Posts - MaplePrimes

Book Flights Easily at the Emirates Airlines Addis...

by: airofficedesks 0

3 hours ago

0 0

Planning your next Emirates flight? Visit the for a smooth, face-to-face booking experience. The friendly staff can help you choose the best itinerary, customize your travel preferences, and answer any queries. Whether you're traveling for business or leisure, you'll get personalized support that makes planning your journey easier and more efficient than ever before.

Announcing the Maple Customer Support Updates...

by: aroche 635 Product: Maple

April 22 2025

12 5

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(4797495082876928); 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().

Notation for bilinear expression for non -linear...

by: janhardo 550

April 13 2025

1 16

The procedure gives the bilinear form of a non-linear pde
Output is not quite in an easy-to-read notation yet , but will come later on as an option in the procedure call of bilinear ( , option)
Depends what further to do with the changed pde in Hirato D-operator form?..using a ansatz ( try function with reason)

>

with(PDEtools): with(DEtools):

## Hirota Bilinear Method
## Bilinear Derivative / Hirota Operator
BD := proc(FF, DD)
    local f, g, x, m, opt;
    if nargs = 1 then
        return `*`(FF[]);
    fi;
    f, g := FF[];
    x, m := DD[];
    opt := args[3..-1];
    if m = 0 then
        return procname(FF, opt);
    fi;
    procname([diff(f, x), g], [x, m-1], opt) - procname([f, diff(g, x)], [x, m-1], opt);
end:

`print/BD` := proc(FF, DD)
    local f, g, x, m, i;
    f, g := FF[];
    f := cat(f, ` `, g);
    g := product(D[args[i][1]]^args[i][2], i = 2..nargs);
    if g <> 1 then
        f := ``(g)*``(f);
    fi;
    f;
end:

## collect(expr, f); first!
getFnumer := proc(df, f, pow::posint := 1)
    local i, g, fdenom;
    if type(df, `+`) then
        g := [op(df)];
        fdenom := map(denom, g);
        for i to nops(fdenom) while fdenom[i] <> f^pow do
        od;
        if i > nops(fdenom) then
            lprint(fdenom);
            error "no term(s) or numer=0 when denom=%1", op(0, f)^pow;
        fi;
        g := numer(g[i]);
        if not type(expand(g), `+`) then
            lprint(g);
            error "Expected more than 1 term about Hirota D-operator";
        fi;
        return g;
    fi;
    lprint(df);
    error "expected 1st argument be type `+`.";
end:

getvarpow := proc(df::function)
    local i, f, var, dif, pow;
    if op(0, df) <> diff then
        lprint(df);
        error "expected diff function";
    fi;
    f := convert(df, D);
    var := [op(f)];
    dif := [op(op([0, 0], f))];
    pow := [0$nops(var)];
    f := op(op(0, f))(var[]);
    for i to nops(var) do
        dif := selectremove(member, dif, {i});
        pow[i] := nops(dif[1]);
        dif := dif[2];
    od;
    pow := zip((x, y) -> [x, y], var, pow);
    pow := remove(has, pow, {0});
    [[f, f], pow[]];
end:

#convert to Hirota Bilinear Form
HBF := proc(df)
    local i, c, f;
    if type(df, `+`) then
        f := [op(df)];
        return map(procname, f);
    fi;
    if type(df, `*`) then
        f := [op(df)];
        f := selectremove(hasfun, f, diff);
        c := f[2];
        f := f[1];
        if nops(f) <> 1 then
            lprint(df);
            error "need only one diff function factor.";
        fi;
        f := f[];
        c := `*`(c[]);
        f := getvarpow(f);
        f := [c, f];
        return f;
    fi;
    if op(0, df) = diff then
        f := getvarpow(df);
        f := [1, f];
        return f;
    fi;
    lprint(df);
    error "unexpected type.";
end:

printHBF := proc(PL::list)
    local j, DD, f, C, tmp, gcdC, i;
    C := map2(op, 1, PL);
    gcdC := 1;
    if nops(C) > 1 then
        tmp := [seq(cat(_Z, i), i = 1..nops(C))];
        gcdC := tmp *~ C;
        gcdC := `+`(gcdC[]);
        gcdC := factor(gcdC);
        tmp := selectremove(has, gcdC, tmp);
        gcdC := tmp[2];
        if gcdC = 0 then
            gcdC := 1;
        fi;
        gcdC := gcdC*content(tmp[1]);
    fi;
    if gcdC <> 1 then
        C := C /~ gcdC;
    fi;
    DD := map2(op, 2, PL);
    f := op(0, DD[1][1][1]);
    DD := map(z -> product(D[z[i][1]]^z[i][2], i = 2..nops(z)), DD);
    DD := zip(`*`, C, DD);
    DD := `+`(DD[]);
    gcdC * ``(DD) * cat(f, ` `, f);
end:

## print Hirota Bilinear Transform
printHBT := proc(uf, u, f, i, j, PL, alpha := 1)
    local DD, g, C, tmp, pl;
    pl := printHBF(PL);
    if j > 0 then
        print(u = 2*alpha*'diff'(ln(f), x$j));
    else
        print(u = 2*alpha*ln(f));
    fi;
    if i > 0 then
        print('diff'(pl/f^2, x$i) = 0);
    else
       print(pl/f^2 = 0);
    fi;
    NULL;
end:

guessdifforder := proc(PL::list, x::name)
    local L, minorder, maxorder, tmp;
    L := map2(op, 2, PL);
    L := map(z -> z[2..-1], L);
    tmp := map(z -> map2(op, 2, z), L);
    tmp := map(z -> `+`(z[]), tmp);
    tmp := selectremove(type, tmp, even);
    minorder := 0;
    if nops(tmp[1]) < nops(tmp[2]) then
        minorder := 1;
    fi;
    tmp := map(z -> select(has, z, {x}), L);
    tmp := map(z -> map2(op, 2, z), tmp);
    if has(tmp, {[]}) then
        maxorder := 0;
    else
        tmp := map(op, tmp);
        maxorder := min(tmp[]);
    fi;
    if type(maxorder - minorder, odd) then
        maxorder := maxorder - 1;
    fi;
    [minorder, maxorder];
end:

guessalpha := proc(Res, uf, u, f, i, j, PL, alpha)
    local tmp, res, pl, flag, k;
    flag := 1;
    tmp := [op(Res)];
    tmp := map(numer, tmp);
    tmp := gcd(tmp[1], tmp[-1]);
    if type(tmp, `*`) then
        tmp := remove(has, tmp, f);
    fi;
    if tmp <> 0 and has(tmp, {alpha}) then
        tmp := solve(tmp/alpha^difforder(uf), {alpha});
        if tmp <> NULL and has(tmp, {alpha}) then
            lprint(tmp);
            for k to nops([tmp]) while flag = 1 do
                res := collect(expand(subs(tmp[k], Res)), f, factor);
                if res = 0 then
                    pl := subs(tmp[k], PL);
                    printHBT(uf, u, f, i, j, pl, rhs(tmp[k]));
                    flag := 0;
                fi;
            od;
        fi;
    fi;
    PL;
end:

Bilinear := proc(uf, u, f, x, alpha)
    local su, h, i, j, g1, CB, PL, gdo, DD, Res;
    if hasfun(uf, int) then
        error "Do not support integral function yet. Please substitute int function.";
    fi;
    for j from 0 to 2 do
        Res := 1;
        su := u = 2*alpha*diff(ln(f), [x$j]);
        h := collect(expand(dsubs(su, uf)), f, factor);
        if hasfun(h, ln) then
            next;
        fi;
        g1 := getFnumer(h, f)/2;
        g1 := expand(g1);
        CB := HBF(g1);
        gdo := guessdifforder(CB, x);
        for i from gdo[1] by 2 to gdo[2] do
            if i = 0 then
                PL := CB;
            else
                PL := HBF(int(g1, x$i));
            fi;
            DD := add(PL[i][1]*BD(PL[i][2][]), i = 1..nops(PL));
            Res := collect(expand(diff(DD/f^2, [x$i]) - h), f, factor);
            if Res = 0 then
                printHBT(uf, u, f, i, j, PL, alpha);
                break;
            elif type(alpha, name) and has(DD, alpha) then
                Res := guessalpha(Res, uf, u, f, i, j, PL, alpha);
            fi;
        od;
        if Res = 0 then
            break;
        fi;
    od;
    PL;
end:

>	with(PDEtools): # Definieer de Boussinesq vergelijking boussinesq := diff(u(x,t),t,t) - diff(u(x,t),x,x) - 3*diff(u(x,t),x,x)^2 - diff(u(x,t),x$4); # Pas de bilineaire transformatie toe Bilinear(boussinesq, u(x,t), f(x,t), x, alpha);

diff(diff(u(x, t), t), t)-(diff(diff(u(x, t), x), x))-3*(diff(diff(u(x, t), x), x))^2-(diff(diff(diff(diff(u(x, t), x), x), x), x))

{alpha = 1}

``(-D[x]^4+D[t]^2-D[x]^2)*`f f`/f(x, t)^2 = 0

(1)

>

>	# Definieer de Kadomtsev–Petviashvili (KP) vergelijking in één regel kp := diff(diff(u(x,y,t),x$3) + 6u(x,y,t)diff(u(x,y,t),x) + diff(u(x,y,t),t), x) + 3delta^2diff(u(x,y,t), y$2): Bilinear(kp, u(x,y,t), f(x,y,t), x, alpha);

{alpha = 1}

u(x, y, t) = 2*(diff(ln(f(x, y, t)), x, x))

diff(``(3*delta^2*D[y]^2+D[x]^4+D[t]*D[x])*`f f`/f(x, y, t)^2, x, x) = 0

(2)

>

printHBF := proc(PL::list) ?

Download dbilinear_proc_def_13-4-2025.mw

I find an error

by: Yuri Zamorano 25

April 11 2025

1 7

In Statistics Library there is an error with PowerFit function. The summarize of results show R-Squared and Adjusted R-Squared with erroneal results. Please could you fix it?

Maple 2025 - much slower than 2024?

by: JanBSDenmark 60 Product: Maple 2025

April 07 2025

5 6

I teach math at the high school level.

I am worried that Maple 2025 appears to be slower than Maple 2024 - in particular for students with older, less strong laptops.

Maple 2025 takes 50% longer to start than Maple 2024 (or Maple 2025 Screen Reader which I expect to be using).

So, on more sluggist student laptops I fear the slowness overall will be an issue - in particular as Maple regularly has to be shutdown and restarted for some of those students.

Further, I really miss the "recompute section !" and the "magniffy" icons on the quest access bar. Having "recompute entire worksheet !!!" seems unwise though. I wish you could costumize the quest access bar.

Overall, from a teaching point of view, I am not at all impressed, sadly.

Approximation of ODEs with cubic splines

by: Kiefer Gerhard 10 Product: Maple 2022

April 06 2025

0 0

Approximation_of_ODEs_with_Cubic_Splines.mw

Errors with formulas from the AI assistant with...

by: C_R 3222 Product: Maple 2025

April 05 2025

1 0

Just an observation.

I was wondering if less obvious errors than in the below can be avoided with future versions of the AI assistant. Maybe a warning that a formula uses special Maple symbols is possible.

Formulas without dimensions are more susceptible to undetected errors.

Deflection of a circular cantilever

(a first attemp with the AI formula assistant)

(1)

AI prompt: Deflection of a circular cantilever with a force applied at the end

Correct formular inserted ->

(2)

AI prompt: Moment of inertia of a circular cross-section

Correct formular inserted ->

(3)

delta = (4/3)*F*L^3/(E*Pi*R^4)

(4)

(5)

delta = 0.1034759757e-8*Units:-Unit(N)*Units:-Unit(m)^3/(Units:-Unit(N/mm^2)*Units:-Unit(mm)^4)

(6)

(7)

The dimension of m^9 for a deflection clearly indicates an error.

A better prompt to avoid this error (caused by automatic simplification) could not be found

Download AI_formula_assistant.mw

P.S.:

This is a real example that happend to me where I did not notice the minus sign in Maples output in equation (1). The error can easily be fixed by adding "local I" as the first statement of the document and the deflection becomes 1 mm.

Maple 2026 - improvement suggestions - global...

by: Christopher2222 5925 Product: Maple

April 04 2025

2 1

I don't have the latest Maple, and I'm sure this isn't in the latest version.

One thing that has been an annoyance for all time, and it gets me time and time again, is not having a global degrees or radians setting.

Of course it needs to be a setting option, otherwise it would break many older worksheets.

The 2025 Maplesoft Physics Updates and its future

by: ecterrab 14400 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.

Bait-and-switch

by: Christian Wolinski 1645 Product: MaplePrimes

March 29 2025

2 3

So, someone just posted spam here. Cleverly designed, the title was an HTTP link that sent the reader to a .in website. Accessing the actual post was impossible, and marking it as spam was impossible as well. Debugging tools had to be used.

The design of this website should be improved to make this impossible. An unalterable icon should be present to connect the reader to the post without relying on the user-defined title.

What is new in Physics in Maple 2025

by: ecterrab 14400 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 14400 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

Announcing Maple 2025

by: Samir Khan 2071 Product: Maple 2025

March 25 2025

14 9

I’m absolutely delighted to announce the launch of Maple 2025!

Although you see a new release every year, new features take anything from a few fast-paced weeks to develop, to months of careful cultivation.

Working on so many features in parallel, each with varying time scales, isn't easy! We have to fastidiously manage and track our work.

So it's easy to lose ourselves in the daily minutiae of software development. To help us maintain perspective, we constantly ask ourselves questions like:

What user problem are we solving and how often does this problem occur?
Can we validate our proposed solution with preliminary user feedback?
Is this a solution to a problem that doesn't exist and will never exist, or are we pre-empting a future need?
Are we offering value to our users?

Given the answers, we course-correct to make sure we stay on track for our central mission - to make you happy, and to keep you coming back year-after-year.

With Maple 2025, I think we've smashed that goal. We have many new features that'll appeal to many different types of users - from students, educators and mathematicians, to engineers, scientists and technical professionals

Let me walk you through some of my personal highlights.

It’ll be difficult for anyone to miss this - Maple 2025 has a new interface! It’s a ribbon-based UI that look clean and contemporary, and helps you find and discover tools more quickly than before.

You have large, meaningful icons.

Items are logically grouped.

The ribbons is contextual. If you click on a plot, you get new tabs for interacting with and drawing on the plot.

A new Education tab collects pedagogical resources that were scattered around the interface in prior releases.

This is the biggest visual overhaul to Maple in many years. We hope you like it!

We also appreciate that changes in look and feel can be divisive. Please rest assured that we will refine and finesse the interface with each successive release; your comments and suggestions are most welcome.

The new interface is available on Windows and Linux, and as a technology preview on Mac.

The right arrow key on my keyboard is wearing out…and it’s all because of Maple. I’m knee deep in Maple nearly every day entering equations, and I’m always using right-arrow to move the cursor. It gets kind of tedious!

This anecdote reflects some investigative work we did. We comprehensively examined our internal library of thousands of Maple worksheets and discovered that these three input patterns are extremely common.

Previously, you’d use the right-arrow key to move the cursor out of the exponential, division or subscript.

Now, in Maple 2025, when you

type ^, /, or enter a literal subscript with a double-underscore,
followed by a number or symbol
and then input another operator (such as +)

the operator is automatically inserted on the baseline (except when y = 1).

Of course, you can also make the cursor return to or stay in the exponent or denominator with a simple keystroke, when that is what is needed.

This is one of those little quality of life refinements that I’m very fond of - it’s a little visual and usability dopamine hit.

The sum command (and its typeset form) now indexes into vectors without you needing to spam unevaluation quotes all over your expression.

Maple 2024	Maple 2025

We’ve been integrating units deeper into the Maple system, release after release. Much of this is driven by our engineering users.

A few releases ago, we made int(numeric) compatible with units. With Maple 2025, you can now numerically differentiate expressions and procedures that have units.

I’m a grizzled thermodynamics hack, so here’s an example in which I calculate the specific heat capacity of water by differentiating enthalpy with respect to temperature (and then confirm the result with the built-in value):

This is in addition to many other improvements to the units experience.

Although this is a part of Maple that I don’t touch often (my colleague Karishma takes point on the education side), I REALLY wish I’d had this when I was struggling with math.

You can now automatically generate unlimited variants of the same problem for students to solve with the Try Another feature, which has been added to Maple’s Check My Work tools (another feature I really could have used!). This is available for many common math principles, including factorization, simplification, integration and more.

This is just one of the improvements in Maple 2025 for teaching and learning.

If you’ve ever found yourself going back and forth (and back and forth) between two large, almost identical-looking Maple expressions, trying to figure out how they are different, you’re going to love this one. ExpressionTools is a new package that lets you compare the differences between two expressions.

I really like the use of color to highlight differences. Less squinting at the screen!

You can now run Maple Flow worksheets from Maple (you don’t need Maple Flow installed to do this). You can send parameters into the Flow worksheet and extract your desired results.

This means you can use the entire flexibility of Maple to analysis and manipulate your Flow worksheet. You could, for example:

Attach a Flow worksheet to a Maple workbook and create an interactive application
Carry out parameter studies of a Flow worksheet by evaluating it over many parameter sets in Maple
Create an Excel interface for a Flow worksheet using the Maple add-in for Excel

Simplify is one of those functions that literally tens of thousands of people use each day. Every time we make an incremental improvement, the cumulative benefits across our entire user base are significant.

We’ve refined simplify in a number of critical ways. For example, simplify now recognizes when exponentials can be profitably converted to hyperbolic trig functions:

The analysis of many scientific phenomena result in Laplace transforms that do not have a symbolic inverse which can be expressed in terms of elementary functions. This includes applications in heat transfer, fluid mechanics, fractional diffusion processes, control systems and electrical transmission.

For example, this monster Laplace transform results from an analysis of voltage on a transmission line:

You can now numerically invert this transform courtesy of an enhancement to inttrans:-invlaplace - a fast quadrature method.

I’ve saved what I think has the most future potential for last.

I’m sure nearly all of you have experimented with the various AI tools. They’re an inevitable part of our present and future, whether we're comfortable with it or not.

This is something we've been mulling over for some time.

In Maple 2019, the DeepLearning package made its debut. This package provides tools for machine learning, supporting operations such as classification and regression using neural networks.
In Maple 2024, we introduced an AI-powered formula lookup feature.

In Maple 2025, we’re giving you an early-stage technology preview of AI-powered document generation.

You can automatically generate worksheet content by prompting an AI, and then gradually refine the content

If you’re an educator, you might want some content that describes applications of calculus. So you might ask the AI “How do I derive the formula for the area of a circle” by entering your prompt into this text box:

This is the worksheet content that may be returned:

If you’re structural engineer who wants to know how to calculate the hardness of concrete, you might ask the AI: “How do I calculate the compressive strength of slow hardening concrete as a function of time? Use the CEB-FIP Model Code 90. Include a worked example with Maple code”.

This worksheet content that could be generated (note the live Maple code):

We’re labelling AI-generated worksheet content as a technology preview. You might see

text that might be misleading (but sounds plausible)
code that doesn’t work (but looks plausible)
or different results each time you click “Generate Document”

For the moment, I would not rely on AI-generated worksheet content without realistic expectations, a healthy dose of scepticism and a modicum of detached analysis. But AI models are rapidly growing in robustness, and we want to position ourselves to best exploit their future potential. The next few years will be VERY exciting.

We can never cover everything in a short blog post like this. So if you want to know more, head on over to the What’s New pages for Maple 2025!

Matt's attempt at Ali-quot Sequences

by: Matt C Anderson 190 Product: Maple 13

March 22 2025

1 1

Hi MaplePrimes,
I made a quick code to calculate Aliquot Sequences.

The code works.

I was inspired by a Numberphile YouTube video.

see

aliquot_sequence_cut_1.mw

aliquot_sequence_cut_1.pdf

This should be helpful.

Regards,

Matt

An Introduction to Limits with Maple

by: Paras31 215 Product: Maple 2024

March 21 2025

2 0

LIMITS

Limits in maths are defined as the values that a function approaches the output for the given input values. Limits play a vital role in calculus and mathematical analysis and are used to define integrals, derivatives, and continuity. It is used in the analysis process, and it always concerns the behavior of the function at a particular point. The limit of a sequence is further generalized in the concept of the limit of a topological net and related to the limit and direct limit in the theory category. Generally, the integrals are classified into two types namely, definite and indefinite integrals. For definite integrals, the upper limit and lower limits are defined properly. Whereas indefinite integrals are expressed without limits, and it will have an arbitrary constant while integrating the function.

Sometimes we can't work something out directly ... but we can see what it should be as we get closer and closer!

Example 1

>

"restart; f(x):=(|x|-3)/(x-3);"

(1)

>

Error, (in f) numeric exception: division by zero

Now 0/0 is a difficulty! We don't really know the value of 0/0 (it is "indeterminate"), so we need another way of answering this.

So instead of trying to work it out for x=3 let's try approaching it closer and closer:

>

(2)

>

(3)

>

(4)

>

(5)

>

(6)

>

(7)

>

(8)

Example 2

Sometimes some functions are not continuous. That is, they appear to be approaching two different values when they are approached from two sides.

>

proc (x) options operator, arrow, function_assign; piecewise(0 < x and x < 2, 1/(2*x-x^2), 2 < x and x <= 3, 2-x, 3 < x and x < 4, x-4, 4 <= x, Pi, undefined) end proc

(9)

>

Suppose we want to approach 2 and see the function’s limit. This naturally leads to directions from which we can approach. Left-hand side and the right-hand side limits.

The right-hand side limit is the value of the function that it takes while approaching it from the right-hand side of the desired point. Similarly, the left-hand side limit is the value of function while approaching it from the left-hand side.

>

(10)

>

(11)

>

(12)

>

(13)

And the ordinary limit "does not exist".

>

(14)

>

(15)

>

(16)

>

(17)

And the ordinary limit "does not exist".

>

Example 3

Estimate the value of the following limit limit(h(x)*where, x = 2), h(x) = piecewise(x <> 2, x+12, x = 2, 4) .

>

"h(x):={[[x+12,x<>2],[4,x=2]];"

proc (x) options operator, arrow, function_assign; piecewise(x <> 2, x+12, x = 2, 4) end proc

(18)

>

(19)

The limit is NOT 2025!Remember from the first example that limits do not care what the function is actually doing at the point in question. Limits are only concerned with what is going on around the point. Since the only thing about the function that we actually changed was its behavior at this will not change the limit.

Example 4

>

proc (x) options operator, arrow, function_assign; piecewise(x < 0, -x+5, 0 <= x, 2*x) end proc

(20)

>

(21)

>

(22)

>

(23)

Example 5

>

proc (x) options operator, arrow, function_assign; piecewise(x < 5, x+4, 5 <= x, x^2-2) end proc

(24)

>

(25)

>

(26)

>

(27)

>

(28)

>

(29)

Example 6

>

proc (x) options operator, arrow, function_assign; piecewise(x <= 1, (x-8)/(x-3), 3 <= x, sqrt(x^2+x+2), undefined) end proc

(30)

>

(31)

>

(32)

>

(33)

>

(34)

Example 7

Estimate the value of the following limit. where, H(t) = piecewise(t < 0, 0, t >= 0, 1)

>

proc (t) options operator, arrow, function_assign; piecewise(t < 0, 0, 0 <= t, 1) end proc

(35)

This function is often called either the Heaviside or step function. We could use a table of values to estimate the limit, but it’s probably just as quick in this case to use the graph so let’s do that. Below is the graph of this function.

>

(36)

>

(37)

We can see from the graph that if we approach from the right side the function is moving in towards a value of 1. Well actually it’s just staying at 1, but in the terminology that we’ve been using in this section it’s moving in towards 1.

Also, if we move in towards from the left the function is moving in towards a value of 0.

According to our definition of the limit the function needs to move in towards a single value as we move in towards (from both sides). This isn’t happening in this case and so in this example we will also say that the limit doesn’t exist.

Download limits.mw

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