal $\Delta(X \times X) \sim X$, and you get mine. The two ones are logically equivalents, it is just a question of primary definition, or if you prefer, of vocabulary. But to my knowledge, mine is the most used.
How can tangent,curvature and torsion be related to the differential geometric concepts of topology, manifolds, etc?
For example, Frenet–Serret formulas are considered as the generalization of higher dimensional Euclidean spaces. These formulas define a non-inertial coordinate system in terms of tangent,curvature and torsion. However understanding these quantities by relating them to the topological spaces, manifolds, etc is required for better visualisation of non linearities.
Viswanath Devan · Indian Institute of Technology Guwahati
@Rogier Brussee & Prof.James Peters: Thank you very much sirs.. @Prof.James: Sir I thanks for the copy of Theodore Shifrin..I find it to be a nice book and the way 'parametrization' has been explained in a simple way makes the book interesting. However what I am perplexed with is that: (a) The book deals with curves and surfaces and now I want to advance the concepts to higher dimensional manifolds, (b) I need a basic understanding of topological spaces the way I can relate to a curve or a surface and (c) Finally it will be really helpful if I can relate to any examples in application form. Thanks n regards.
Who first defined orthogonal polynomials?
It is known that Legendre and Chebyshev have contributions in the field.
Yamilet Quintana · Simon Bolívar University
The explicit formula of the Lagrange interpolating polynomial was first published by Waring (1779), rediscovered by Euler in 1783, and published by Lagrange in 1795 (cf. Jeffreys, H. and Jeffreys, B. S. "Lagrange's Interpolation Formula." §9.011 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 260, 1988.) With respect to the general theory of orthogonal polynomials really started with the investigations of Chebyshev and Stieltjes (see for instance, The impact of Stieltjes' work on continued fractions and orthogonal polynomials by Walter Van Assche, and the references therein. http://arxiv.org/pdf/math/9307220v1.pdf)
Anyone know how to add to Archimedes' calculus and square root methods?
Http://planetmath.org/archimedescalculus
Milo Gardner · California State University, Fullerton
footnote: Fibonacci’s square root of 17 method was appropriately cited as used by Galileo though not properly analyzed in every detail. Fibonacci guessed (4 + 1/8)^2 = (17 + 1/64) , and Fibonacci reduced the estimated 1/64 error by finding an inverse proportion:1/64 x 8/66 = 1/528 which meant (4 + 1/8 - 1/528)^2 = (2177/528)^2 = 17.000003 is accurate to (1/528)^2 per Archimedes and not Newton. III CONCLUSION Unit fraction square root was formalized by 2050 BCE and used by Egyptians, Greeks, Arabs, medieval scribes and as late as Galileo. The method estimated irrational square roots of N by 1-step, 2-step, 3-step and 4-steps methods. Step 1 guessed quotients (Q) and remainders (R) = n/(2Q) with n = (N - Q^2). Step 2, 3, and 4 reduced error 1, 2 and 3 associated with the previous step by dividing by 2(Q + R).
What is a good way to teach the linear equation to non mathematics students?
Suppose we want to teach the Linear Equation to other student who have don't know the mathematics. They do not have mathematics background.
Qefsere Doko Gjonbalaj · University of Prishtina
For students who don't have mathematics background "Painless Algebra" by Lynette Long is very good textbook.
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?
Joseph Uphoff · The Institute Of Martial Arts, Journal Of Regional Criticism
Things such as these can be found in Archeology Magazine or Biblical Archeology Review. In the library, lots of old books, some in foreign languages, begin each paragraph with a special and elaborate capital letter that gives prescience of the content of the paragraph.
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
Enrico Scalas · University of Sussex
The book by Podlubny is a nice starting point, but one could also use these lecture notes by Mainardi and Gorenflo: http://www.fracalmo.org/download/fmrg10.pdf The value of the fractional derivative of a constant just depends on the definition of the pseudo-differential operator. I am not sure there is a simple dynamic interpretation.
How does one find the values of coefficients from a system of the following equations?
We have the following two equations, where A_1, A_2, B_1 and B_2 are the coefficients that we are interested in to find the values. We need to calculate them so that, for example, p_1(R) and A_1 p_1(q) + B_1 p_2(q) represent the same function. We could do this by requiring that, at some point R_0, both sides of this equation have the same value and the same derivative.
Oscar Chavoya-Aceves · Independent
I am not sure that your problem is well posed.
What should we teach under the derivative concept?
What should be the concepts underlying the derivative subject in calculus education? What each concept weighs in importance?
Sevgi Sofuoğlu · Middle East Technical University
Thank you very much all and especially Richard Kolacinski. My concern is not how to teach derivative but what to teach, and what are key concepts. I seek for people who teach calculus, or wrote a calculus book or interested in calculus education. I also think we miss the geometric part of it.
Convergence and divergence of the following integrals?
By using comparison tests, how we can explain the convergence and divergence of these two integrals?
Tauqeer Hussain Shah · Linnaeus University
@ Anton Boitsev Yes, For second case we have \neq [\alpha_1; \alpha_2]. Please rectify me if I am wrong, in the attached file you can see what I interpret you.
For what condition will the inequalities be true?
In the attached file, there are two inequalities. When will these two different inequalities be true for real value of w?
Tauqeer Hussain Shah · Linnaeus University
Yes, you are right I do agree with you.
We know that the integral of f(x)=1/x from 1 to infinity doesn't exist (it tends to be infinite).
This means that the area of the region bound by the graph of f and x-axis and x=1 is infinite. But we know that the volume of the solid obtained by rotating the above mentioned area around the x-axis is finite. How do we explain this paradox in calculus?
There is no paradox. One thing is the surface area of a solid which, in this case is infinity, and other thing is its volume which is finite. You can enclose a finite volume by a surface with an area as large as you want (with the obvious physical limitations). In fact this is the theoretical foundation of heat dissipators or heat radiators which have a very large area but a finite (even very small) enclosed volume. If you want to see another simple equivalent example in 2D (plane region with a finite area but a border of infinite length) take a look at the Koch snowflake curve (http://en.wikipedia.org/wiki/Koch_snowflake)
e^(z_1)+e^(z_2)=0 if and only if Re(z_1)=Re(z_2) and Im(z_2-z_1)=pi(2n+1), where z_1, z_2 are complex numbers. Is there a similar statement for the sum of 4 exponents e^(z_1)+e^(z_2)+e^(z_3)+e^(z_4)=0 ?
This question is connected with the oscillations of differential equations. Since every solution of the linear equation with constant coefficients is the linear combination of exponents this combination is oscillatory if and and only if it has infinitely many zeros.
Gro Hovhannisyan · Kent State University
Hey guys, recently I published a paper (you may find it here) about oscillations of n-th order equations, where I gave thanks to everybody for a good discussion of my questions.
I need to evaluate the definite integral in the attached pdf file. Any suggestion?
The definite integral of the first derivative of a function is trivial. What if the argument is a non integer power of the first derivative of a function? any reduction formula? Thanks
Miroslav Pavlović · University of Belgrade
In is not clear what "evaluate" means. For example, if f(x)=x^3+x, then \int [f '(x)]^{1/3} dx is not an elementary function.
Is Taylor expansion related to Helmholtz decomposition?
The Taylor expansion of a vector field $f(x)$ to the order of one is $$f(x)=f(x_0)+Jf(x_0)\cdot\Delta x+o(\Delta x)$$ where $Jf$ is Jacobian of the vector field and $\Delta x=x-x_0$. Suppose we decompose it into symmetric and skew-symmetric part such that $$S=\frac{Jf+(Jf)^T}{2}$$ $$A=\frac{Jf-(Jf)^T}{2}$$ Then $$f(x)\approx f(x_0)+S\Delta x+A\Delta x$$ Could we regard $f(x_0)$ as a translation, $S\Delta x$ as an expansion or a contradiction along axes of eigen directions for it resembles strain tensor and $A\Delta x$ a rotation since skew-symmetric matrix is infinitesimal of rotation matrix? If it is correct, I'm a bit confused with, say operator $A$, what object it acts on? Otherwise, does this decomposition have any practical meaning in terms of vector field or velocity field?
Fractional Differentiation?
How do we physically interpret d^n f(x)/dx^n, where n is a fraction. And what kind of physical systems/problems generally give rise to such differentiations?
Rogier Brussee · Hogeschool van Utrecht
It is easier to understand if you do a Fourier transform first . Then a fractional derivative becomes just a fractional power of the wavenumber (in fact fractional derivatives are defined this way). More generally, we can take a not necessarily linear or polynomial function of the wave number. Non linear functions of the wavenumber occur naturally as dispersion relations. For example in solid state physics the momentum of a phonon is almost always a non linear function of its wavenumber.
What is the difference in usage, for writing the differential before or after the integrand?
In many (or most) calculus or introductory books of analysis, when an integral (definite or indefinite) is written, one mostly finds the notation, where the integral symbol (with/without limits) is followed by a function known as the integrand, and at the end the symbol of the differential of the variable you are integrating into. However, in more than one paper of mathematical physics, I have met instances where the integrand and the differential operator exchange places (Jizba et al, 2013, Eq. 2 for example, included in this question). What is the difference in usage of either case?
Nicolas Guarin-Zapata · Purdue University
I think that is just another way of write it that is pretty common in some physics literature. For example: http://en.wikipedia.org/wiki/Density_functional_theory
Can some one advise on how to integrate such kinds of functions?
In the attached image, its a kind of integral. I would like to solve this integral.
Ehsan Karamad · University of Toronto
The indefinite integral seems hard. However, the definite integral in (-inf, inf) can be done by taking "w" as a complex number and assuming integration over a half circle with the real axis as diameter and the arc being some "R e^{i \theta}" where "R->inf". To solve it then, you will need the residual values which is straightforward given the fact that you already know the poles.
What is bilateral Laplace Transform?
How to define bilateral laplace transform? What is application of bilatrral LT? Can anyone suggest good material for studying bilateral Laplace Transform?
Narendrakumar Dasre · Ramrao Adik Institute of Technology
Thanks friends!!!
• Ariel M. Salort asked a question:
Does anybody know if there exists a min-max characterization for the eigenvalues of the fractional laplacian?
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What are the convergence and divergence of the integral(|dF/dw|^2|dw/dz|dw)?
I would like to know about the convergence and divergence of the integral described above. Where dF/dw and dw/dz are the conformal transformation from one plane to another plane. In the attached file, one can see the integral.
Shiuh-Hwa Shyu · WuFeng University
Don't quite understand what you mean the singularity. Once you derive to the results, they will tell you whether x=0 is an issue or not. Be aware that the integral you write is an even function. So, it can be written as 2* integral(|x|^p dx) from 0 to +1.
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.
Victor Shestopal · Institute for Theoretical and Experimental Physics
P>-1