Applications, Examples and Libraries

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This Saturday is Canada’s 150th birthday. As you can imagine, the country has been paying a lot more attention to this year’s anniversary than our usual low key approach, and as a Canadian company, we at Maplesoft decided to join in the fun.

And what better way for Maplesoft to celebrate Canada’s birthday than to create a maple leaf in Maple! 

So here is a maple leaf inspired by the Canada 150 logo, which was created by Ariana Cuvin, a student at the University of Waterloo and former co-op here at Maplesoft:

Here’s the code to reproduce this plot:

p:=thickness=5,color="#DC2828":
plots:-display(

    plot([[-.216,-.216],[0,0],[-.216,.216],[-.81,0],[-.216,-.216]],p),  
    plot([[-.55,.095],[-.733,.236],[-.49,.245]],p),
    plot([[-.376,0],[0,0],[0,.376],[-.705,.705],[-.376,0]],p),
    plot([[-.342,.536],[-.355,.859],[-.138,.622]],p),
    plot([[-.267,.267],[0,0],[.267,.267],[0,1],[-.267,.267]],p),
    plot([[.342,.536],[.355,.859],[.138,.622]],p),
    plot([[0.,.376],[0,0],[.376,0],[.705,.705],[0.,.376]],p),
    plot([[.55,.095],[.733,.236],[.49,.245]],p),
    plot([[.216,.216],[0,0],[.216,-.216],[.81,0],[.216,.216]],p),
    plot([[0,-.5],[0,0]],p),

scaling=constrained,view=[-1..1,-.75..1.25],axes=box);

 

Know other ways to plot a maple leaf in Maple?  If so, please share them below - we’d love to see them!

 

Physics

 

 

Maple provides a state-of-the-art environment for algebraic and tensorial computations in Physics, with emphasis on ensuring that the computational experience is as natural as possible.

 

The theme of the Physics project for Maple 2017 has been the consolidation of the functionality introduced in previous releases, together with significant enhancements and new functionality in General Relativity, in connection with classification of solutions to Einstein's equations and tensor representations to work in an embedded 3D curved space - a new ThreePlusOne  package. This package is relevant in numerical relativity and a Hamiltonian formulation of gravity. The developments also include first steps in connection with computational representations for all the objects entering the Standard Model in particle physics.

Classification of solutions to Einstein's equations and the Tetrads package

 

In Maple 2016, the digitizing of the database of solutions to Einstein's equations  was finished, added to the standard Maple library, with all the metrics from "Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; and Herlt, E., Exact Solutions to Einstein's Field Equations". These metrics can be loaded to work with them, or change them, or searched using g_  (the Physics command representing the spacetime metric that also sets the metric to your choice in one go) or using the command DifferentialGeometry:-Library:-MetricSearch .


In Maple 2017, the Physics:-Tetrads  package has been vastly improved and extended, now including new commands like PetrovType  and SegreType  to classify these metrics, and the TransformTetrad  now has an option canonicalform to automatically derive a transformation and put the tetrad in canonical form (reorientation of the axis of the local system of references), a relevant step in resolving the equivalence between two metrics.

Examples

 

Petrov and Segre types, tetrads in canonical form

   

Equivalence for Schwarzschild metric (spherical and Kruskal coordinates)

 

Formulation of the problem (remove mixed coordinates)

   

Solving the Equivalence

   

The ThreePlusOne (3 + 1) new Maple 2017 Physics package

 

ThreePlusOne , is a package to cast Einstein's equations in a 3+1 form, that is, representing spacetime as a stack of nonintersecting 3-hypersurfaces Σ. This 3+1 description is key in the Hamiltonian formulation of gravity as well as in the study of gravitational waves, black holes, neutron stars, and in general to study the evolution of physical system in general relativity by running numerical simulations as traditional initial value (Cauchy) problems. ThreePlusOne includes computational representations for the spatial metric gamma[i, j] that is induced by g[mu, nu] on the 3-dimensional hypersurfaces, and the related covariant derivative, Christoffel symbols and Ricci and Riemann tensors, the Lapse, Shift, Unit normal and Time vectors and Extrinsic curvature related to the ADM equations.

 

The following is a list of the available commands:

 

ADMEquations

Christoffel3

D3_

ExtrinsicCurvature

gamma3_

Lapse

Ricci3

Riemann3

Shift

TimeVector

UnitNormalVector

 

 

The other four related new Physics  commands:

 

• 

Decompose , to decompose 4D tensorial expressions (free and/or contracted indices) into the space and time parts.

• 

gamma_ , representing the three-dimensional metric tensor, with which the element of spatial distance is defined as  `#mrow(msup(mi("dl"),mrow(mo("⁢"),mn("2"))),mo("="),msub(mi("γ",fontstyle = "normal"),mrow(mi("i"),mo(","),mi("j"))),mo("⁢"),msup(mi("dx"),mi("i")),mo("⁢"),msup(mi("dx"),mi("j")))`.

• 

Redefine , to redefine the coordinates and the spacetime metric according to changes in the signature from any of the four possible signatures(− + + +), (+ − − −), (+ + + −) and ((− + + +) to any of the other ones.

• 

EnergyMomentum , is a computational representation for the energy-momentum tensor entering Einstein's equations as well as their 3+1 form, the ADMEquations .

 

Examples

 

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

`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (x, y, z, t)}

 

`Systems of spacetime Coordinates are: `*{X = (x, y, z, t)}

 

[coordinatesystems = {X}]

(2.1.1)

with(ThreePlusOne)

`Setting lowercaselatin_is letters to represent space indices `

 

0, "%1 is not a command in the %2 package", ThreePlusOne, Physics

 

`Changing the signature of spacetime from `(`- - - +`)*` to `(`+ + + -`)*` in order to match the signature customarily used in the ADM formalism`

 

[ADMEquations, Christoffel3, D3_, ExtrinsicCurvature, Lapse, Ricci3, Riemann3, Shift, TimeVector, UnitNormalVector, gamma3_]

(2.1.2)

Note the different color for gamma[mu, nu], now a 4D tensor representing the metric of a generic 3-dimensional hypersurface induced by the 4D spacetime metric g[mu, nu]. All the ThreePlusOne tensors are displayed in black to distinguish them of the corresponding 4D or 3D tensors. The particular hypersurface gamma[mu, nu] operates is parameterized by the Lapse  alpha and the Shift  beta[mu].

The induced metric gamma[mu, nu]is defined in terms of the UnitNormalVector  n[mu] and the 4D metric g[mu, nu] as

gamma3_[definition]

Physics:-ThreePlusOne:-gamma3_[mu, nu] = Physics:-ThreePlusOne:-UnitNormalVector[mu]*Physics:-ThreePlusOne:-UnitNormalVector[nu]+Physics:-g_[mu, nu]

(2.1.3)

where n[mu] is defined in terms of the Lapse  alpha and the derivative of a scalar function t that can be interpreted as a global time function

UnitNormalVector[definition]

Physics:-ThreePlusOne:-UnitNormalVector[mu] = -Physics:-ThreePlusOne:-Lapse*Physics:-D_[mu](t)

(2.1.4)

The TimeVector  is defined in terms of the Lapse  alpha and the Shift  beta[mu] and this vector  n[mu] as

TimeVector[definition]

Physics:-ThreePlusOne:-TimeVector[mu] = Physics:-ThreePlusOne:-Lapse*Physics:-ThreePlusOne:-UnitNormalVector[mu]+Physics:-ThreePlusOne:-Shift[mu]

(2.1.5)

The ExtrinsicCurvature  is defined in terms of the LieDerivative  of  gamma[mu, nu]

ExtrinsicCurvature[definition]

Physics:-ThreePlusOne:-ExtrinsicCurvature[mu, nu] = -(1/2)*Physics:-LieDerivative[Physics:-ThreePlusOne:-UnitNormalVector](Physics:-ThreePlusOne:-gamma3_[mu, nu])

(2.1.6)

The metric gamma[mu, nu]is also a projection tensor in that it projects 4D tensors into the 3D hypersurface Σ. The definition for any 4D tensor that is also a 3D tensor in Σ, can thus be written directly by contracting their indices with gamma[mu, nu]. In the case of Christoffel3 , Ricci3  and Riemann3,  these tensors can be defined by replacing the 4D metric g[mu, nu] by gamma[mu, nu] and the 4D Christoffel symbols GAMMA[mu, nu, alpha] by the ThreePlusOne GAMMA[mu, nu, alpha] in the definitions of the corresponding 4D tensors. So, for instance

Christoffel3[definition]

Physics:-ThreePlusOne:-Christoffel3[mu, nu, alpha] = (1/2)*Physics:-ThreePlusOne:-gamma3_[mu, `~beta`]*(Physics:-d_[alpha](Physics:-ThreePlusOne:-gamma3_[beta, nu], [X])+Physics:-d_[nu](Physics:-ThreePlusOne:-gamma3_[beta, alpha], [X])-Physics:-d_[beta](Physics:-ThreePlusOne:-gamma3_[nu, alpha], [X]))

(2.1.7)

Ricci3[definition]

Physics:-ThreePlusOne:-Ricci3[mu, nu] = Physics:-d_[alpha](Physics:-ThreePlusOne:-Christoffel3[`~alpha`, mu, nu], [X])-Physics:-d_[nu](Physics:-ThreePlusOne:-Christoffel3[`~alpha`, mu, alpha], [X])+Physics:-ThreePlusOne:-Christoffel3[`~beta`, mu, nu]*Physics:-ThreePlusOne:-Christoffel3[`~alpha`, beta, alpha]-Physics:-ThreePlusOne:-Christoffel3[`~beta`, mu, alpha]*Physics:-ThreePlusOne:-Christoffel3[`~alpha`, nu, beta]

(2.1.8)

Riemann3[definition]

Physics:-ThreePlusOne:-Riemann3[mu, nu, alpha, beta] = Physics:-g_[mu, lambda]*(Physics:-d_[alpha](Physics:-ThreePlusOne:-Christoffel3[`~lambda`, nu, beta], [X])-Physics:-d_[beta](Physics:-ThreePlusOne:-Christoffel3[`~lambda`, nu, alpha], [X])+Physics:-ThreePlusOne:-Christoffel3[`~lambda`, upsilon, alpha]*Physics:-ThreePlusOne:-Christoffel3[`~upsilon`, nu, beta]-Physics:-ThreePlusOne:-Christoffel3[`~lambda`, upsilon, beta]*Physics:-ThreePlusOne:-Christoffel3[`~upsilon`, nu, alpha])

(2.1.9)

When working with the ADM formalism, the line element of an arbitrary spacetime metric can be expressed in terms of the differentials of the coordinates dx^mu, the Lapse , the Shift  and the spatial components of the 3D metric gamma3_ . From this line element one can derive the relation between the Lapse , the spatial part of the Shift , the spatial part of the gamma3_  metric and the g[0, j] components of the 4D spacetime metric.

For this purpose, define a tensor representing the differentials of the coordinates and an alias  dt = `#msup(mi("dx"),mn("0"))`

Define(dx[mu])

`Defined objects with tensor properties`

 

{Physics:-ThreePlusOne:-D3_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-ThreePlusOne:-Ricci3[mu, nu], Physics:-ThreePlusOne:-Shift[mu], Physics:-d_[mu], dx[mu], Physics:-g_[mu, nu], Physics:-ThreePlusOne:-gamma3_[mu, nu], Physics:-gamma_[i, j], Physics:-ThreePlusOne:-Christoffel3[mu, nu, alpha], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-ThreePlusOne:-Riemann3[mu, nu, alpha, beta], Physics:-ThreePlusOne:-TimeVector[mu], Physics:-ThreePlusOne:-ExtrinsicCurvature[mu, nu], Physics:-ThreePlusOne:-UnitNormalVector[mu], Physics:-SpaceTimeVector[mu](X)}

(2.1.10)

"alias(dt = dx[~0]):"

The expression for the line element in terms of the Lapse  and Shift   is (see [2], eq.(2.123))

ds^2 = (-Lapse^2+Shift[i]^2)*dt^2+2*Shift[i]*dt*dx[`~i`]+gamma_[i, j]*dx[`~i`]*dx[`~j`]

ds^2 = (-Physics:-ThreePlusOne:-Lapse^2+Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`])*dt^2+2*Physics:-ThreePlusOne:-Shift[i]*dt*dx[`~i`]+Physics:-gamma_[i, j]*dx[`~i`]*dx[`~j`]

(2.1.11)

Compare this expression with the 3+1 decomposition of the line element in an arbitrary system. To avoid the automatic evaluation of the metric components, work with the inert form of the metric %g_

ds^2 = %g_[mu, nu]*dx[`~mu`]*dx[`~nu`]

ds^2 = %g_[mu, nu]*dx[`~mu`]*dx[`~nu`]

(2.1.12)

Decompose(ds^2 = %g_[mu, nu]*dx[`~mu`]*dx[`~nu`])

ds^2 = %g_[0, 0]*dt^2+%g_[0, j]*dt*dx[`~j`]+%g_[i, 0]*dt*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`]

(2.1.13)

The second and third terms on the right-hand side are equal

op(2, rhs(ds^2 = dt^2*%g_[0, 0]+dt*%g_[0, j]*dx[`~j`]+dt*%g_[i, 0]*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`])) = op(3, rhs(ds^2 = dt^2*%g_[0, 0]+dt*%g_[0, j]*dx[`~j`]+dt*%g_[i, 0]*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`]))

%g_[0, j]*dt*dx[`~j`] = %g_[i, 0]*dt*dx[`~i`]

(2.1.14)

subs(%g_[0, j]*dt*dx[`~j`] = %g_[i, 0]*dt*dx[`~i`], ds^2 = dt^2*%g_[0, 0]+dt*%g_[0, j]*dx[`~j`]+dt*%g_[i, 0]*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`])

ds^2 = %g_[0, 0]*dt^2+2*%g_[i, 0]*dt*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`]

(2.1.15)

Taking the difference between this expression and the one in terms of the Lapse  and Shift  we get

simplify((ds^2 = dt^2*%g_[0, 0]+2*dt*%g_[i, 0]*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`])-(ds^2 = (-Physics:-ThreePlusOne:-Lapse^2+Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`])*dt^2+2*Physics:-ThreePlusOne:-Shift[i]*dt*dx[`~i`]+Physics:-gamma_[i, j]*dx[`~i`]*dx[`~j`]))

0 = (Physics:-ThreePlusOne:-Lapse^2-Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]+%g_[0, 0])*dt^2+2*dx[`~i`]*(%g_[i, 0]-Physics:-ThreePlusOne:-Shift[i])*dt-dx[`~i`]*dx[`~j`]*(Physics:-gamma_[i, j]-%g_[i, j])

(2.1.16)

Taking coefficients, we get equations for the Shift , the Lapse  and the spatial components of the metric gamma3_

eq[1] := coeff(coeff(rhs(0 = (Physics:-ThreePlusOne:-Lapse^2-Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]+%g_[0, 0])*dt^2+2*dx[`~i`]*(%g_[i, 0]-Physics:-ThreePlusOne:-Shift[i])*dt-dx[`~i`]*dx[`~j`]*(Physics:-gamma_[i, j]-%g_[i, j])), dt), dx[`~i`]) = 0

2*%g_[i, 0]-2*Physics:-ThreePlusOne:-Shift[i] = 0

(2.1.17)

eq[2] := coeff(rhs(0 = (Physics:-ThreePlusOne:-Lapse^2-Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]+%g_[0, 0])*dt^2+2*dx[`~i`]*(%g_[i, 0]-Physics:-ThreePlusOne:-Shift[i])*dt-dx[`~i`]*dx[`~j`]*(Physics:-gamma_[i, j]-%g_[i, j])), dt^2)

Physics:-ThreePlusOne:-Lapse^2-Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]+%g_[0, 0]

(2.1.18)

eq[3] := coeff(coeff(rhs(0 = (Physics:-ThreePlusOne:-Lapse^2-Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]+%g_[0, 0])*dt^2+2*dx[`~i`]*(%g_[i, 0]-Physics:-ThreePlusOne:-Shift[i])*dt-dx[`~i`]*dx[`~j`]*(Physics:-gamma_[i, j]-%g_[i, j])), dx[`~i`]), dx[`~j`]) = 0

-Physics:-gamma_[i, j]+%g_[i, j] = 0

(2.1.19)

Using these equations, these quantities can all be expressed in terms of the time and space components of the 4D metric g[0, 0] and g[i, j]

isolate(eq[1], Shift[i])

Physics:-ThreePlusOne:-Shift[i] = %g_[i, 0]

(2.1.20)

isolate(eq[2], Lapse^2)

Physics:-ThreePlusOne:-Lapse^2 = Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]-%g_[0, 0]

(2.1.21)

isolate(eq[3], gamma_[i, j])

Physics:-gamma_[i, j] = %g_[i, j]

(2.1.22)

References

 
  

[1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

  

[2] Alcubierre, M., Introduction to 3+1 Numerical Relativity, International Series of Monographs on Physics 140, Oxford University Press, 2008.

  

[3] Baumgarte, T.W., Shapiro, S.L., Numerical Relativity, Solving Einstein's Equations on a Computer, Cambridge University Press, 2010.

  

[4] Gourgoulhon, E., 3+1 Formalism and Bases of Numerical Relativity, Lecture notes, 2007, https://arxiv.org/pdf/gr-qc/0703035v1.pdf.

  

[5] Arnowitt, R., Dese, S., Misner, C.W., The Dynamics of General Relativity, Chapter 7 in Gravitation: an introduction to current research (Wiley, 1962), https://arxiv.org/pdf/gr-qc/0405109v1.pdf

  

 

Examples: Decompose, gamma_

 

restartwith(Physics)NULL

Setup(mathematicalnotation = true)

[mathematicalnotation = true]

(2.2.1)

Define  now an arbitrary tensor A

Define(A)

`Defined objects with tensor properties`

 

{A, Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}

(2.2.2)

So A^mu is a 4D tensor with only one free index, where the position of the time-like component is the position of the different sign in the signature, that you can query about via

Setup(signature)

[signature = `- - - +`]

(2.2.3)

To perform a decomposition into space and time, set - for instance - the lowercase latin letters from i to s to represent spaceindices and

Setup(spaceindices = lowercase_is)

[spaceindices = lowercaselatin_is]

(2.2.4)

Accordingly, the 3+1 decomposition of A^mu is

Decompose(A[`~mu`])

Array(%id = 18446744078724512334)

(2.2.5)

The 3+1 decomposition of the inert representation %g_[mu,nu] of the 4D spacetime metric; use the inert representation when you do not want the actual components of the metric appearing in the output

Decompose(%g_[mu, nu])

Matrix(%id = 18446744078724507998)

(2.2.6)

Note the position of the component %g_[0, 0], related to the trailing position of the time-like component in the signature "(- - - +)".

Compare the decomposition of the 4D inert with the decomposition of the 4D active spacetime metric

g[]

g[mu, nu] = (Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -1, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -1, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1}))

(2.2.7)

Decompose(g_[mu, nu])

Matrix(%id = 18446744078724494270)

(2.2.8)

Note that in general the 3D space part of g[mu, nu] is not equal to the 3D metric gamma[i, j] whose definition includes another term (see [1] Landau & Lifshitz, eq.(84.7)).

gamma_[definition]

Physics:-gamma_[i, j] = -Physics:-g_[i, j]+Physics:-g_[0, i]*Physics:-g_[0, j]/Physics:-g_[0, 0]

(2.2.9)

The 3D space part of -g[`~mu`, `~nu`] is actually equal to the 3D metric "gamma[]^(i,j)"

"gamma_[~,definition];"

Physics:-gamma_[`~i`, `~j`] = -Physics:-g_[`~i`, `~j`]

(2.2.10)

To derive the formula  for the covariant components of the 3D metric, Decompose into 3+1 the identity

%g_[`~alpha`, `~mu`]*%g_[mu, beta] = KroneckerDelta[`~alpha`, beta]

%g_[`~alpha`, `~mu`]*%g_[mu, beta] = Physics:-KroneckerDelta[beta, `~alpha`]

(2.2.11)

To the side, for illustration purposes, these are the 3 + 1 decompositions, first excluding the repeated indices, then excluding the free indices

Eq := Decompose(%g_[`~alpha`, `~mu`]*%g_[mu, beta] = Physics[KroneckerDelta][beta, `~alpha`], repeatedindices = false)

Matrix(%id = 18446744078132963318)

(2.2.12)

Eq := Decompose(%g_[`~alpha`, `~mu`]*%g_[mu, beta] = Physics[KroneckerDelta][beta, `~alpha`], freeindices = false)

%g_[0, beta]*%g_[`~alpha`, `~0`]+%g_[i, beta]*%g_[`~alpha`, `~i`] = Physics:-KroneckerDelta[beta, `~alpha`]

(2.2.13)

Compare with a full decomposition

Eq := Decompose(%g_[`~alpha`, `~mu`]*%g_[mu, beta] = Physics[KroneckerDelta][beta, `~alpha`])

Matrix(%id = 18446744078724489454)

(2.2.14)

Eq is a symmetric matrix of equations involving non-contracted occurrences of `#msup(mi("g"),mrow(mn("0"),mo(","),mn("0")))`, `#msup(mi("g"),mrow(mi("j"),mo(","),mo("0")))` and `#msup(mi("g"),mrow(mi("j"),mo(","),mi("i")))`. Isolate, in Eq[1, 2], `#msup(mi("g"),mrow(mi("j"),mo(","),mo("0")))`, that you input as %g_[~j, ~0], and substitute into Eq[1, 1]

"isolate(Eq[1, 2], `%g_`[~j, ~0]);"

%g_[`~j`, `~0`] = -%g_[i, 0]*%g_[`~j`, `~i`]/%g_[0, 0]

(2.2.15)

subs(%g_[`~j`, `~0`] = -%g_[i, 0]*%g_[`~j`, `~i`]/%g_[0, 0], Eq[1, 1])

-%g_[0, k]*%g_[i, 0]*%g_[`~j`, `~i`]/%g_[0, 0]+%g_[i, k]*%g_[`~j`, `~i`] = Physics:-KroneckerDelta[k, `~j`]

(2.2.16)

Collect `#msup(mi("g"),mrow(mi("j"),mo(","),mi("i")))`, that you input as %g_[~j, ~i]

collect(-%g_[0, k]*%g_[i, 0]*%g_[`~j`, `~i`]/%g_[0, 0]+%g_[i, k]*%g_[`~j`, `~i`] = Physics[KroneckerDelta][k, `~j`], %g_[`~j`, `~i`])

(-%g_[0, k]*%g_[i, 0]/%g_[0, 0]+%g_[i, k])*%g_[`~j`, `~i`] = Physics:-KroneckerDelta[k, `~j`]

(2.2.17)

Since the right-hand side is the identity matrix and, from , `#msup(mi("g"),mrow(mi("i"),mo(","),mi("j")))` = -`#msup(mi("γ",fontstyle = "normal"),mrow(mi("i"),mo(","),mi("j")))`, the expression between parenthesis, multiplied by -1, is the reciprocal of the contravariant 3D metric `#msup(mi("γ",fontstyle = "normal"),mrow(mi("i"),mo(","),mi("j")))`, that is the covariant 3D metric gamma[i, j], in accordance to its definition for the signature `- - - +`

gamma_[definition]

Physics:-gamma_[i, j] = -Physics:-g_[i, j]+Physics:-g_[0, i]*Physics:-g_[0, j]/Physics:-g_[0, 0]

(2.2.18)

NULL

References

 
  

[1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

Example: Redefine

   

Tensors in Special and General Relativity

 

A number of relevant changes happened in the tensor routines of the Physics package, towards making the routines pack more functionality both for special and general relativity, as well as working more efficiently and naturally, based on Maple's Physics users' feedback collected during 2016.

New functionality

 
• 

Implement conversions to most of the tensors of general relativity (relevant in connection with functional differentiation)

• 

New setting in the Physics Setup  allows for specifying the cosmologicalconstant and a default tensorsimplifier


 

Download PhysicsMaple2017.mw

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

In this file you will be able to observe and analyze how the exercises and problems of Kinematics and Dynamics are solved using the commands and operators through a very well-structured syntax; Allowing me to save time and use it in interpretation. I hope you can share and spread to break the traditional and unnecessary myths. Only for Engineering and Science. Share if you like.

In Spanish.

Kinematics_using_syntax_in_Maple.mw

Lenin Araujo Castillo

Ambassador of Maple

The MapleCloud now provides the GRTensorIII package for component tensor calculations in general relativity. This package is an update of the established GRTensorII package, last updated in 1999.  In early June, I had the opportunity to present this update at the recent Atlantic General Relativity meeting. The GR community was delighted to have an updated version that works well with more recent versions of Maple. Several talks specifically called out the key role GRTensorIII had in establishing the results presented.

GRTensorIII supports the calculation of the standard curvature objects in relativity. It also provides the command which allows the definition of new tensor objects via a simple definition string (without programming). This command is the key reason for GRTensor’s continued use in the GR community. GRTensorIII also provides a new, more direct API to define a spacetime via the command. This command allows for the metric or line element definition within a worksheet – removing the need for storing the metric in a file and allowing example worksheets to be self-contained. GRTensorIII has significant new functionality to support the definition of hyper-surfaces and the calculation of junction conditions and thin-shell stress energy. An extensive series of example worksheets based on Eric Poisson’s “A Relativist’s Toolkit” are included.

GRTensorIII is available from the MapleCloud window within Maple in the list of available packages or via a download from Maple. It can be loaded directly via the command line as: 

It is also available on github. The source code is available on github in a separate repository: grtensor3src .  

GRTensorIII was developed in collaboration with Kayll Lake and Denis Pollney. I would like to thank Maple for access to the beta program and Eric Poisson for testing and feedback.

The OrthogonalExpansions package is a collection of commands to compute one-dimensional and multi-dimensional orthogonal series expansions of a function. The expansions are evaluated symbolically, numerically or in an inert form.

In particular, the package includes six modifications of the one-dimensional Fourier series, as well the multi-dimensional Fourier, cosine and sine series, one- and multi-dimensional series expansions of classical orthogonal polynomials.


The package includes MixedSeries command to compute multi-dimensional mixed orthogonal series expansions. Mixed series are created from the one-dimensional and multi-dimensional series that are available in the package.


The package also includes GramSchmidtL2 command for computing symbolically or numerically an orthonormal set of one- and multi-variable functions.

 

I've developed a framework in Maple for playing with cyclic division algebras. I have methods for multiplication and passing to and from:

The standard basis over the base field in terms of x^i*y^j and:
The standard matrix basis for the central simple algebra when taking coefficients in the splitting field.

I have it done in the case of degree 7, but could generalise it to any prime degree if this is of use to anyone.

We just posted a submission to the Maple Application Center that I thought people might be interested in. Mathematics for Chemistry isn't a typical application - it's an e-book, written in Maple by J.F. Ogilvie. It covers both standard mathematics topics chemistry students are expected to know,  such as calculus and linear algebra, as well as chemistry-specific topics like chemical equilbrium, quantum chemistry, and nuclear-magnetic-resonance spectra.

There's lots of interesting content on the Application Center, and the range of topics is always fascinating, but it's not every day I see an entire e-book come across my desk(top)!

If you are interested, you can find it on the Application Center, here: Mathematics for Chemistry

eithne

While preparing for a recent webinar, I ran across something that didn't behave the same way in Maple 2017 as it did in previous releases. In particular, it was the failure of the over-dot notation for t-derivatives to display with the over-dot. Turns out that this is due to a change in behavior of typesetting that was detailed in the What's New page for Maple 2017, a page I had looked at many times in the last few months, but apparently didn't comprehend fully. The details are below.

Prior to Maple 2017, under the aegis of extended typesetting, the following two lines of code would alert Maple that the over-dot notation for t-derivatives should be used in the output display.

However, this changed in Maple 2017. Extended typesetting is now the default, but these two lines of code are no longer sufficient to induce Maple to display the over-dot in output. Indeed, we would now have

as output. The change is documented in the following paragraph

lifted from the help page 

Thus, it now takes the additional command

to induce Maple to display the over-dot notation in output.

I must confess that, even though I pored over the "What's New" pages for Maple 2017, I completely missed the import of this change to typesetting. I stumbled over the issue while preparing for an upcoming webinar, and frantically sent out help calls to the developers back in the building. Fortunately, I was quickly set straight on the matter, but was disappointed in my own reading of all the implications of the typesetting changes in Maple 2017. So perhaps this note will alert other users to the changes, and to the help page wherein one finds those essential bits of information needed to complete the tasks we set for ourselves.

And one more thing - I was cautioned that the "= true" was essential. Without it, the command would act as a query, echoing the present state of the setting, and not making the desired change to the setting.
 

A question was raised recently on Stewart Gough platforms.  I decided to tidy up some old code to show platform position and leg lengths for any given displacement.
 

restart

``

Hexapod Setup Data

 

RotZ := proc (delta) options operator, arrow; Matrix(1 .. 3, 1 .. 3, {(1, 1) = cos(delta), (1, 2) = -sin(delta), (1, 3) = 0, (2, 1) = sin(delta), (2, 2) = cos(delta), (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1}, datatype = anything, storage = rectangular, order = Fortran_order, subtype = Matrix) end proc

a[1] := Vector(3, [.5, 3.0, 0]); a[2] := evalf(RotZ(20*((1/180)*Pi)).a[1]); a[3] := evalf(RotZ(100*((1/180)*Pi)).a[2]); a[4] := evalf(RotZ(20*((1/180)*Pi)).a[3]); a[5] := evalf(RotZ(100*((1/180)*Pi)).a[4]); a[6] := evalf(RotZ(20*((1/180)*Pi)).a[5])

b[1] := evalf(.7*RotZ(-40*((1/180)*Pi)).a[1]); b[2] := evalf(RotZ(100*Pi*(1/180)).b[1]); b[3] := evalf(RotZ(20*Pi*(1/180)).b[2]); b[4] := evalf(RotZ(100*Pi*(1/180)).b[3]); b[5] := evalf(RotZ(20*Pi*(1/180)).b[4]); b[6] := evalf(RotZ(100*Pi*(1/180)).b[5])

Zeroposn := Vector(3, [0, 0, 3])

Tx := Vector(3, [1, 0, 0]); Ty := Vector(3, [0, 1, 0]); Tz := Vector(3, [0, 0, 1])

``

``

NULL

Procedures

 

PlatPosn := proc (x := 0, y := 0, z := 0, alpha := 0, beta := 0, delta := 0) local i, v, Rot, L1, L2, L3, L4, L5, L6; global txn, tyn, tzn, ctrp; description "Calculates the platform position in the Global Coordinates, Unit normals and Leg Lengths"; v := Vector(3, [x, y, z]); ctrp := Zeroposn+v; Rot := Matrix(1 .. 3, 1 .. 3, {(1, 1) = cos(delta)*cos(beta), (1, 2) = -sin(delta)*cos(alpha)+cos(delta)*sin(beta)*sin(alpha), (1, 3) = sin(delta)*sin(alpha)+cos(delta)*sin(beta)*cos(alpha), (2, 1) = sin(delta)*cos(beta), (2, 2) = cos(delta)*cos(alpha)+sin(delta)*sin(beta)*sin(alpha), (2, 3) = -cos(delta)*sin(alpha)+sin(delta)*sin(beta)*cos(alpha), (3, 1) = -sin(beta), (3, 2) = cos(beta)*sin(alpha), (3, 3) = cos(beta)*cos(alpha)}, datatype = anything, storage = rectangular, order = Fortran_order, subtype = Matrix); for i to 6 do bn || i := Zeroposn+v+Rot.b[i] end do; txn := Rot.Tx; tyn := Rot.Ty; tzn := Rot.Tz; print(" Platform centre Global", ctrp); print(" Platform corner Co-ords Global", bn1, bn2, bn3, bn4, bn5, bn6); print("Platform Triad Vectors  ", "X green ", txn, "Y blue", tyn, "Z red ", tzn); L1 := sqrt((bn1-a[1])^%T.(bn1-a[1])); L2 := sqrt((bn2-a[2])^%T.(bn2-a[2])); L3 := sqrt((bn3-a[3])^%T.(bn3-a[3])); L4 := sqrt((bn4-a[4])^%T.(bn4-a[4])); L5 := sqrt((bn5-a[5])^%T.(bn5-a[5])); L6 := sqrt((bn6-a[6])^%T.(bn6-a[6])); print("Leg Lengths"); print("L1= ", L1); print("L2= ", L2); print("L3= ", L3); print("L4= ", L4); print("L5= ", L5); print("L6= ", L6) end proc

``

PlatPlot := proc () local Base, Platformdisplacement, picL1, picL2, picL3, picL4, picL5, picL6; global tx0, ty0, tz0; description "Displays the Hexapod"; Base := plots:-polygonplot3d(Matrix([a[1], a[2], a[3], a[4], a[5], a[6]], datatype = float), color = black, transparency = .5); Platformdisplacement := plots:-polygonplot3d(Matrix([seq(bn || i, i = 1 .. 6)]), color = cyan, transparency = .5); picL1 := plots:-arrow(a[1], bn || 1-a[1], colour = green); picL2 := plots:-arrow(a[2], bn || 2-a[2], colour = blue); picL3 := plots:-arrow(a[3], bn || 3-a[3], colour = blue); picL4 := plots:-arrow(a[4], bn || 4-a[4], colour = blue); picL5 := plots:-arrow(a[5], bn || 5-a[5], colour = blue); picL6 := plots:-arrow(a[6], bn || 6-a[6], colour = orange); tx0 := plots:-arrow(ctrp, txn, colour = green); ty0 := plots:-arrow(ctrp, tyn, colour = blue); tz0 := plots:-arrow(ctrp, tzn, colour = red); plots:-display(Base, picL1, picL2, picL3, picL4, picL5, picL6, Platformdisplacement, tx0, ty0, tz0, axes = box, labels = [X, Y, Z], scaling = constrained) end proc

``

NULL

``

``

PlatPosn()

"L6= ", 3.586394355

(1)

PlatPlot()

 

NULL

PlatPosn(.52, -.89, .7, .2, .67, .3)

"L6= ", 3.055217994

(2)

PlatPlot()

 

NULL

NULL

 

NULL

print('tzn' = LinearAlgebra:-CrossProduct(txn, tyn), `= `, tzn)

tzn = Vector[column](%id = 18446744074564617750), `= `, Vector[column](%id = 18446744074328082542)

(3)

``

``NULL

NULL

Rotation Matrices

NULL

``

 

RotZ := Matrix(3, 3, {(1, 1) = cos(delta), (1, 2) = -sin(delta), (1, 3) = 0, (2, 1) = sin(delta), (2, 2) = cos(delta), (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1})

RotY := Matrix(3, 3, {(1, 1) = cos(beta), (1, 2) = 0, (1, 3) = sin(beta), (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (3, 1) = -sin(beta), (3, 2) = 0, (3, 3) = cos(beta)})

RotX := Matrix(3, 3, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = cos(alpha), (2, 3) = -sin(alpha), (3, 1) = 0, (3, 2) = sin(alpha), (3, 3) = cos(alpha)})

NULL

ROT := RotZ.RotY.RotX

Matrix(%id = 18446744074564619310)

(4)

``

``

``


 

Download Reverse_Kinematics_Stewart_Gough_Platform.mw

With this application the components of the acceleration can be calculated. The components of the acceleration in scalar and vector of the tangent and the normal. In addition to the curvilinear kinetics in polar coordinates. It can be used in different engineers, especially mechanics, civilians and more.

In Spanish.

Kinematics_Curvilinear.mw

Lenin Araujo Castillo

Ambassador of Maple

 

 

Maple 2017 has launched!

Maple 2017 is the result of hard work by an enthusiastic team of developers and mathematicians.

As ever, we’re guided by you, our users. Many of the new features are of a result of your feedback, while others are passion projects that we feel you will find value in.

Here’s a few of my favourite enhancements. There’s far more that’s new - see What’s New in Maple 2017 to learn more.

 

MapleCloud Package Manager

Since it was first introduced in Maple 14, the MapleCloud has made thousands of Maple documents and interactive applications available through a web interface.

Maple 2017 completely refreshes the MapleCloud experience. Allied with a new, crisp, interface, you can now download and install user-created packages.

Simply open the MapleCloud interface from within Maple, and a mouse click later, you see a list of user-created packages, continuously updated via the Internet. Two clicks later, you’ve downloaded and installed a package.

This completely bypasses the traditional process of searching for and downloading a package, copying to the right folder, and then modifying libname in Maple. That was a laborious process, and, unless I was motivated, stopped me from installing packages.

The MapleCloud hosts a growing number of packages.

Many regular visitors to MaplePrimes are already familiar with Sergey Moiseev’s DirectSearch package for optimization, equation solving and curve fitting.

My fellow product manager, @DSkoog has written a package for grouping data into similar clusters (called ClusterAnalysis on the Package Manager)

Here’s a sample from a package I hacked together for downloading maps images using the Google Maps API (it’s called Google Maps and Geocoding on the Package Manager).

You’ll also find user-developed packages for exploring AES-based encryption, orthogonal series expansions, building Maple shell scripts and more.

Simply by making the process of finding and installing packages trivially easy, we’ve opened up a new world of functionality to users.

Maple 2017 also offers a simple method for package authors to upload workbook-based packages to the MapleCloud.

We’re engaging with many package authors to add to the growing list of packages on the MapleCloud. We’d be interested in seeing your packages, too!

 

Advanced Math

We’re committed to continually improving the core symbolic math routines. Here area few examples of what to expect in Maple 2017.

Resulting from enhancements to the Risch algorithm, Maple 2017 now computes symbolic integrals that were previously intractable

Groeber:-Basis uses a new implementation of the FGLM algorithm. The example below runs about 200 times faster in Maple 2017.

gcdex now uses a sparse primitive polynomial remainder sequence together.  For sparse structured problems the new routine is orders of magnitude faster. The example below was previously intractable.

The asympt and limit commands can now handle asymptotic cases of the incomplete Γ function where both arguments tend to infinity and their quotient remains finite.

Among several improvements in mathematical functions, you can now calculate and manipulate the four multi-parameter Appell functions.

 

Appel functions are of increasing importance in quantum mechanics, molecular physics, and general relativity.

pdsolve has seen many enhancements. For example, you can tell Maple that a dependent variable is bounded. This has the potential of simplifying the form of a solution.

 

Plot Builder

Plotting is probably the most common application of Maple, and for many years, you’ve been able to create these plots without using commands, if you want to.  Now, the re-designed interactive Plot Builder makes this process easier and better.

When invoked by a context menu or command on an expression or function, a panel slides out from the right-hand side of the interface.

 

Generating and customizing plots takes a single mouse click. You alter plot types, change formatting options on the fly and more.

To help you better learn Maple syntax, you can also display the actual plot command.

Password Protected Content

You can distribute password-protected executable content. This feature uses the workbook file format introduced with Maple 2016.

You can lock down any worksheet in a Workbook. But from any other worksheet, you can send (author-specified) parameters into the locked worksheet, and extract (author-specified) results.

 

Plot Annotations

You can now get information to pop up when you hover over a point or a curve on a plot.

In this application, you see the location and magnitude of an earthquake when you hover over a point

Here’s a ternary diagram of the color of gold-silver-copper alloys. If you let your mouse hover over the points, you see the composition of the points

Plot annotations may seem like a small feature, but they add an extra layer of depth to your visualizations. I’ve started using them all the time!

 

Engineering Portal

In my experience, if you ask an engineer how they prefer to learn, the vast majority of them will say “show me an example”. The significantly updated Maple Portal for Engineers does just that, incorporating many more examples and sample applications.  In fact, it has a whole new Application Gallery containing dozens of applications that solve concrete problems from different branches of engineering while illustrating important Maple techniques.

Designed as a starting point for engineers using Maple, the Portal also includes information on math and programming, interface features for managing your projects, data analysis and visualization tools, working with physical and scientific data, and a variety of specialized topics.

 

Geographic Data

You can now generate and customize world maps. This for example, is a choropleth of European fertility rates (lighter colors indicate lower fertility rates)

You can plot great circles that show the shortest path between two locations, show varying levels of detail on the map, and even experiment with map projections.

A new geographic database contains over one million locations, cross-referenced with their longitude, latitude, political designation and population.

The database is tightly linked to the mapping tools. Here, we ask Maple to plot the location of country capitals with a population of greater than 8 million and a longitude lower than 30.

 

There’s much more to Maple 2017. It’s a deep, rich release that has something for everyone.

Visit What’s New in Maple 2017 to learn more.

This worksheet is designed to develop engineering exercises with Maple applications. You should know the theory before using these applications. It is designed to solve problems faster. I hope you use something that is fully developed with embedded components.

In Spanish

Vector_Force.mw

Lenin Araujo Castillo

Ambassador Of Maple

 

 

Let us consider 

sol := pdsolve({diff(u(x, t), t)-(diff(v(x, t), x))+u(x, t)+v(x, t) = (1+t)*x+(x-1)*t^2, diff(v(x, t), t)-(diff(u(x, t), x))+u(x, t)+v(x, t) = (1+t)*x*t+(2*x-1)*t}, {u(0, t) = 0, u(x, 0) = 0, v(0, t) = 0, v(x, 0) = 0}, time = t, numeric, timestep = 0.1e-1, spacestep = 0.1e-1, range = 0 .. 1); 
sol:-plot3d(v(x, t), x = 0 .. 1, t = 0 .. 1);

A nice plot similar to the one produced by Mma (see the  attached pdf file pdesystem.pdf) is expected. 
The exact solutions u(x,t)=x*t,v(x,t)=x*t^2 are known

pdetest({u(x, t) = x*t, v(x, t) = x*t^2}, {diff(u(x, t), t)-(diff(v(x, t), x))+u(x, t)+v(x, t) =
(1+t)*x+(x-1)*t^2, diff(v(x, t), t)-(diff(u(x, t), x))+u(x, t)+v(x, t) = (1+t)*x*t+(2*x-1)*t});
                              {0}

But the wrong result

               module() ... end module         
Error, (in pdsolve/numeric/plot3d) unable to compute solution for t>HFloat(0.26000000000000006):
solution becomes undefined, problem may be ill posed or method may be ill suited to solution

is obtained. Also 

sol:-plot3d(v(x, t), x = 0 .. 1, t = 0 ..0.1);


 

The plot 

sol:-plot3d(v(x, t), x = 0 .. .5, t = 0 .. .1);

is not better.

Here is a problem from SEEMOUS 2017 (South Eastern European Mathematical Olympiad for University Students)
which Maple can solve (with a little help).

For k a fixed nonnegative integer, compute:

Sum( binomial(i,k) * ( exp(1) - Sum(1/j!, j=0..i) ), i=k..infinity );

(It is the last one, theoretically the most difficult.)

Application that allows us to measure the reliability of a group of data through a row and columns called cronbach alpha at the same time to measure the correlation of items through the pearson correlation of even and odd items. It can run on maple 18 to maple 2017. This will be useful when we are developing a thesis in the statistical part.

In Spanish

StatisticsSocialCronbachPearson.zip

Lenin Araujo Castillo

Ambassador of Maple

 

 

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