MaplePrimes - Questions, Posts, help, musings, answers ... all things Maple and math

Ask a Question

Create a Post

Featured Post

Samir Khan

2071

Announcing Maple 2025

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!

March 25

14 9

Read Full Post

Featured Post

ecterrab

14370

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

(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

(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.