MaplePrimes Commons General Technical Discussions

Announcing the Maple Customer Support Updates...

by: aroche 560 Product: Maple

5 hours ago

5 1

Maplesoft now has a new approach to providing customer support for Maple users! The Maple Customer Support Updates allows Maplesoft to provide important updates to our customers as fast as possible. These updates contain a series of improvements and fixes to any area of the Maple library, enabling a rapid response for customer reports and requests. When a Maple user reports a bug or weakness, or requests some missing functionality that can be addressed with an update to the Maple library, such an update can now be provided immediately after the fix or improvement is developed. Furthermore, the update will not just be available to that customer who reported it, but also to any other Maple users who wishes to use them. Of course, not all reports will be able to be addressed quickly, and for those that are, it will be up to the developer's discretion whether to make the fix or improvement available via these new Maple Customer Support Updates. Please note that these Updates may contain experimental elements that could change in subsequent official releases.

The updates are available as a workbook containing a Maple library file that can be downloaded and installed from the Maple Cloud. To install the Maple Customer Support Updates from the Maple Cloud,

Click the MapleCloud icon in the upper-right corner of the Maple GUI window and select Packages.
Find the Maple Customer Support Updates package and click the Install button, the last one under Actions.
To check for new versions of Maple Customer Support Updates, click the MapleCloud icon and select Updates. If the cloud icon in the Actions column of Maple Customer Support Updates has the word Update beside it, then you can click on it to download a new update.

To make the process of installing and maintaining the Maple Customer Support Updates as smooth as possible, we've also introduced a new Maple library package, SupportTools, with 3 commands, Update, Version, and RemoveUpdates.

Load the SupportTools package:

(1)

Check which version is currently installed:

$`The Customer Support Updates version in the MapleCloud is 10. The version installed in this computer is 9 created April 22, 2025, 15:14 hours Eastern Time, found in the directory C:\Users\Austin\Maple\toolbox\2025\Maple Customer Support Updates\lib\Maple`$

(2)

Update to the latest version (you could also call Update(latest)):

Warning, You have just upgraded from version 9 to version 10 of the Customer Support Updates. In order to have this version active, please close Maple entirely, then open Maple and enter SupportTools:-Version() to confirm the active version.

Check the version again:

(3)

Remove all updates for this release of Maple (except for those installing the SupportTools package itself):

Warning, You have just reverted to version 4 of the Customer Support Updates. This version contains no actual updates other than the SupportTools package itself. In order to verify this, please close Maple entirely, then open Maple and enter SupportTools:-Version() to verify that the version number is 4.

Note: You can also specify which version to install by supplying the version number as the argument to the Update command:

Warning, You have just upgraded from version 4 to version 10 of the Customer Support Updates. In order to have this version active, please close Maple entirely, then open Maple and enter SupportTools:-Version() to confirm the active version.

Download SupportTools.mw

In Maple 2025.0, the SupportTools package is not installed by default. For the first installation, you can also run the command
PackageTools:-Install(5137472255164416); instead of installing it from the Maple Cloud.

The Maple Customer Support Updates were inspired by and modelled after the existing Physics Updates which many Maple users may be famiilar with already. Going forward, Physics Updates will only contain changes to the Physics package itself. All other library updates will be available via the Maple Customer Support Updates. For compatibility with the pre-existing Physics:-Version command, calling SupportTools:-Version(n) is equivalent to calling SupportTools:-Update(n), and similarly SupportTools:-Version(latest) and SupportTools:-Update(latest) are both equivalent to the single call SupportTools:-Update().

The 2025 Maplesoft Physics Updates and its future

by: ecterrab 14375 Product: Maple 2025

April 01 2025

8 4

The Maplesoft Physics Updates, introduced over a decade ago, brought with them an innovative concept: to deliver fixes and new developments continuously, as soon as they enter the development version of the Maple library for the next release. A key aspect of this initiative was prioritizing the resolution of issues reported on MaplePrimes, ensuring that fixes became available to everyone within 24 to 48 hours. Initially focused solely on the Physics package, the scope of the updates quickly expanded to include other parts of the Maple library and the Typesetting system.

This initiative, which I developed outside regular work hours, aimed to enhance the Maple experience—where issues encountered in daily use could be resolved almost immediately, minimizing disruptions and benefiting the entire user community through shared updates.

As of January 1st, I have stepped away from my role at Maplesoft and have been increasingly involved in activities unrelated to Maple. This raises the question of what will happen with the Physics Updates for Maple 2025 and after.

The Physics project remains a unique and personally meaningful endeavor for me. So, for now, I will continue to dedicate some time to these Updates—but only for the Physics package, not for other parts of the library. As before, these fixes and developments will be included in the Physics Updates only after they have been integrated into the development version of Maple’s official library for the next release. In that sense, they will continue to be Maplesoft updates.

On that note, the first release of the Physics Updates for Maple 2025—focused solely on the Physics package—went out today as version 1854. To install it, the first time open Maple 2025 and use the Maplecloud toolbar -> Packages, or else input PackageTools:-Install(5137472255164416). Any next time, just enter Physics:-Version(latest)

As for fixes beyond the Physics package, I understand that Maplesoft is exploring the possibility of offering something similar to what was previously delivered through the Maplesoft Physics Updates.

All the best

PS: to install the last version of the Maplesoft Physics Updates for Maple 2024, open Maple and input Physics:-Version(1852), not 1853.

Edgardo S. Cheb-Terrab
Physics, Differential Equations, and Mathematical Functions
Maplesoft Emmeritus
Research and Education—passionate about all that.

What is new in Physics in Maple 2025

by: ecterrab 14375 Product: Maple

March 27 2025

8 1

What is new in Physics in Maple 2025

This post is, basically, the page of what is new in Physics distributed with Maple 2025. Why post it? Because this time we achieved a result that is a breakthrough/milestone in Computer Algebra: for the first time we can systematically compute Einstein's equations from first principles, using a computer, starting from a Lagrangian for gravity. This is something to celebrate.

Although being able to perform this computation on a computer algebra sheet is relevant mostly for physicists working in general relativity, the developments performed in the tensor manipulation and functional differentiation routines of the Maple system to cover this computation were tremendous, in number of lines of code, efforts and time, resulting in improvements not just fore general relativity but for physics all around, and in an area out of reach of neural network AIs . Other results, like the new routines for linearizing gravity, requested other times here in Mapleprimes, are also part of the relevant novelties.

Lagrange Equations and simplification of tensorial expressions in curved spacetimes

Linearized Gravity

Relative Tensors

New Physics:-Library commands

See Also

Lagrange Equations and simplification of tensorial expressions in curved spacetimes

LagrangeEquations is a Physics command introduced in 2023 taking advantage of the functional differentiation capabilities of the Physics package . This command can handle tensors and vectors of the Physics package as well as derivatives using vectorial differential operators (see d_ and Nabla ), works by performing functional differentiation (see Fundiff ), and handles 1^st, and higher order derivatives of the coordinates in the Lagrangian automatically. LagrangeEquations receives an expression representing a Lagrangian and returns a sequence of Lagrange equations with as many equations as coordinates are indicated. The number of parameters can also be many. For example, in electrodynamics, the "coordinate" is a tensor field , there are then four coordinates, one for each of the values of the index , and there are four parameters .

New in Maple 2025, the "coordinates" can now also be the components of the metric tensor in a curved spacetime, in which case the equations returned are Einstein's equations. Also new, instead of a coordinate or set of them, you can pass the keyword EnergyMomentum, in which case the output is the conserved energy-momentum tensor of the physical model represented by the given Lagrangian L.

Examples

(1)

The model in classical field theory and corresponding field equations, as in previous releases

>

(2)

>

(1/2)*Physics:-d_[mu](Phi(X), [X])*Physics:-d_[`~mu`](Phi(X), [X])-(1/2)*m^2*Phi(X)^2+(1/4)*lambda*Phi(X)^4

(3)

Lagrange's equations

>

(4)

New: The energy-momentum tensor can be computed as the Lagrange equations taking the metric as the coordinate, not equating to 0 the result, but multiplying the variation of the action "(delta S)/(delta g[]^(mu,nu))" by (in flat spacetimes ). For that purpose, you can use the EnergyMomentum keyword. You can optionally indicate the indices to be used in the output as well as their covariant or contravariant character

>

Physics:-EnergyMomentum[mu, nu] = (-(1/4)*lambda*Phi(X)^4+(1/2)*m^2*Phi(X)^2-(1/2)*Physics:-d_[`~beta`](Phi(X), [X])*Physics:-d_[beta](Phi(X), [X]))*Physics:-g_[mu, nu]+Physics:-d_[mu](Phi(X), [X])*Physics:-d_[nu](Phi(X), [X])

(5)

To further compute using the above as the definition for , you can use the Define command

>

${Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-EnergyMomentum[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(6)

After which the system knows about the symmetry properties and the components of

>

Physics:-EnergyMomentum[mu, nu] = (-(1/4)*lambda*Phi(X)^4+(1/2)*m^2*Phi(X)^2-(1/2)*Physics:-d_[`~beta`](Phi(X), [X])*Physics:-d_[beta](Phi(X), [X]))*Physics:-g_[mu, nu]+Physics:-d_[mu](Phi(X), [X])*Physics:-d_[nu](Phi(X), [X])

(7)

>

(8)

>

Physics:-EnergyMomentum[mu, nu] = Matrix(%id = 36893488151952536988)

(9)

New: LagrangeEquations takes advantage of the extension of Fundiff to compute functional derivatives in curved spacetimes introduced for Maple 2025, and so it also handles the case of a scalar field in a curved spacetime. Set for instance an arbitrary metric

>

Physics:-g_[mu, nu] = Matrix(%id = 36893488152257720060)

(10)

For the action to be a true scalar in spacetime, the Lagrangian density now needs to be multiplied by the square root of the determinant of the metric

>

(-%g_[determinant])^(1/2)*((1/2)*Physics:-d_[mu](Phi(X), [X])*Physics:-d_[`~mu`](Phi(X), [X])-(1/2)*m^2*Phi(X)^2+(1/4)*lambda*Phi(X)^4)

(11)

New: With the extension of the tensorial simplification algorithms for curved spacetimes, the Lagrange equations can be computed arriving directly to the compact form

>

(12)

Comparing with the result (4) for the same Lagrangian in a flat spacetime, we see the only difference is that the dAlembertian is now expressed in terms of covariant derivatives D_ .

The EnergyMomentum tensor is computed in the same way as when the spacetime is flat

>

Physics:-EnergyMomentum[mu, nu] = (-(1/4)*lambda*Phi(X)^4+(1/2)*m^2*Phi(X)^2-(1/2)*Physics:-d_[`~beta`](Phi(X), [X])*Physics:-d_[beta](Phi(X), [X]))*Physics:-g_[mu, nu]+Physics:-d_[mu](Phi(X), [X])*Physics:-d_[nu](Phi(X), [X])

(13)

General Relativity

New: the most significant development in LagrangeEquations is regarding General Relativity. It can now compute Einstein's equations directly from the Lagrangian, not using tabulated cases, and properly handling several (traditional or not) alternative ways of presenting the Lagrangian.

Einstein's equations concern the case of a curved spacetime with metric as, for instance, the general case of an arbitrary metric set lines above. In the Lagrangian formulation, the coordinates of the problem are the components of the metric , and as in the case of electrodynamics the parameters are the spacetime coordinates . The simplest case is that of Einstein's equation in vacuum, for which the Lagrangian density is expressed in terms of the trace of the Ricci tensor by

>

(14)

Einstein's equations in vacuum:

>

-(1/2)*Physics:-g_[mu, nu]*Physics:-Ricci[alpha, `~alpha`]+Physics:-Ricci[mu, nu] = 0

(15)

where in the above instead of passing as second argument, we passed to get the equations using those free indices. The tensorial equation computed is also the definition of the Einstein tensor

>

Physics:-Einstein[mu, nu] = -(1/2)*Physics:-g_[mu, nu]*Physics:-Ricci[alpha, `~alpha`]+Physics:-Ricci[mu, nu]

(16)

The Lagrangian used to compute Einstein's equations (15) contains first and second derivatives of the metric. To see that, rewrite L in terms of Christoffel symbols

>

(17)

Recalling the definition

>

Physics:-Christoffel[alpha, mu, nu] = (1/2)*Physics:-d_[nu](Physics:-g_[alpha, mu], [X])+(1/2)*Physics:-d_[mu](Physics:-g_[alpha, nu], [X])-(1/2)*Physics:-d_[alpha](Physics:-g_[mu, nu], [X])

(18)

in the two terms containing derivatives of Christoffel symbols contain second order derivatives of . Now, it is always possible to add a total spacetime derivative to without changing Einstein's equations (assuming the variation of the metric in the corresponding boundary integrals vanishes), and in that way, in this particular case of , obtain a Lagrangian involving only 1st order derivatives. The total derivative, expressed using the inert command to see it before the differentiation operation is performed, is

>

(19)

Adding this term to , performing the differentiation operation and simplifying we get

>

(20)

>

(21)

>

(22)

which is a Lagrangian depending only on 1st order derivatives of the metric through Christoffel symbols. As expected, the equations of motion resulting from this Lagrangian are the same Einstein equations computed in (15)

>

-(1/2)*Physics:-Ricci[iota, `~iota`]*Physics:-g_[mu, nu]+Physics:-Ricci[mu, nu] = 0

(23)

To illustrate the new Maple 2025 tensorial simplification capabilities note that is no just ≡ (17) after discarding its two terms involving derivatives of Christoffel symbols. To verify this, split into the terms containing or not derivatives of Christoffel

>

(24)

Comparing, the total derivative TD≡ (19) is not just , but

>

(25)

Things like these, , can now be verified directly with the new tensorial simplification capabilities: take the left-hand side minus the right-hand side, evaluate the inert derivative and simplify to see the equality is true

>

(26)

>

(27)

>

(28)

That said, it is also true that results in the Lagrangian , and since the equations of movement don't depend on the sign of the Lagrangian, for this Lagrangian adding the term happens to be equivalent to just discarding the terms of involving derivatives of Christoffel symbols.

Also new in Maple 2025, due to the extension of Fundiff to compute in curved spacetimes, it is now also possible to compute Einstein's equations from first principles by constructing the action,

>

Int(Int(Int(Int((-%g_[determinant])^(1/2)*Physics:-Ricci[alpha, `~alpha`], x = -infinity .. infinity), y = -infinity .. infinity), z = -infinity .. infinity), t = -infinity .. infinity)

(29)

and equating to zero the functional derivative with respect to the metric. To avoid displaying the resulting large expression, end the input line with ":"

>

Simplifying this result, we get an expression in terms of Christoffel symbols and its derivatives

>

(30)

In this result, we see derivatives of Christoffel symbols, expressed using the D_ command for covariant differentiation. Although, such objects have not the geometrical meaning of a covariant derivative, computationally, they here represent what would be a covariant derivative if the Christoffel symbols were a tensor. For example,

>

(31)

With this computational meaning for the derivatives of Christoffel symbols appearing in (30), rewrite EEC≡(30) in terms of the Ricci and Riemann tensors. For that, consider the definition

>

(32)

Rewrite the noncovariant derivatives in terms of derivatives using the computational representation (31), simplify and isolate one of them

>

(33)

>

(34)

>

(35)

Analogously, derive an expression to rewrite derivatives of Christoffel symbols using the Riemann tensor

>

(36)

>

(37)

>

(38)

>

(39)

Substitute these two equations, in sequence, into Einstein's equations EEC≡(30)

>

(40)

Simplify to arrive at the traditional compact form of Einstein's equations

>

(1/2)*Physics:-Ricci[chi, `~chi`]*Physics:-g_[`~alpha`, `~beta`]-Physics:-Ricci[`~alpha`, `~beta`] = 0

(41)

Linearized Gravity

Generally speaking, linearizing gravity is about discarding in Einstein's field equations the terms that are quadratic in the metric and its derivatives, an approximation valid when the gravitational field is weak (the deviation from a flat Minkowski spacetime is small). Linearizing gravity is used, e.g. in the study of gravitational waves. In the context of Maple's Physics , the formulation of linearized gravity can be done using the general relativity tensors that come predefined in Physics plus a new in Maple 2025 Physics:-Library:-Linearize command.

In what follows it is shown how to linearize the Ricci tensor and through it Einstein 's equations. To compare results, see for instance the Wikipedia page for Linearized gravity. Start setting coordinates, you could use Cartesian, spherical, cylindrical, or define your own.

>

restart; with(Physics); Setup(coordinates = cartesian)

(42)

The default metric when Physics is loaded is the Minkowski metric, representing a flat (no curvature) spacetime

>

Physics:-g_[mu, nu] = Matrix(%id = 36893488151987392132)

(43)

The weakly perturbed metric

Suppose you want to define a small perturbation around this metric. For that purpose, define a perturbation tensor , that in the general case depends on the coordinates and is not diagonal, the only requirement is that it is symmetric (to have it diagonal, change symmetric by diagonal; to have it constant, change by )

>

h[mu, nu] = Matrix(%id = 36893488152568729716)

(44)

In the above it is understood that , so that quadratic or higher powers of it or its derivatives can be approximated to 0 (discarded). Define the components of accordingly

>

${Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], h[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(45)

Define also a tensor representing the unperturbed Minkowski metric

>

eta[mu, nu] = Matrix(%id = 36893488151987392132)

(46)

>

${Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], h[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(47)

The weakly perturbed metric is given by

>

(48)

Make this be the definition of the metric

>

Physics:-g_[mu, nu] = Matrix(%id = 36893488152066920564)

${Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], eta[mu, nu], Physics:-g_[mu, nu], Physics:-gamma_[i, j], h[mu, nu], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(49)

Linearizing the Ricci tensor

The linearized form of the Ricci tensor is computed by introducing this weakly perturbed metric (48) in the expression of the Ricci tensor as a function of the metric. This can be accomplished in different ways, the simpler being to use the conversion network between tensors , but for illustration purposes, showing steps one at time, a substitution of definitions one into the other one is used

>

(50)

>

Physics:-Christoffel[`~alpha`, mu, nu] = (1/2)*Physics:-g_[`~alpha`, `~beta`]*(Physics:-d_[nu](Physics:-g_[beta, mu], [X])+Physics:-d_[mu](Physics:-g_[beta, nu], [X])-Physics:-d_[beta](Physics:-g_[mu, nu], [X]))

(51)

>

(52)

Introducing (48)≡, and also the inert form of the Ricci tensor to facilitate simplification some steps below,

>

(53)

This expression contains several terms quadratic in the small perturbation and its derivatives. The new in Maple 2025 routine to filter out those terms is Physics:-Library:-Linearize , which requires specifying the symbol representing the small quantities

>

(54)

Important: in this result, is the flat Minkowski metric, not the perturbed metric . However, in the context of a linearized formulation, raises and lowers tensor indices the same way as . Hence, to further simplify contracted products of in (54) , it is practical to reintroduce representing that Minkowski metric and simplify using the internal algorithms for a flat metric

>

Physics:-g_[mu, nu] = Matrix(%id = 36893488152066895868)

(55)

To proceed simplifying, replace in the expression (54) for the Ricci tensor the intermediate Minkowski by

>

(56)

Simplifying, results in the linearized form of the Ricci tensor shown in the Wikipedia page for Linearized gravity.

>

%Ricci[mu, nu] = -(1/2)*Physics:-d_[mu](Physics:-d_[nu](h[tau, `~tau`], [X]), [X])-(1/2)*Physics:-dAlembertian(h[mu, nu], [X])+(1/2)*Physics:-d_[nu](Physics:-d_[tau](h[mu, `~tau`], [X]), [X])+(1/2)*Physics:-d_[mu](Physics:-d_[tau](h[nu, `~tau`], [X]), [X])

(57)

Linearizing Einstein's equations

Einstein's equations are the components of Einstein's tensor , whose definition in terms of the Ricci tensor is

>

Physics:-Einstein[mu, nu] = Physics:-Ricci[mu, nu]-(1/2)*Physics:-g_[mu, nu]*Physics:-Ricci[alpha, `~alpha`]

(58)

Compute the trace directly from the linearized form (57) of the Ricci tensor,

>

%Ricci[mu, nu]*Physics:-g_[`~mu`, `~nu`] = (-(1/2)*Physics:-d_[mu](Physics:-d_[nu](h[tau, `~tau`], [X]), [X])-(1/2)*Physics:-dAlembertian(h[mu, nu], [X])+(1/2)*Physics:-d_[nu](Physics:-d_[tau](h[mu, `~tau`], [X]), [X])+(1/2)*Physics:-d_[mu](Physics:-d_[tau](h[nu, `~tau`], [X]), [X]))*Physics:-g_[`~mu`, `~nu`]

(59)

>

(60)

The linearized Einstein equations are constructed reproducing the definition (58) using (57) and (60)

>

(61)

which is the same formula shown in the Wikipedia page for Linearized gravity.

You can now redefine the general introduced in (44) in different ways (see discussion in the Wikipedia page), or, depending on the case, just substitute your preferred gauge in this formula (61) for the general case. For example, the condition for the Harmonic gauge also known as Lorentz gauge reduces the linearized field equations to their simplest form

>

Physics:-d_[mu](h[`~mu`, nu], [X]) = (1/2)*Physics:-d_[nu](h[alpha, `~alpha`], [X])

(62)

>

(63)

>

%Ricci[mu, nu]-(1/2)*Physics:-g_[mu, nu]*%Ricci[alpha, `~alpha`] = -(1/2)*Physics:-dAlembertian(h[mu, nu], [X])+(1/4)*Physics:-dAlembertian(h[alpha, `~alpha`], [X])*Physics:-g_[mu, nu]

(64)

Relative Tensors

In General Relativity, the context of a curved spacetime, it is sometimes necessary to work with relative tensors, for which the transformation rule under a transformation of coordinates involves powers of the determinant of the transformation - see Chapter 4 of "Lovelock, D., and Rund, H. Tensors, Differential Forms and Variational Principles, Dover, 1989." Physics in Maple 2025 includes a complete, new implementation of relative tensors.

To indicate that a tensor being defined is relative pass its relative weight. For example, set a curved spacetime,

>

restart; with(Physics); g_[sc]

$`Default differentiation variables for d_, D_ and dAlembertian are:`*{X = (r, theta, phi, t)}$

Physics:-g_[mu, nu] = Matrix(%id = 36893488152573138332)

(65)

Define now two tensors of one index, one of them being relative

>

${Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], T[mu], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[i, j], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(66)

>

${Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], R[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], T[mu], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[i, j], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(67)

Transformation of Coordinates

Consider a transformation of coordinates, from spherical to where

>

TR := r = (1+m/(2*rho))^2*rho

r = (1+(1/2)*m/rho)^2*rho

(68)

The transformed components of and are, respectively,

>

Vector[column](%id = 36893488151820263660)

(69)

>

Vector[column](%id = 36893488151823155676)

(70)

where, when comparing both results, we see that the transformed components for are all multiplied by with and is the determinant of the transformation:

>

Matrix(%id = 36893488151964220700)

(71)

>

J = (1/4)*(-m^2+4*rho^2)/rho^2

(72)

Relative weight

The relative weight of a scalar, tensor or tensorial expression can be computed using the Physics:-Library:-GetRelativeWeight command. For the two tensors and used above,

>

(73)

>

(74)

The relative weight of a tensor does not depend on the covariant or contravariant character of its indices

>

(75)

The LeviCivita tensor is a special case, has its relative weight defined when Physics is loaded, and because in a curved spacetime it is not a tensor its relative weight depends on the covariant or contravariant character of its indices

>

(76)

>

(77)

The relative weight w of a product is equal to the sum of relative weights of each factor

>

(78)

>

(79)

The relative weight w of a power is equal to the relative weight of the base multiplied by the power

>

1/(R[mu]*R[`~mu`])

(80)

(81)

The relative weight w of a sum is equal to the relative weight of one of its terms and exists if all the terms have the same w.

>

(82)

>

(83)

The relative weight of any determinant is always equal to 2

>

(84)

>

(85)

Relative Term in covariant derivatives

When computing the covariant derivative of a relative scalar, tensor or tensorial expression that has non-zero relative weight w, a relative term is added, that can be computed using the Physics:-Library:-GetRelativeWeight command.

>

(86)

>

(87)

Consequently,

>

(88)

To understand this zero value on the right-hand side, express the left-hand side in terms of d_

>

(89)

evaluate the inert %d_

>

(90)

The factor in parentheses is equal to , where the covariant derivative of the metric is equal to zero, so

>

(91)

Consider the covariant derivative of and defined in (66) and (67)

>

(92)

>

(93)

The corresponding covariant derivatives

>

(94)

>

%D_[mu](T[`~mu`](X)) = 2*T[`~mu`](X)*Physics:-d_[mu](r, [X])/r+Physics:-d_[mu](theta, [X])*cos(theta)*T[`~mu`](X)/sin(theta)+Physics:-d_[mu](T[`~mu`](X), [X])

(95)

>

(96)

>

%D_[mu](R[`~mu`](X)) = 2*R[`~mu`](X)*Physics:-d_[mu](r, [X])/r+Physics:-d_[mu](theta, [X])*cos(theta)*R[`~mu`](X)/sin(theta)+Physics:-d_[mu](R[`~mu`](X), [X])-Physics:-Christoffel[`~nu`, mu, nu]*R[`~mu`](X)

(97)

where in the above we see the additional (relative) term

>

(98)

New Physics:-Library commands

ConvertToF , Linearize , GetRelativeTerm , GetRelativeWeight.

Examples

•

ConvertToF receives an algebraic expression involving tensors and/or tensor functions and rewrites them in terms of the tensor of name F when that is possible. This routine is similar, however more general than the standard convert which only handles the existing conversion network for the tensors of General Relativity in that ConvertToF also uses any tensor definition you introduce using Define , expressing a tensor in terms of others.

Load any curved spacetime metric automatically setting the coordinates

restart; with(Physics); g_[sc]

Physics:-g_[mu, nu] = Matrix(%id = 36893488152573203868)

(99)

For example, rewrite the Christoffel symbols in terms of the metric g_ ; this works as in previous releases

>

Physics:-Christoffel[mu, alpha, beta] = (1/2)*Physics:-d_[beta](Physics:-g_[alpha, mu], [X])+(1/2)*Physics:-d_[alpha](Physics:-g_[beta, mu], [X])-(1/2)*Physics:-d_[mu](Physics:-g_[alpha, beta], [X])

(100)

Define a representing the 4D electromagnetic potential as a function of the coordinates and representing the electromagnetic field tensors

>

${A[mu], Physics:-D_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[i, j], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(101)

>

${A[mu], Physics:-D_[mu], Physics:-Dgamma[mu], F[mu, nu], Physics:-Psigma[mu], Physics:-Ricci[mu, nu], Physics:-Riemann[mu, nu, alpha, beta], Physics:-Weyl[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[i, j], Physics:-Christoffel[mu, nu, alpha], Physics:-Einstein[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}$

(102)

Rewrite the following expression in terms of the electromagnetic potential

>

(103)

In the example above, the output is similar to this other one

>

(104)

The rewriting, however, works also with tensorial expressions

>

(105)

>

(106)

•	Linearize receives a tensorial expression T and an indication of the small quantities h in T , and discards terms quadratic or of higher order in h. For an example of this new routine in action, see the section Linearized Gravity above.

•	GetRelativeTerm and GetRelativeWeight are illustrated in the section Relative Tensors above.

	See Also
	Index of New Maple 2025 Features , Physics , Computer Algebra for Theoretical Physics, The Physics project, The Physics Updates

Download New_in_Physics_2025.mw

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

Computing Einstein's equations using variational...

by: ecterrab 14375 Product: Maple

March 27 2025

8 1

Computing Einstein's equations using variational principles

Edgardo S. Cheb-Terrab

Freddy Baudine

One of the biggest challenges for a computer algebra system is to compute Einstein's equations from first principles, by equating to zero the functional derivative of the Action for gravity, without using human-shortcuts or tricks. Developments during 2024, appearing as new in Maple 2025, broke through that barrier and now the LagrangeEquations Physics command, introduced in 2023, can perform that elusive computation of Einstein's equations, not using tabulated cases, properly handling several (traditional or not) alternative ways of presenting the Lagrangian (the integrand in the Action), taking advantage of the functional differentiation capabilities of the Physics package .

The computation can be performed in one call to Physics:-LagrangeEquations, or in steps interactively using the Physics:-Fundiff and Physics:-Simplify commands that include newly implemented capabilities for simplifying tensorial expressions in curved spacetimes. This is an exciting breakthrough/milestone in Computer Algebra, also in an area out of reach of neural network AIs.

Einstein's equations are a system of second order nonlinear coupled partial differential equations for the 10 components of the spacetime metric

>

Physics:-g_[mu, nu] = Matrix(%id = 36893488152225885228)

(1)

In the Lagrangian formulation, the coordinates of the problem are the 10 components of the metric,

>

(2)

and the parameters of the variational problem are the spacetime coordinates .

The traditional Lagrangian

The simplest case is that of Einstein's equation in vacuum, for which the Lagrangian density is expressed in terms of the trace of the Ricci tensor and the determinant of the metric by

>

(3)

What was almost a dream in previous years, Einstein's equations can now be computed in one simple instruction as the Lagrange equations for this Lagrangian, in the traditional compact form, taking the components of the metric tensor as the coordinates (for an interactive, step by step computation, see further below)

>

-(1/2)*Physics:-g_[mu, nu]*Physics:-Ricci[alpha, `~alpha`]+Physics:-Ricci[mu, nu] = 0

(4)

The tensorial equation computed is also the definition of the Einstein tensor

>

Physics:-Einstein[mu, nu] = -(1/2)*Physics:-g_[mu, nu]*Physics:-Ricci[alpha, `~alpha`]+Physics:-Ricci[mu, nu]

(5)

A Lagrangian depending only on first derivatives of the metric

The Lagrangian used to compute Einstein's equations (4) contains first and second derivatives of the metric; the latter make the computation significantly more complicated. The second order derivatives, however, can be removed from the formulation. To see that, rewrite L in terms of Christoffel symbols

>

(6)

Recalling the definition

>

Physics:-Christoffel[alpha, mu, nu] = (1/2)*Physics:-d_[nu](Physics:-g_[alpha, mu], [X])+(1/2)*Physics:-d_[mu](Physics:-g_[alpha, nu], [X])-(1/2)*Physics:-d_[alpha](Physics:-g_[mu, nu], [X])

(7)

in the two terms containing derivatives of Christoffel symbols contain second order derivatives of .

Now, it is always possible to add a total spacetime derivative to without changing Einstein's equations (assuming the variation of the metric in the corresponding boundary integrals vanishes), and in that way, in this particular case of , obtain a Lagrangian involving only 1st order derivatives. The total derivative, expressed using the inert command to see it before the differentiation operation is performed, is

>

(8)

Adding this term to , performing the differentiation operation and simplifying we get

>

(9)

>

(10)

>

(11)

which is a Lagrangian depending only on 1st order derivatives of the metric through Christoffel symbols. As expected, the equations of motion resulting from this Lagrangian are the same Einstein equations computed in (4); this computation can also now be performed in a single call

>

-(1/2)*Physics:-Ricci[iota, `~iota`]*Physics:-g_[mu, nu]+Physics:-Ricci[mu, nu] = 0

(12)

Simplification capabilities developed to cover these computations

To illustrate the new Maple 2025 tensorial simplification capabilities note that is no just ≡ (6) after discarding its two terms involving derivatives of Christoffel symbols. To verify this, split into the terms containing or not derivatives of Christoffel

>

(13)

Comparing, the total derivative TD≡ (8) is not just , but

>

(14)

Things like these, , can now be verified directly with the new tensorial simplification capabilities: take the left-hand side minus the right-hand side, evaluate the inert derivative and simplify to see the equality is true

>

(15)

>

(16)

>

(17)

That said, it is also true that results in the Lagrangian , and since the equations of movement don't depend on the sign of the Lagrangian, for this Lagrangian adding the term happens to be equivalent to just discarding the terms of involving derivatives of Christoffel symbols.

Derivation step by step using functional differentiation (variational principle)

Also new in Maple 2025, due to the extension of Fundiff to compute in curved spacetimes, it is now possible to compute Einstein's equations from first principles by constructing the action,

>

Int(Int(Int(Int((-%g_[determinant])^(1/2)*Physics:-Ricci[alpha, `~alpha`], x = -infinity .. infinity), y = -infinity .. infinity), z = -infinity .. infinity), t = -infinity .. infinity)

(18)

and equating to zero the functional derivative with respect to the metric. To avoid displaying the resulting large expression, end the input line with ":"

>

Simplifying this result, we get an expression in terms of Christoffel symbols and its derivatives

>

(19)

In this result, we see derivatives of Christoffel symbols, expressed using the D_ command for covariant differentiation. Although, such objects have not the geometrical meaning of a covariant derivative, computationally, they here represent what would be a covariant derivative if the Christoffel symbols were a tensor. For example,

>

(20)

With this computational meaning for the derivatives of Christoffel symbols appearing in (19), rewrite EEC≡(19) in terms of the Ricci and Riemann tensors. For that, consider the definition

>

(21)

Rewrite the noncovariant derivatives in terms of derivatives using the computational representation (20), simplify and isolate one of them

>

(22)

>

(23)

>

(24)

Analogously, derive an expression to rewrite derivatives of Christoffel symbols using the Riemann tensor

>

(25)

>

(26)

>

(27)

>

(28)

Substitute these two equations, in sequence, into Einstein's equations EEC≡(19)

>

(29)

Simplify to arrive at the traditional compact form of Einstein's equations

>

(1/2)*Physics:-Ricci[chi, `~chi`]*Physics:-g_[`~alpha`, `~beta`]-Physics:-Ricci[`~alpha`, `~beta`] = 0

(30)

Download Einstein_Equations_From_First_Principles.mw

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

We’re back!

by: eithne 2052 Product: MaplePrimes

March 19 2025

5 7

As some of you may have noticed, MaplePrimes was down for a few days, and some customers may have encountered a few problems with our web site, as well.

As you can see, MaplePrimes is up and running again. We are still experiencing a few lingering issues on our main web site, but we expect these will be resolved shortly.

Apologies to those who were inconvenienced by the outage. We appreciate your patience.

AI and writing Maple code

by: Harry Garst 259 Product: Maple

February 18 2025

3 0

After a long chat with ChatGPT I finally received a fully working code for a proc for a generalized Woodbury Identity for the inversion of the sum of two or more positive definite matrices.
I was inspired by a family member who is a trained professional programmer, who told me that in his professional work he uses ChatGTP for an initial draft of his program.
I did find out that ChatGTP makes errors: from simple ones like writing 'Simplify' instead of 'simplify' to serious conceptual errors, for example in recursive loops. However, ChatGTP seems to 'understand' the error after given specific feedback. Although, this does not mean that the next proposal does not contain the same logical error. But after a long chat I received a nice proc that seems to work.
My second surprise was that Gemini suggested a formula for the generalized Woodbury lemma that was unknown to me, and I was unable to find on Scholar Google or https://math.stackexchange.com. Based on a special case of that formula, I was able to write the second proc myself.
In conclusion, to start working it can be helpful to collaborate with AI friend, a little patience may help, AI may not be astute as someone on Mapleprimes wrote, but neither am I. I am now retired, and it is fun to play with Maple and AI.
By the way, the search term Woodbury did not give a single hit on Mapleprimes.With_a_little_help_from_my_friends.mw
kind regards,Harry

Major deficiency in Physics[Vectors]; Distinct...

by: nmacsai 260 Product: Maple 2024

December 12 2024

0 0

Major deficiency in Physics[Vectors]; Distinct sets of basis vectors are not recognized!

You can't define vectors in alternative bases like: {\hat{i}',\hat{j}',\hat{k}'} or {\hat{i}_{1},\hat{j}_{2},\hat{k}_{3}}.

This deficiency has been around for a while. I have found other posts regarding this problem.

The deficiency greatly reduces the allowable calculations with Physics[Vector].

Are there any plans to fix this?

Here is my example which shows this deficiency in more detail.

physics_vectors_and_multiple_unit_vectors.mw

[`&x`, `+`, `.`, Assume, ChangeBasis, ChangeCoordinates, CompactDisplay, Component, Curl, DirectionalDiff, Divergence, Gradient, Identify, Laplacian, Nabla, Norm, ParametrizeCurve, ParametrizeSurface, ParametrizeVolume, Setup, Simplify, `^`, diff, int]

(1)

Crucial Deficiency in Physics[Vectors]

I can only guess the purpose of the Physics[Vectors] package from reviewing it's corresponding help documentation. My interpretation of the documentation leads me to believe that the package is best used for generating vector equation formulas in different coordinate bases of a SINGLE coordinate system.

This means one can easily generate position vector expressions such as:

(1.1)

Cylindrical Position Vector

The position vector in a cylindrical basis is given by:

(1.1.1)

(1.1.2)

Spherical Position Vector

(1.2.1)

(1.2.2)

As is known from the vector analysis of curvilinear coordinate systems the basis vectors can depend on the coordinates in question.

In cylindrical, the basis vectors are

(1.2)

(1.3)

and in spherical, the basis vectors are

(1.4)

(1.5)

(1.6)

Example of this Deficiency using Biot-Savart Law

Biot-Savart law can be used to calculate a magnetic field due to a current carrying wire. The deficiency in question can be observed by explicity constructing the integrand in the Biot-Savart integral defined below.

In electrodynamics, quantum mechanics and applied mathematics, it is common practice to define a position of observation by a vector and a position of the source responsible for generating the field by a vector .

It is just as common to define the difference in these vectors as

(1.3.1)

and thus

(1.3.2)

as found in the integrand of the Biot-Savart integral.

It suffices to consider in a cylindrical basis for this argument.

The observation position is:

The source position is:

The deficiency in question arises because MAPLE cannot define multiple unit vectors in distinct bases such as or . These pairs of unit vectors arise naturally, as shown above in Biot-Savart law.

If we look at and generally, they look like:

If the bases vectors and are Cartesian and are not related related through rotations so that

and so,

the radial unit vectors in cylindrical are then,

In typical problems, the anglular location of the observation point, φ, is distinct from the angular location of the source, and so under this condition, .

Consider the classic problem of the magnetic field due to a circular current carrying wire. Surely, the angular coordinate of one location of the current carrying wire is different from the angular coordinate of an observation point hovering above and off-axis on the other side of the current carrying wire. See figure below.

Therefore,

What happens in MAPLE when you try to define a second distinct unit vector ?

One can easily find .

(1.3.3)

If you try to define ...

(1.3.4)

So using a prime doesn't work.

You could try a numbered subscript...

(1.3.5)

but that doesn't work.

You could try an indexed unit vector...

(1.3.6)

which can be define but is not recognized by Physics[Vectors] since...

Error, (in Physics:-Vectors:-Identify) incorrect indexed use of a unit vector: _rho[2]

And so it's just not possible with the current implementation.

Download physics_vectors_and_multiple_unit_vectors.mw

Maple Transactions Autumn Issue 2024 is now up

by: rcorless 710 Product: Maple

October 31 2024

8 0

Maple Transactions has just published the Autumn 2024 issue at mapletransactions.org

From the header:

This Autumn Issue contains a "Puzzles" section, with some recherché questions, which we hope you will find to be fun to think about. The Borwein integral (not the Borwein integral of XKCD fame, another one) set out in that section is, so far as we know, open: we "know" the value of the integral because how could the identity be true for thousands of digits but yet not be really true? Even if there is no proof. But, Jon and Peter Borwein had this wonderful paper on Strange Series and High Precision Fraud showing examples of just that kind of trickery. So, we don't know. Maybe you will be the one to prove it! (Or prove it false.)

We also have some historical papers (one by a student, discussing the work of his great grandfather), and another paper describing what I think is a fun use of Maple not only to compute integrals (and to compute them very rapidly) but which actually required us to make an improvement to a well-known tool in asymptotic evaluation of integrals, namely Watson's Lemma, just to explain why Maple is so successful here.

Finally, we have an important paper on rational interpolation, which tells you how to deal well with interpolation points that are not so well distributed.

Enjoy the issue, and keep your contributions coming.

About the inverse kinematics problem. The simplest...

by: one man 1335 Product: Maple

August 29 2024

2 8

This is another attempt to tell about one way to solve the problem of inverse kinematics of a manipulator.
We have a flat three-link manipulator. Its movement is determined by changing three angles - these are three control parameters. 1. the first link rotates around the black fixed point, 2. the second link rotates around the extreme movable point of the first link, 3. the third link − around the last point of the second link. These movable points are red. (The order of the links is from thick to thin.) The working point is green. For example, we need it to move along a circle. But the manipulator has one extra mobility (degree of freedom), that is, the problem has an infinite number of solutions. We have the ability to remove this extra degree of freedom mathematically. And this can also be done in an infinite number of ways.
Let us give two examples where the same manipulator performs the same movement of the working point in different ways. In one case the last red point moves in a straight line, and in the other case it moves in an ellipse. The result is the same. In the corresponding program texts, the manipulator model is described by a system of nonlinear equations f1, f2, f3, f4, f5 relative to the coordinates of the ends of the links (very easy to understand). The specific additional connection that takes away one degree of freedom is highlighted in blue. Equation of a circle in red color.
1.mw

2.mw

And as an elective. The same circle was obtained using a spatial 3-link manipulator with 5 degrees of freedom. In the last text, blue and red colors perform the same functions as in the previous texts.
3.mw

Introducing VerifyTools

by: aroche 560 Product: Maple

August 21 2024

5 2

VerifyTools is a package that has been available in Maple for roughly 24 years, but until now it has never been documented, as it was originally intended for internal use only. Documentation for it will be included in the next release of Maple. Here is a preview:

VerifyTools is similar to the TypeTools package. A type is essentially a predicate that a single expression can either satisfy or not. Analogously, a verification is a predicate that applies to a pair of expressions, comparing them. Just as types can be combined to produce compound types, verifications can also be combined to produce compund verifications. New types can be created, retrieved, queried, or deleted using the commands AddType, GetType (or GetTypes), Exists, and RemoveType, respectively. Similarly in the VerifyTools package we can create, retrieve, query or delete verifications using AddVerification, GetVerification (or GetVerifications), Exists, and RemoveVerification.

The package command VerifyTools:-Verify is also available as the top-level Maple command verify which should already be familiar to expert Maple users. Similarly, the command VerifyTools:-IsVerification is also available as a type, that is,

VerifyTools:-IsVerification(ver);

will return the same as

type(ver, 'verification');

The following examples show what can be done with these commands. Note that in each example where the Verify command is used, it is equivalent to the top-level Maple command verify. (Also note that VerifyTools commands shown below will be slightly different compared to the Maple2024 version):

>	with(VerifyTools):

Suppose we want to create a verification which will checks that the length of a result has not increased compared to the expected result. We can do this using the AddVerification command:

>	AddVerification(length_not_increased, (a, b) -> evalb(length(a) <= length(b)));

First, we can check the existence of our new verification and get its value:

>	Exists(length_not_increased);

>	GetVerification(length_not_increased);

For named verifications, IsVerification is equivalent to Exists (since names are only recognized as verifications if an entry exists for them in the verification database):

>	IsVerification(length_not_increased);

On the other hand, a nontrivial structured verification can be checked with IsVerification,

>	IsVerification(boolean = length_not_increased);

whereas Exists only accepts names:

>	Exists(boolean = length_not_increased);

Error, invalid input: VerifyTools:-Exists expects its 1st argument, x, to be of type symbol, but received boolean = length_not_increased

The preceding command using Exists is also equivalent to the following type call:

>	type(boolean = length_not_increased, verification);

Now, let's use the new verification:

>	Verify(x + 1/x, (x^2 + 1)/x, length_not_increased);

>	Verify((x^2 + 1)/x, x + 1/x, length_not_increased);

Finally, let's remove the verification:

>	RemoveVerification(length_not_increased);

>	Exists(length_not_increased);

>	GetVerification(length_not_increased);

Error, (in VerifyTools:-GetVerification) length_not_increased is not a recognized verification

GetVerifications returns the list of all verifications known to the system:

>	GetVerifications();

Download VerificationTools_blogpost.mw

Austin Roche
Software Architect
Mathematical Software
Maplesoft

Maple Conference 2024 - Registration is now open

by: KaskaKowalska 65 Products: Maple , Maple Learn

July 19 2024

5 0

We are pleased to announce that the registration for the Maple Conference 2024 is now open.

Like the last few years, this year’s conference will be a free virtual event. Please visit the conference page for more information on how to register.

This year we are offering a number of new sessions, including more product training options and an Audience Choice session.
You can find an overview of the program on the Sessions page. Those who register before September 10th, 2024 will have a chance to vote for the topics they want to learn more about during the Audience Choice session.

We hope to see you there!

Maple Conference 2024: Application reminder

by: KaskaKowalska 65 Products: Maple , Maple Learn

July 05 2024

0 0

This is a reminder that presentation applications for the Maple Conference are due July 17, 2024.

The conference is a a free virtual event and will be held on October 24 and 25, 2024.

We are inviting submissions of presentation proposals on a range of topics related to Maple, including Maple in education, algorithms and software, and applications. We also encourage submission of proposals related to Maple Learn. You can find more information about the themes of the conference and how to submit a presentation proposal at the Call for Participation page.

I encourage all of you here in the Maple Primes community to consider joining us for this event, whether as a presenter or an attendee!

Kaska Kowalska
Contributed Program Co-Chair

Maple Conference 2023 Proceedings

by: rcorless 710 Product: Maple

July 04 2024

3 0

The Proceedings of the Maple Conference 2023 is now out, at

mapletransactions.org

The presentations these are based on (and more) can be found at https://www.maplesoft.com/mapleconference/2023/full-program.aspx#schedule .

There are several math research papers using Maple, an application paper by an undergraduate student, an engineering application paper, and an interesting geometry teaching paper.

Please have a look, and don't forget to register for the Maple Conference 2024.

Maple Conference 2024: Call for Participation

by: KaskaKowalska 65 Products: Maple , Maple Learn

June 17 2024

6 0

We are happy to announce another Maple Conference to be held October 24 and 25, 2024!

It will be a free virtual event again this year, and it will be an excellent opportunity to meet other members of the Maple community and get the latest news about our products. More importantly, it's a chance for you to share the work you've been doing with Maple and Maple Learn.

We have just opened the Call for Participation. We are inviting submissions of presentation proposals on a range of topics related to Maple, including Maple in education, algorithms and software, and applications. We also encourage submission of proposals related to Maple Learn.

You can find more information about the themes of the conference and how to submit a presentation proposal at the Call for Participation page. Applications are due July 17, 2024.

Presenters will have the option to submit papers and articles to a special Maple Conference issue of the Maple Transactions journal after the conference.

Registration for attending the conference will open in July. Watch for further announcements in the coming weeks.

I encourage all of you here in the Maple Primes community to consider joining us for this event, whether as a presenter or an attendee!

Kaska Kowalska
Contributed Program Co-Chair

Discover What's New in Maple 2024!

by: Karishma 805 Product: Maple 2024

March 06 2024

12 2

A blue triangle with white text

Description automatically generated

Attention Maple enthusiasts! It gives me great pleasure to announce Maple 2024! Maple 2024 brings together a collection of new features and enhancements carefully designed to enrich your mathematical explorations. Maple 2024 is the result of a lot of hard work by a lot of people, and there is far more in it than I can cover here. But I’d like to share with you some of my favorite features in this release.

AI Formula Assistant

The AI Formula Assistant in Maple 2024 is undoubtedly the feature that excites me the most, especially considering how often I’m asked the question: 'When will Maple include AI features?' This assistant serves as your new mathematical companion and will change the way you look up and enter formulas and equations. Driven by advanced AI technology, it presents a range of relevant options based on your search query. Alongside suggestions, you'll also receive detailed explanations for each formula and its parameters so you can select the one you need, and then you can insert the formula into your document at a click of a button, as a proper Maple expression.

A screenshot of a computer

Description automatically generated

NaturalLanguage Package

The Formula Assistant is built on top of the new NaturalLanguage package, which integrates powerful language models like GPT-4 and ChatGPT from OpenAI into Maple. With this feature, you can leverage large language models to process natural language within Maple. Ask the AI to explain concepts, provide additional details, find specific Maple commands, and more. Of course, since this is a Maple package, you can also use it as a basis to build your own AI-powered applications inside Maple. We’re really looking forward to seeing what you will do with it!

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Description automatically generated

Argument Completion
We’ve had a lot of requests from people who wanted Maple’s command completion features to do even more, and I’m happy to say that Maple 2024 delivers. The new argument completion feature in Maple 2024 is poised to significantly enhance your experience with commands. For many users, including myself, not being aware of all the options a command takes is a challenge, often leading me to refer to help pages for clarification. With argument completion, that's no longer a concern. Just enter the command with the help of the existing command completion feature, then automatic argument completion takes over to guide you through the rest. Give it a try with the 'plot' command!

A screenshot of a computer

Description automatically generated

Check My Work

A personal favorite feature that's gotten even better in Maple 2024 is Check My Work. As someone who has tutored students, I vividly recall their stress before exams, often receiving emails and text messages from them seeking last-minute help. At the time, I found myself wishing the students had a way to check their work themselves, so we would all be less stressed! So I was super excited when we added the first Check My Work feature a couple of years ago, and am very happy that it gets better ever year. In Maple 2024, we’ve expanded its capabilities to support problems involving factoring, simplification, and limits.

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Description automatically generated

Scrollable Matrices:
This feature will definitely resonate with many of the engineers in the Maple Primes community. If you're someone who works with worksheets containing large matrices, you've likely wished that you could scroll the matrices inside your document instead of having to launch a separate matrix browser. With Maple 2024, your wish has come true.

A white sheet with numbers and lines

Description automatically generated with medium confidence

Color Bars

And finally, for those of you who appreciated the addition of color bars in Maple 2023 but wanted to see them extended to more 2D and 3D plots, you'll be delighted to know that this is exactly what we’ve done. We’ve also added new customization options, providing you with greater control over appearance.

A close-up of a graph

Description automatically generated

This is just a partial glimpse of what's new in Maple 2024. For a comprehensive overview, visit What’s New in Maple 2024.

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