## Equidistant surface

by: Maple 15

Example of the equidistant surface at a distance of 0.25 to the surface
x3
-0.1 * (sin (4 * x1) + sin (3 * x2 + x3) + sin (2 * x2)) = 0
Constructed on the basis of universal parameterization of surfaces.

equidistant_surface.mw

## Numerov algorithm in Maple

Maple 18

Hi there, fellow primers, it's good to be back after almost 5 years! I just want to share a worksheet on Numerov's algorithm in Maple using procedures as I've recently found out that google could not find any Maple procedure that implements Numerov's algorithm to solve ODEs.   numerov.mw   Reference.pdf

## Computer Algebra in Theoretical Physics: the IOP...

by: Maple

Below is the worksheet with the whole material presented yesterday in the webinar, “Applying the power of computer algebra to theoretical physics”, broadcasted by the “Institute of Physics” (IOP, England). The material was very well received, rated 4.5 out of 5 (around 30 voters among the more than 300 attendants), and generated a lot of feedback. The webinar was recorded so that it is possible to watch it (for free, of course, click the link above, it will ask you for registration, though, that’s how IOP works).

Anyway, you can reproduce the presentation with the worksheet below (mw file linked at the end, or the corresponding pdf also linked with all the input lines executed). As usual, to reproduce the input/output you need to have installed the latest version of Physics, available in the Maplesoft R&D Physics webpage.

 Why computer algebra? ... and why computer algebra? We can concentrate more on the ideas instead of on the algebraic manipulations   We can extend results with ease   We can explore the mathematics surrounding a problem   We can share results in a reproducible way
 Representation issues that were preventing the use of computer algebra in Physics Notation and related mathematical methods that were missing: coordinate free representations for vectors and vectorial differential operators, covariant tensors distinguished from contravariant tensors, functional differentiation, relativity differential operators and sum rule for tensor contracted (repeated) indices Bras, Kets, projectors and all related to Dirac's notation in Quantum Mechanics   Inert representations of operations, mathematical functions, and related typesetting were missing:   inert versus active representations for mathematical operations ability to move from inert to active representations of computations and viceversa as necessary hand-like style for entering computations and textbook-like notation for displaying results   Key elements of the computational domain of theoretical physics were missing:   ability to handle products and derivatives involving commutative, anticommutative and noncommutative variables and functions ability to perform computations taking into account custom-defined algebra rules of different kinds (commutator, anticommutator and bracket rules, etc.)

Examples

 The Maple computer algebra environment

Classical Mechanics

 Inertia tensor for a triatomic molecule

Classical Field Theory

 *The field equations for the  model
 *Maxwell equations departing from the 4-dimensional Action for Electrodynamics
 *The Gross-Pitaevskii field equations for a quantum system of identical particles

Quantum mechanics

 *The quantum operator components of   satisfy
 Quantization of the energy of a particle in a magnetic field

Unitary Operators in Quantum Mechanics

 *Eigenvalues of an unitary operator and exponential of Hermitian operators

Properties of unitary operators

Consider two set of kets  and , each of them constituting a complete orthonormal basis of the same space.

One can always build an unitary operator  that maps one basis to the other, i.e.:

 *Verify that  implies on
 *Show that is unitary
 *Show that the matrix elements of  in the  and   basis are equal
 Show that  and have the same spectrum

Schrödinger equation and unitary transform

Consider a ket  that solves the time-dependant Schrödinger equation:

and consider

,

where  is a unitary operator.

Does  evolves according a Schrödinger equation

and if yes, which is the expression of ?

 Solution

Translation operators using Dirac notation

In this section, we focus on the operator

 Settings
 The Action (translation) of the operator  on a ket
 Action of  on an operator

General Relativity

 *Exact Solutions to Einstein's Equations

*"Physical Review D" 87, 044053 (2013)

Given the spacetime metric,

a) Compute the Ricci and Weyl scalars

b) Compute the trace of

where  is some function of the radial coordinate,  is the Ricci tensor,  is the covariant derivative operator and  is the stress-energy tensor

c) Compute the components of the traceless part of   of item b)

d) Compute an exact solution to the nonlinear system of differential equations conformed by the components of   obtained in c)

Background: paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by P. Fiziev.

 a) The Ricci and Weyl scalars
 b) The trace of
 b) The components of the traceless part of
 c) An exact solution for the nonlinear system of differential equations conformed by the components of

*The Equivalence problem between two metrics

From the "What is new in Physics in Maple 2016" page:

 In the Maple PDEtools package, you have the mathematical tools - including a complete symmetry approach - to work with the underlying [Einstein’s] partial differential equations. [By combining that functionality with the one in the Physics and Physics:-Tetrads package] you can also formulate and, depending on the metrics also resolve, the equivalence problem; that is: to answer whether or not, given two metrics, they can be obtained from each other by a transformation of coordinates, as well as compute the transformation.
 Example from: A. Karlhede, "A Review of the Geometrical Equivalence of Metrics in General Relativity", General Relativity and Gravitation, Vol. 12, No. 9, 1980
 *Equivalence for Schwarzschild metric (spherical and Krustal coordinates)

Tetrads and Weyl scalars in canonical form

Generally speaking a canonical form is obtained using transformations that leave invariant the tetrad metric in a tetrad system of references, so that theWeyl scalars are fixed as much as possible (conventionally, either equal to 0 or to 1).

Bringing a tetrad in canonical form is a relevant step in the tackling of the equivalence problem between two spacetime metrics.

The implementation is as in "General Relativity, an Einstein century survey", edited by S.W. Hawking (Cambridge) and W. Israel (U. Alberta, Canada), specifically Chapter 7 written by S. Chandrasekhar, page 388:

 Residual invariance Petrov type I 0 1 0 none Petrov type II 0 0 1 0 none Petrov type III 0 0 0 1 0 none Petrov type D 0 0 0 0 remains invariant under rotations of Class III Petrov type N 0 0 0 0 1 remains invariant under rotations of Class II

The transformations (rotations of the tetrad system of references) used are of Class I, II and III as defined in Chandrasekar's chapter - equations (7.79) in page 384, (7.83) and (7.84) in page 385. Transformations of Class I can be performed with the command Physics:-Tetrads:-TransformTetrad using the optional argument nullrotationwithfixedl_, of Class II using nullrotationwithfixedn_ and of Class III by calling TransformTetrad(spatialrotationsm_mb_plan, boostsn_l_plane), so with the two optional arguments simultaneously.

The determination of appropriate transformation parameters to be used in these rotations, as well as the sequence of transformations happens all automatically by using the optional argument, canonicalform of TransformTetrad .

 >
 (7.4.1)
 Petrov type I
 Petrov type II
 Petrov type III
 Petrov type N
 Petrov type D

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

## Rotational motion mechanism

by: Maple 15

Rotational motion mechanism with quasi stops
02rep.pdf
DIMA.mw

## Equivalence problem in General Relativity

by: Maple

Formulating and solving the equivalence problem for Schwarzschild metric in a simple case

In connection with the digitizing in Maple 2016 of the database of solutions to Einstein's equations of the book Exact Solutions to Einstein Field Equations. I was recently asked about a statement found in the "What is new in Physics in Maple 2016" page:

 In the Maple PDEtools package, you have the mathematical tools - including a complete symmetry approach - to work with the underlying [Einstein’s] partial differential equations. [By combining that functionality with the one in the Physics and Physics:-Tetrads package] you can also formulate and, depending on the metrics also resolve, the equivalence problem; that is: to answer whether or not, given two metrics, they can be obtained from each other by a transformation of coordinates, as well as compute the transformation.

This question posed is a reasonable one: "could you please provide one example?" This post provides that example.

First of all the existing science behind: in my opinion, the main reference regarding the equivalence problem is at the paper "A Review of the Geometrical Equivalence of Metrics in General Relativity", General Relativity and Gravitation, Vol. 12, No. 9, 1980, by A. Karlhede (University of Stockholm). This approach got refined later by others and, generally speaking, it is currently know as the Cartan-Karlhede method, summarized in chapter 9.2 of the book Exact Solutions to Einstein Field Equations. whose solutions were all digitized within the Physics and DifferentialGeometry packages for Maple 2016. This method of Chapter 9.2 (see also Tetrads and Weyl scalars in canonical form, Mapleprimes post), however, is not the only approach to the problem, and sometimes simpler methods can handle the problem faster, or just in simpler forms.

The example worked out below is actually the example from Karlhede's paper just mentioned, on pages 704 - 706: "Show that the Schwarzschild metric and its form written in terms of isotropic spherical coordinates are equivalent, and derive the transformation that relates them". Because this problem happens to be simple for nowadays computer algebra, below I also tackle it modified, slightly more difficult variants of it. The approach shown works for more complicated cases as well.

Below we tackle Karlhede's paper-problem using: one PDEtools command, the Physics:-TransformCoordinates, the Physics:-Weyl command to compute the Weyl scalars and the Physics:-Tetrads:-PetrovType to see the Petrov type of the metrics involved. The transformation resolving the equivalence is explicitly derived.

 >
 (1)

To formulate the problem, set first some symbols to represent the changed metric, changed mass and changed coordinates - no mathematics at this point

 >
 (2)

Set now a new coordinates system, call it Y, involving the new coordinates (in the paper they are represented with a tilde on top of the letters)

 >
 (3)

According to eq.(7.6) of the paper, the line element of Schwarzschild solution in isotropic spherical coordinates is given by

 >
 (4)

Set this to be the metric

 >

Check it out

 >
 (5)

In connection with the transformation used further below, compute now the Petrov type and the Weyl scalars for this metric, just to have an idea of what is behind this metric.

 >
 (6)
 >
 (7)

We see that the Weyl scalars are already in canonical form (see post in Mapleprimes about canonical forms): only  and the important thing: it depends on only one coordinate,  .

Now: we want to see if this metric (5) is equivalent to Schwarzschild metric in standard spherical coordinates

 >
 (8)

The equivalence we want to resolve is regarding an arbitrary relationship between the masses used in (5) and (8) and a generic change of variables from X to Y

 >
 (9)

Using a differential equation mindset, the formulation of the equivalence between (8) and (5) under the transformation (9) is actually simple: change variables in (8), using (9) and the Physics:-TransformCoordinates command (this is the command that changes variables in tensorial expressions), then equate the result to (5), then try to solve the problem for the unknowns ,  and .

We note at this point, however, that the Weyl scalars for Schwarzschild metric in this standard form (8) are also in canonical form of Petrov type D and also depend on only one variable, r

 >
 (10)
 >
 (11)

The fact that the Weyl scalars in both cases ((7) and (11)) are in canonical form (only  ) and in both cases this scalar depends on only one coordinate is already an indicator that the transformation involved changes only one variable in terms of the other one. So one could just search for a transformation of the form  and resolve the problem instantly. Still, to make the problem slightly more general, consider instead a generic transformation for r in terms of all of

 >
 (12)
 >
 (13)

Transform the  coordinates in the metric (because of having used PDEtools:-declare, derivatives of the unknowns R are displayed indexed, for compact notation)

 >
 (14)

Proceed equating (14) to (5) to obtain a set of equations that entirely formulates the problem

 >
 (15)

This problem, shown in Karlhede's paper as the example of the approach he summarized, is solvable using the differential equation commands of PDEtools (in this case casesplit) in one go and no time, obtaining the same solution shown in the paper with equation number (7.10), the problem actually admits two solutions

 >
 (16)

By all means this does not mean this differential equation approach is better than the general approach mentioned in the paper (also in section 9.2 of the Exact Solutions book). This presentation above only makes the point of the paragraph mentioned at the beginning of this worksheet "... [in Maple 2016] you can also formulate and, depending on the the metrics also resolve, the equivalence problem; that is: to answer whether or not, given two metrics, they can be obtained from each other by a transformation of coordinates, as well as compute the transformation."

In any case this problem above is rather easy for the computer. Consider a slightly more difficult problem, where . For example:

 >
 (17)

Tackle now the same problem

 >
 (18)

The solutions to the equivalence between (17) and (5) are then given by

 >
 (19)

Moreover, despite that the Weyl scalars suggest that a transformation of only one variable is sufficient to solve the problem, one could also consider a more general transformation, of more variables. Provided we exclude  (because there is  around and that would take us to solve differential equations for , that involve things like ), and also to speed up matters let's remove the change in , consider an arbitrary change in r and t

 >
 (20)
 >
 (21)

So our transformation now involve two arbitrary variables, each one depending on all the four coordinates, and a more complicated function . Change variables (because of having used PDEtools:-declare, derivatives of the unknowns R and  are displayed indexed, for compact notation)

 >
 (22)

Construct the set of Partial Differential Equations to be tackled

 >
 (23)

Solve the problem running a differential elimination (actually without solving any differential equations): there are more than two solutions

 >
 (24)

Consider for instance the first one

 >
 (25)

Compute the actual solution behind this case :

 >
 (26)

The fact that the time t appears defined in terms of the transformed time  involving an arbitrary constant is expected: the time does not enter the metric, it only enters through derivatives of  entering the Jacobian of the transformation used to change variables in tensorial expressions (the metric) in (22).

Summary: the approach shown above, based on formulating the problem for the transformation functions of the equivalence and solving for them the differential equations using the commands in PDEtools, after restricting the generality of the transformation functions by looking at the form of the Weyl scalars, works well for other cases too, specially now that, in Maple 2016, the Weyl scalars can be expressed also in canonical form in one go (see previous Mapleprimes post on "Tetrads and Weyl scalars in canonical form").  Also important: in Maple 2016 it is present the functionality necessary to implement the approach of section 9.2 of the Exact solutions book as well.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

## Tetrads and Weyl scalars in canonical form

by: Maple

Tetrads and Weyl scalars in canonical form

The material below is about a new development that didn't arrive in time for the launch of Maple 2016 (March) and that complements in a relevant way the ones introduced in Physics in Maple 2016. It is at topic in general relativity, the computation of a canonical form of a tetrad, so that, generally speaking (skipping a technical description) the Weyl scalars are fixed as much as possible (either equal to 0 or to 1) regarding transformations that leave invariant the tetrad metric in a tetrad system of references. Bringing a tetrad in canonical form is a relevant step in the tackling of the equivalence problem between two spacetime metrics (Mapleprimes post), and it is relevant in connection with the digitizing in Maple 2016 of the database of solutions to Einstein's equations of the book Exact Solutions to Einstein Field Equations.

The reference for this development is the book "General Relativity, an Einstein century survey", edited by S.W. Hawking (Cambridge) and W. Israel (U. Alberta, Canada), specifically Chapter 7 written by S. Chandrasekhar, and more specifically exploring what is said in page 388 about the Petrov classification.

A canonical form for the tetrad and Weyl scalars admits alternate forms; the implementation is as implicit in page 388:

 Residual invariance Petrov type I 0 1 0 none Petrov type II 0 0 1 0 none Petrov type III 0 0 0 1 0 none Petrov type D 0 0 0 0 remains invariant under rotations of Class III Petrov type N 0 0 0 0 1 remains invariant under rotations of Class II

The transformations (rotations of the tetrad system of references) used are of Class I, II and III as defined in Chandrasekar's chapter - equations (7.79) in page 384, (7.83) and (7.84) in page 385. Transformations of Class I can be performed with the command Physics:-Tetrads:-TransformTetrad using the optional argument nullrotationwithfixedl_, of Class II using nullrotationwithfixedn_ and of Class III by calling TransformTetrad(spatialrotationsm_mb_plan, boostsn_l_plane), so with the two optional arguments simultaneously.

In this development, a new optional argument, canonicalform got implemented to TransformTetrad so that the whole sequence of three transformations of Classes I, II and III is performed automatically, in one go. Regarding the canonical form of the tetrad, the main idea is that from the change in the Weyl scalars one can derive the parameters entering tetrad transformations that result in a canonical form of the tetrad.

 >
 (1)

(Note the Tetrads:-PetrovType command, unfinished in the first release of Maple 2016.) To run the following computations you need to update your Physics library to the latest version from the Maplesoft R&D Physics webpage, so with this datestamp or newer:

 >
 (2)

An Example of Petrov type I

There are six Petrov types: I, II, III, D, N and O. Start with a spacetime metric of Petrov type "I"  (the numbers always refer to the equation number in the "Exact solutions to Einstein's field equations" textbook)

 >
 (3)

The Weyl scalars

 >
 (4)

... there is abs around. Let's assume everything is positive to simplify formulas, use Capital Physics:-Assume  (the lower case assume  command redefines the assumed variables, so it is not compatible with Physics, DifferentialGeometry and VectorCalculus among others).

 >
 (5)

The scalars are now simpler, although still not in "canonical form" because  and .

 >
 (6)

The Petrov type

 >
 (7)

 >
 (8)

into another tetrad such that the Weyl scalars are in canonical form, which for Petrov "I" type happens when  and .

 >
 (9)

Despite the fact that the result is a much more complicated tetrad, this is an amazing result in that the resulting Weyl scalars are all fixed (see below).  Let's first verify that this is indeed a tetrad, and that now the Weyl scalars are in canonical form

 >
 (10)

Set (9) to be the tetrad in use and recompute the Weyl scalars

 >

Inded we now have  and

 >
 (11)

So Weyl scalars computed after setting the canonical tetrad (9) to be the tetrad in use are in canonical form. Great! NOTE: computing the canonicalWeyl scalars is not really the difficult part, and within the code, these scalars (11) are computed before arriving at the tetrad (9). What is really difficult (from the point of view of computational complexity and simplifications) is to compute the actual canonical form of the tetrad (9).

An Example of Petrov type II

Consider this other solution to Einstein's equation (again, the numbers in g_[[24,37,7]] always refer to the equation number in the "Exact solutions to Einstein's field equations" textbook)

 >
 (12)

Check the Petrov type

 >
 (13)

 >
 (14)

results in Weyl scalars not in canonical form:

 >
 (15)

For Petrov type "II", the canonical form is as for type "I" but in addition  Again let's assume positive, not necessary, but to get simpler formulas around

 >
 (16)

Compute now a canonical form for the tetrad, to be used instead of (14)

 >
 (17)

Set this tetrad and check the Weyl scalars again

 >
 >
 (18)

This result (18) is fantastic. Compare these Weyl scalars with the ones (15) before transforming the tetrad.

An Example of Petrov type III

 >
 (19)
 >
 (20)

The Petrov type and the original tetrad

 >
 (21)
 >
 (22)

This tetrad results in the following scalars

 >
 (23)

that are not in canonical form, which for Petrov type III is as in Petrov type II but in addition we should have .

Compute now a canonical form for the tetrad

 >
 (24)

Set this one to be the tetrad in use and recompute the Weyl scalars

 >
 >
 (25)

Great!

An Example of Petrov type N

 >
 (26)
 >
 (27)
 >
 (28)

The original tetrad and related Weyl scalars are not in canonical form:

 >
 (29)
 >
 (30)

For Petrov type "N", the canonical form has  and all the other .

Compute a canonical form, set it to be the tetrad in use and recompute the Weyl scalars

 >
 (31)
 >
 >
 (32)

All as expected.

An Example of Petrov type D

 >
 (33)
 >
 (34)
 >
 (35)

The default tetrad and related Weyl scalars are not in canonical form, which for Petrov type "D" is with  and all the other

 >
 (36)
 >
 (37)

Transform the  tetrad, set it and recompute the Weyl scalars

 >
 (38)
 >
 >
 (39)

Again the expected canonical form of the Weyl scalars, and  remains invariant under transformations of Class III.

An Example of Petrov type O

Finally an example of type "O". This corresponds to a conformally flat spacetime, for which the Weyl tensor (and with it all the Weyl scalars) vanishes. So the code just interrupts with "not implemented for conformally flat spactimes of Petrov type O"

 >
 (40)
 >
 (41)

The Weyl tensor and its scalars all vanish:

 >
 (42)
 >
 (43)
 >
 >

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

## Acceptable Maple solution?

by: Maple

In a recent post, the following inequality was proved with Maple:

(a,b,c,d >= 0).

Here is another direct proof attempt.

f:=(a+b+c+d)^2*(a*b+a*c+a*d+b*c+b*d+c*d)^2-144*(a^2+b^2+c^2+d^2)*a*b*c*d:
g:=expand(eval(f,d=1)):
s:=minimize(g, [a=0..infinity,b=0..infinity,c=0..infinity]):
length(s);   # huge
304856
map(evalf@evalf[500],s);

map(evalf@evalf[1000],s);

So, Maple returns the expression

min(0,r1,r2,r3)

where r1,r2,r3 are huge expressions containing RootOfs. In order to evaluate them, several hundreds of digits are needed.
The solution seems to be correct, but the question is: may we (mathematically) accept it? What do you think?

## Maple T.A. 2016 is here!

by:

I’m pleased to announce the release of Maple T.A. 2016, our online assessment system.

For this release, we put a lot of effort into streamlining the authoring experience. We worked closely with customers to find out how they authored content, the places where they found the interface awkward, the tasks that took longer than they should have, and what they’d like to see changed. Then we made it better.

Right away you’ll notice that questions and assignments are no longer in separate places in Maple T.A. All your content is stored in a convenient location that makes it simple to browse your content. Contextual navigation, filtering options, sorting tools, question details, drag and drop organization, combined import feature, and more make it easier than ever to find and organize your content. The Maple T.A. Cloud also sees improvements. Not only can questions be shared, but assignments and entire course modules can be as well.

For question creation, we consolidated all question authoring into the question designer, so you have a single starting point no matter what kind of question you want to create. We have also refined the text editor to help authors find the tools they need to modify their questions. This includes embedding external content, importing questions from the repository, text formatting options, and more.

Of course, once you have questions, you’ll want to put them into an assignment, and assignment creation is now easier than ever. A key change is that you can now create and modify questions while you are creating an assignment, without having to leave the assignment editor. There are also changes to how you preview questions, set properties, and even save your assignments, all of which contribute to making assignment creation simpler and faster.

Of course, there’s more than just a significantly improved author workflow. Here are some highlights:

• Assignment groups for efficient organization, both in the content repository and on the class homepage.
• Easy-to-create sorting questions – no coding required!
• HTML questions, which can be authored directly in the question designer.
• Clickable image questions are Java-free and easier to author.
• Maximum word counts and other improvements to the essay question type.
• A new scanned document feature lets instructors upload and even grade scanned documents.
• Officially certified LTI integration for connectivity with a wide range of course management systems

See What’s New in Maple T.A. 2016 for more information on these and other new features.

Jonny Zivku
Maplesoft Product Manager, Online Education Products

## Inequality proved with Maple. II

Maple

How to prove the inequality  , assuming that the  variables are nonnegative? That hard question  was asked by arqady in dxdy and answered  by himself  in a complicated way. Maple proves the inequality by the LagrangeMultipliers command which is strong. I think these calculations cannot be done by hand at all. Without loss of generality one may assume . Then

restart:with(Student[MultivariateCalculus]):

ans := [LagrangeMultipliers((a+b+c+d)*(a*b+a*c+a*d+b*c+b*d+c*d)-12*sqrt((a^2+b^2+c^2+d^2)*a*b*c*d), [a+b+c+d-1], [a, b, c, d], output = detailed)]:

We have to remove complex solutions by
ans1:=remove(c -> has(evalf(c), I),ans):

The next big output is  only partly seen in the post (look in the attached file for the whole one).

 [[a = 1/6, b = 1/2, c = 1/6, d = 1/6, lambda[1] = 0, -12*sqrt((a^2+b^2+c^2+d^2)*a*b*c*d)+(b+c+d)*a^2+(b^2+(3*c+3*d)*b+c^2+3*c*d+d^2)*a+(d+c)*b^2+(c^2+3*c*d+d^2)*b+c^2*d+c*d^2 = 0],[a = 1/4, b = 1/4, c = 1/4, d = 1/4, lambda[1] = 0, -12*sqrt((a^2+b^2+c^2+d^2)*a*b*c*d)+(b+c+d)*a^2+(b^2+(3*c+3*d)*b+c^2+3*c*d+d^2)*a+(d+c)*b^2+(c^2+3*c*d+d^2)*b+c^2*d+c*d^2 = 0],[a = 13/72-(1/216)*sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(1/216)*sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))), b = 11/24+(1/72)*sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))-(1/72)*sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))), c = 13/72-(1/216)*sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(1/216)*sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))), d = 13/72-(1/216)*sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(1/216)*sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))), lambda[1] = -(5/36)*(sqrt(2)*(sqrt(3)*(sqrt(13397)-(71/27)*(11548+108*sqrt(13397))^(1/3)-(103/540)*(11548+108*sqrt(13397))^(2/3)+2887/27)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))-15*sqrt(13397)+(355/9)*(11548+108*sqrt(13397))^(1/3)+(109/36)*(11548+108*sqrt(13397))^(2/3)-14435/9)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))-(133/15)*(11548+108*sqrt(13397))^(2/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(2*((sqrt(13397)+2374/45)*(11548+108*sqrt(13397))^(1/3)+(103/5)*sqrt(13397)+(449/90)*(11548+108*sqrt(13397))^(2/3)+132727/45))*sqrt(3))/((11548+108*sqrt(13397))^(2/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))), -12*sqrt((a^2+b^2+c^2+d^2)*a*b*c*d)+(b+c+d)*a^2+(b^2+(3*c+3*d)*b+c^2+3*c*d+d^2)*a+(d+c)*b^2+(c^2+3*c*d+d^2)*b+c^2*d+c*d^2 = -(13/46656)*(((2/13)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+sqrt(2)*(sqrt(3)*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))-(11/13)*(11548+108*sqrt(13397))^(1/3)-(2/13)*(11548+108*sqrt(13397))^(2/3)+568/13))*sqrt(5)*sqrt((sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))-sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))-33)*(sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))-sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+39)*(sqrt(2)*(sqrt(3)*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(11/5)*(11548+108*sqrt(13397))^(1/3)+(2/5)*(11548+108*sqrt(13397))^(2/3)-568/5)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))-(216/5)*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))-(328/5*((11548+108*sqrt(13397))^(1/3)+(5/164)*(11548+108*sqrt(13397))^(2/3)-355/41))*sqrt(3))/((11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))))-(180/13)*sqrt(2)*(sqrt(3)*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(11/5)*(11548+108*sqrt(13397))^(1/3)+(2/5)*(11548+108*sqrt(13397))^(2/3)-568/5)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))-(15552/13)*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(11808/13*((11548+108*sqrt(13397))^(1/3)+(5/164)*(11548+108*sqrt(13397))^(2/3)-355/41))*sqrt(3))/((11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))] (1)

evalf(ans2);

 (2)

Indeed, the minimum value of the target function is exactly 0. Quod erat demonstrantum.

## The mechanism of transport of the material. Animation...

by: Maple 15

The mechanism of transport of the material of the sewing machine M 1022 class: mathematical animation.   BELORUS.mw

## Inequality proved with Maple

Maple

How to prove the inequality  provided , ? That problem was posed  by Israeli mathematician nicked by himself as arqady in Russian math forum and was not answered there.I know how to prove that with Maple and don't know how to prove that without Maple. Neither  nor  work here. The difficulty consists in the nonlinearity both the target function and the main constraint. The first step is to linearize the main constraint and the second step is to reduce the number of variables to one.

restart; A := eval(x^(4*y)+y^(4*x), [x = sqrt(u), y = sqrt(v)]);

 (1)

B := expand(A);

 (2)

C := eval(B, u = 2-v);

 (3)

It is more or less clear that the plot of F is symmetric wrt  the straight line v=1. This motivates the following change of variable  to obtain an even function.

F := simplify(expand(eval(C, v = z+1)), symbolic, power);

 (4)

The plots suggest the only maximim of F at z=0 and its concavity.

Student[Calculus1]:-FunctionPlot(F, z = -1 .. 1);

Student[Calculus1]:-FunctionPlot(diff(F, z, z), z = -1 .. 1);

As usually, numeric global solvers cannot prove certain inequalities. However, the GlobalSearch command of the DirectSearch package indicates the only local maximum of  F and F''.

Digits := 25; DirectSearch:-GlobalSearch(F, {z = -1 .. 1}, maximize, solutions = 3, tolerances = 10^(-15)); DirectSearch:-GlobalSearch(diff(F, z, z), {z = -1 .. 1}, maximize, solutions = 3, tolerances = 10^(-15));

 (5)

The series command confirms a local maximum of F at z=0.

series(F, z, 6);

 (6)

The extrema command indicates only the value of F at a critical point, not outputting its position.

extrema(F, z); extrema(F, z, 's');

 (7)

solve(F = 2);

 (8)

DirectSearch:-SolveEquations(F = 2, {z = -1 .. 1}, AllSolutions, solutions = 3);

 (9)

DirectSearch:-SolveEquations(F = 2, {z = -1 .. 1}, AllSolutions, solutions = 3, assume = integer);

 (10)

PS. I see my proof needs an additional explanation. The DirectSearch command establishes the only both local and global  maximum of F is located at z= -1.98*10^(-13) up to default error 10^(-9). After that  the series command confirms a local maximum at z=0. Combining these, one draws the conclusion that the global maximum is placed exactly at z=0 and equals 2. In order to confirm that the only real root of F=2 at z=0  is found approximately and exactly by the DirectSearch.

## Maple: Software Engineering Solutions

Maple 2016

Maple is a scientific software based on Computational Algebraic System (SAC) which has enabled this work entirely solve applied to Civil Engineering, Mechanical and Mecatrónica.The present problems in education, research and engineering are developed with static work sheets ie coding used innecesaria.Maple proposed models are shown below with an innovative structure; with the method of graphics algorithms and embedded components; putting aside the traditional and obsolete syntax; using dynamic worksheets as viable and optimal solutions to interpret and explain problems Ingineering.Design Advanced Analysis Tools (Applied Mathematics) Sophisticated Applications (efficient algorithms) and Multiple deployment options (different styles); this allowed generate math apps (applications engineering); can be interactive on the internet without the need to have the software installed on our computer; This way our projects can be used with a vision of sustainability around the world. Resulting in the generation of data and curves; which in turn will help you make better decisions analytical and predictive modeling in manufacturing and 3D objects; which would lead to new patterns of contrasting solutions.

ECI_2016.pdf

ECI_2016v_full.mw

Lenin Araujo Castillo

by: Maple 17

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations.
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism.  The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf

by: Maple 17

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations.
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism.  The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf

by: Maple 17

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations.
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism.  The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf

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