ecterrab

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Yes, use the Physics package...

Answered: ecterrab 14540

April 21 2023

1 0

As soon as you set a prefix for entering noncommutative objects, eg. Setup(noncommutativeprefix = Z), you can work with symbolic/general/unknown matrices, e.g. Z1 * Z2 - Z2 * Z1 without receiving 0, including the possibility of setting commutator and anticommutator rules for them. See the help pages for Physics:-`*` (the multiplication operator that is at work after you input with(Physics)) and also for Physics:-Setup. Additionally, you have Physics:-Inverse to represent (or compute) the multiplicative inverse, Physics:-Gtaylor for computing series involving these non or anti-commutative objects, and Physics:-diff for differentiatiing expressions involving them and Physics:-Simplify to simplify these algebraic expressions. If at some point you prefer to actually compute products of matrices, use Physics:-`.`. The corresponding help pages have illustrative examples.

Alternatively, you can compute with tensors with two indices representing these matrices (use genericindices to represent matrices of unknown dimensions see ?Physics:-Setup), and if you use this approach, also, Physics:-KroneckerDelta indexed with the indices of generic type you set represents a generic identity matrix.

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

No and Yes....

Answered: ecterrab 14540

April 12 2023

3 0

No, you cannot have two Riemann tensors "set at the same time".

It is however easy to have as many "Riemann tensors" as you want for different metrics; not the way you suggest but as follows.

Start setting any non-flat metric, only to be able to work with formulas that do not automatically evaluate to zero when there is no curvature. E.g.

Physics:-g_[mu, nu] = Matrix(%id = 36893488152008277756)

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So now, how is the Riemann tensor computed?

(2)

Express everything in terms of the metric

(3)

You see you have a sort of template here. Define a new tensor R and call g_ as G

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

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And that is basically all. This approach is of use not only for the Riemann tensor but for everything that has a definition - pre-existing one or that you construct. To compute the components of this R tensor, indicate (define) to represent whatever metric (4 x 4 symmetric matrix) you want, and the formula above represents the corresponding Riemann. For example,

$G[mu, nu] = (Matrix(4, 4, {(1, 1) = r^2/(e^2-2*m*r+r^2), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = (-e^2+2*m*r-r^2)/r^2}))$

G[mu, nu] = Matrix(%id = 36893488152081444124)

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

(7)

These are the components of R

${R[1, 1, 1, 1] = 0, R[1, 1, 1, 2] = 0, R[1, 1, 1, 3] = 0, R[1, 1, 1, 4] = 0, R[1, 1, 2, 1] = 0, R[1, 1, 2, 2] = 0, R[1, 1, 2, 3] = 0, R[1, 1, 2, 4] = 0, R[1, 1, 3, 1] = 0, R[1, 1, 3, 2] = 0, R[1, 1, 3, 3] = 0, R[1, 1, 3, 4] = 0, R[1, 1, 4, 1] = 0, R[1, 1, 4, 2] = 0, R[1, 1, 4, 3] = 0, R[1, 1, 4, 4] = 0, R[1, 2, 1, 1] = 0, R[1, 2, 1, 2] = (2*m-r)*r^3*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 2, 1, 3] = 0, R[1, 2, 1, 4] = 0, R[1, 2, 2, 1] = -(2*m-r)*r^3*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 2, 2, 2] = 0, R[1, 2, 2, 3] = 0, R[1, 2, 2, 4] = 0, R[1, 2, 3, 1] = 0, R[1, 2, 3, 2] = 0, R[1, 2, 3, 3] = 0, R[1, 2, 3, 4] = 0, R[1, 2, 4, 1] = 0, R[1, 2, 4, 2] = 0, R[1, 2, 4, 3] = 0, R[1, 2, 4, 4] = 0, R[1, 3, 1, 1] = 0, R[1, 3, 1, 2] = 0, R[1, 3, 1, 3] = (2*m-r)*r^3*sin(theta)^2*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 3, 1, 4] = 0, R[1, 3, 2, 1] = 0, R[1, 3, 2, 2] = 0, R[1, 3, 2, 3] = 0, R[1, 3, 2, 4] = 0, R[1, 3, 3, 1] = -(2*m-r)*r^3*sin(theta)^2*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 3, 3, 2] = 0, R[1, 3, 3, 3] = 0, R[1, 3, 3, 4] = 0, R[1, 3, 4, 1] = 0, R[1, 3, 4, 2] = 0, R[1, 3, 4, 3] = 0, R[1, 3, 4, 4] = 0, R[1, 4, 1, 1] = 0, R[1, 4, 1, 2] = 0, R[1, 4, 1, 3] = 0, R[1, 4, 1, 4] = (-2*m*r^11+(3*e^2+20*m^2)*r^10+(-32*e^2*m-80*m^3)*r^9+(5*e^4+140*e^2*m^2+160*m^4)*r^8+(-42*e^4*m-312*e^2*m^3-160*m^5)*r^7+(e^6+136*e^4*m^2+352*e^2*m^4+64*m^6)*r^6+(-4*e^6*m-200*e^4*m^3-160*e^2*m^5)*r^5+(-5*e^8+4*e^6*m^2+112*e^4*m^4)*r^4+24*e^8*m*r^3+(-4*e^10-29*e^8*m^2)*r^2+10*e^10*m*r-e^12)/(r^4*(e^2-2*m*r+r^2)^5), R[1, 4, 2, 1] = 0, R[1, 4, 2, 2] = 0, R[1, 4, 2, 3] = 0, R[1, 4, 2, 4] = 0, R[1, 4, 3, 1] = 0, R[1, 4, 3, 2] = 0, R[1, 4, 3, 3] = 0, R[1, 4, 3, 4] = 0, R[1, 4, 4, 1] = (2*m*r^11+(-3*e^2-20*m^2)*r^10+(32*e^2*m+80*m^3)*r^9+(-5*e^4-140*e^2*m^2-160*m^4)*r^8+(42*e^4*m+312*e^2*m^3+160*m^5)*r^7+(-e^6-136*e^4*m^2-352*e^2*m^4-64*m^6)*r^6+(4*e^6*m+200*e^4*m^3+160*e^2*m^5)*r^5+(5*e^8-4*e^6*m^2-112*e^4*m^4)*r^4-24*e^8*m*r^3+(4*e^10+29*e^8*m^2)*r^2-10*e^10*m*r+e^12)/(r^4*(e^2-2*m*r+r^2)^5), R[1, 4, 4, 2] = 0, R[1, 4, 4, 3] = 0, R[1, 4, 4, 4] = 0, R[2, 1, 1, 1] = 0, R[2, 1, 1, 2] = -4*(m-(1/2)*r)^2*r^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[2, 1, 1, 3] = 0, R[2, 1, 1, 4] = 0, R[2, 1, 2, 1] = r^2*(e^2-m*r)*(2*m-r)^2/(e^2-2*m*r+r^2)^3, R[2, 1, 2, 2] = 0, R[2, 1, 2, 3] = 0, R[2, 1, 2, 4] = 0, R[2, 1, 3, 1] = 0, R[2, 1, 3, 2] = 0, R[2, 1, 3, 3] = 0, R[2, 1, 3, 4] = 0, R[2, 1, 4, 1] = 0, R[2, 1, 4, 2] = 0, R[2, 1, 4, 3] = 0, R[2, 1, 4, 4] = 0, R[2, 2, 1, 1] = 0, R[2, 2, 1, 2] = 0, R[2, 2, 1, 3] = 0, R[2, 2, 1, 4] = 0, R[2, 2, 2, 1] = 0, R[2, 2, 2, 2] = 0, R[2, 2, 2, 3] = 0, R[2, 2, 2, 4] = 0, R[2, 2, 3, 1] = 0, R[2, 2, 3, 2] = 0, R[2, 2, 3, 3] = 0, R[2, 2, 3, 4] = 0, R[2, 2, 4, 1] = 0, R[2, 2, 4, 2] = 0, R[2, 2, 4, 3] = 0, R[2, 2, 4, 4] = 0, R[2, 3, 1, 1] = 0, R[2, 3, 1, 2] = 0, R[2, 3, 1, 3] = 0, R[2, 3, 1, 4] = 0, R[2, 3, 2, 1] = 0, R[2, 3, 2, 2] = 0, R[2, 3, 2, 3] = r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[2, 3, 2, 4] = 0, R[2, 3, 3, 1] = 0, R[2, 3, 3, 2] = -r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[2, 3, 3, 3] = 0, R[2, 3, 3, 4] = 0, R[2, 3, 4, 1] = 0, R[2, 3, 4, 2] = 0, R[2, 3, 4, 3] = 0, R[2, 3, 4, 4] = 0, R[2, 4, 1, 1] = 0, R[2, 4, 1, 2] = 0, R[2, 4, 1, 3] = 0, R[2, 4, 1, 4] = 0, R[2, 4, 2, 1] = 0, R[2, 4, 2, 2] = 0, R[2, 4, 2, 3] = 0, R[2, 4, 2, 4] = -4*(m-(1/2)*r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[2, 4, 3, 1] = 0, R[2, 4, 3, 2] = 0, R[2, 4, 3, 3] = 0, R[2, 4, 3, 4] = 0, R[2, 4, 4, 1] = 0, R[2, 4, 4, 2] = (2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[2, 4, 4, 3] = 0, R[2, 4, 4, 4] = 0, R[3, 1, 1, 1] = 0, R[3, 1, 1, 2] = 0, R[3, 1, 1, 3] = -4*(m-(1/2)*r)^2*sin(theta)^2*r^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[3, 1, 1, 4] = 0, R[3, 1, 2, 1] = 0, R[3, 1, 2, 2] = 0, R[3, 1, 2, 3] = 0, R[3, 1, 2, 4] = 0, R[3, 1, 3, 1] = sin(theta)^2*r^2*(2*m-r)^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[3, 1, 3, 2] = 0, R[3, 1, 3, 3] = 0, R[3, 1, 3, 4] = 0, R[3, 1, 4, 1] = 0, R[3, 1, 4, 2] = 0, R[3, 1, 4, 3] = 0, R[3, 1, 4, 4] = 0, R[3, 2, 1, 1] = 0, R[3, 2, 1, 2] = 0, R[3, 2, 1, 3] = 0, R[3, 2, 1, 4] = 0, R[3, 2, 2, 1] = 0, R[3, 2, 2, 2] = 0, R[3, 2, 2, 3] = -r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[3, 2, 2, 4] = 0, R[3, 2, 3, 1] = 0, R[3, 2, 3, 2] = r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[3, 2, 3, 3] = 0, R[3, 2, 3, 4] = 0, R[3, 2, 4, 1] = 0, R[3, 2, 4, 2] = 0, R[3, 2, 4, 3] = 0, R[3, 2, 4, 4] = 0, R[3, 3, 1, 1] = 0, R[3, 3, 1, 2] = 0, R[3, 3, 1, 3] = 0, R[3, 3, 1, 4] = 0, R[3, 3, 2, 1] = 0, R[3, 3, 2, 2] = 0, R[3, 3, 2, 3] = 0, R[3, 3, 2, 4] = 0, R[3, 3, 3, 1] = 0, R[3, 3, 3, 2] = 0, R[3, 3, 3, 3] = 0, R[3, 3, 3, 4] = 0, R[3, 3, 4, 1] = 0, R[3, 3, 4, 2] = 0, R[3, 3, 4, 3] = 0, R[3, 3, 4, 4] = 0, R[3, 4, 1, 1] = 0, R[3, 4, 1, 2] = 0, R[3, 4, 1, 3] = 0, R[3, 4, 1, 4] = 0, R[3, 4, 2, 1] = 0, R[3, 4, 2, 2] = 0, R[3, 4, 2, 3] = 0, R[3, 4, 2, 4] = 0, R[3, 4, 3, 1] = 0, R[3, 4, 3, 2] = 0, R[3, 4, 3, 3] = 0, R[3, 4, 3, 4] = -4*(m-(1/2)*r)^2*sin(theta)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[3, 4, 4, 1] = 0, R[3, 4, 4, 2] = 0, R[3, 4, 4, 3] = sin(theta)^2*(2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[3, 4, 4, 4] = 0, R[4, 1, 1, 1] = 0, R[4, 1, 1, 2] = 0, R[4, 1, 1, 3] = 0, R[4, 1, 1, 4] = (2*m*r^11+(-3*e^2-20*m^2)*r^10+(40*e^2*m+80*m^3)*r^9+(-13*e^4-196*e^2*m^2-160*m^4)*r^8+(114*e^4*m+456*e^2*m^3+160*m^5)*r^7+(-17*e^6-368*e^4*m^2-512*e^2*m^4-64*m^6)*r^6+(108*e^6*m+520*e^4*m^3+224*e^2*m^5)*r^5+(-11*e^8-228*e^6*m^2-272*e^4*m^4)*r^4+(48*e^8*m+160*e^6*m^3)*r^3+(-4*e^10-53*e^8*m^2)*r^2+10*e^10*m*r-e^12)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), R[4, 1, 2, 1] = 0, R[4, 1, 2, 2] = 0, R[4, 1, 2, 3] = 0, R[4, 1, 2, 4] = 0, R[4, 1, 3, 1] = 0, R[4, 1, 3, 2] = 0, R[4, 1, 3, 3] = 0, R[4, 1, 3, 4] = 0, R[4, 1, 4, 1] = (-2*m*r^11+(3*e^2+20*m^2)*r^10+(-40*e^2*m-80*m^3)*r^9+(13*e^4+196*e^2*m^2+160*m^4)*r^8+(-114*e^4*m-456*e^2*m^3-160*m^5)*r^7+(17*e^6+368*e^4*m^2+512*e^2*m^4+64*m^6)*r^6+(-108*e^6*m-520*e^4*m^3-224*e^2*m^5)*r^5+(11*e^8+228*e^6*m^2+272*e^4*m^4)*r^4+(-48*e^8*m-160*e^6*m^3)*r^3+(4*e^10+53*e^8*m^2)*r^2-10*e^10*m*r+e^12)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), R[4, 1, 4, 2] = 0, R[4, 1, 4, 3] = 0, R[4, 1, 4, 4] = 0, R[4, 2, 1, 1] = 0, R[4, 2, 1, 2] = 0, R[4, 2, 1, 3] = 0, R[4, 2, 1, 4] = 0, R[4, 2, 2, 1] = 0, R[4, 2, 2, 2] = 0, R[4, 2, 2, 3] = 0, R[4, 2, 2, 4] = (e^2-2*m*r+r^2)*(e^2-m*r)/r^4, R[4, 2, 3, 1] = 0, R[4, 2, 3, 2] = 0, R[4, 2, 3, 3] = 0, R[4, 2, 3, 4] = 0, R[4, 2, 4, 1] = 0, R[4, 2, 4, 2] = -(e^2-2*m*r+r^2)*(e^2-m*r)/r^4, R[4, 2, 4, 3] = 0, R[4, 2, 4, 4] = 0, R[4, 3, 1, 1] = 0, R[4, 3, 1, 2] = 0, R[4, 3, 1, 3] = 0, R[4, 3, 1, 4] = 0, R[4, 3, 2, 1] = 0, R[4, 3, 2, 2] = 0, R[4, 3, 2, 3] = 0, R[4, 3, 2, 4] = 0, R[4, 3, 3, 1] = 0, R[4, 3, 3, 2] = 0, R[4, 3, 3, 3] = 0, R[4, 3, 3, 4] = (e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4, R[4, 3, 4, 1] = 0, R[4, 3, 4, 2] = 0, R[4, 3, 4, 3] = -(e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4, R[4, 3, 4, 4] = 0, R[4, 4, 1, 1] = 0, R[4, 4, 1, 2] = 0, R[4, 4, 1, 3] = 0, R[4, 4, 1, 4] = 0, R[4, 4, 2, 1] = 0, R[4, 4, 2, 2] = 0, R[4, 4, 2, 3] = 0, R[4, 4, 2, 4] = 0, R[4, 4, 3, 1] = 0, R[4, 4, 3, 2] = 0, R[4, 4, 3, 3] = 0, R[4, 4, 3, 4] = 0, R[4, 4, 4, 1] = 0, R[4, 4, 4, 2] = 0, R[4, 4, 4, 3] = 0, R[4, 4, 4, 4] = 0}$

(8)

If you want to ignore those with right-hand side equal to zero, go with

${R[1, 2, 1, 2] = (2*m-r)*r^3*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 2, 2, 1] = -(2*m-r)*r^3*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 3, 1, 3] = (2*m-r)*r^3*sin(theta)^2*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 3, 3, 1] = -(2*m-r)*r^3*sin(theta)^2*(2*e^6-4*e^4*m^2-6*e^4*m*r+4*e^4*r^2+8*e^2*m^3*r+4*e^2*m^2*r^2-10*e^2*m*r^3+3*e^2*r^4-8*m^4*r^2+12*m^3*r^3-6*m^2*r^4+m*r^5)/(e^2-2*m*r+r^2)^5, R[1, 4, 1, 4] = (-2*m*r^11+(3*e^2+20*m^2)*r^10+(-32*e^2*m-80*m^3)*r^9+(5*e^4+140*e^2*m^2+160*m^4)*r^8+(-42*e^4*m-312*e^2*m^3-160*m^5)*r^7+(e^6+136*e^4*m^2+352*e^2*m^4+64*m^6)*r^6+(-4*e^6*m-200*e^4*m^3-160*e^2*m^5)*r^5+(-5*e^8+4*e^6*m^2+112*e^4*m^4)*r^4+24*e^8*m*r^3+(-4*e^10-29*e^8*m^2)*r^2+10*e^10*m*r-e^12)/(r^4*(e^2-2*m*r+r^2)^5), R[1, 4, 4, 1] = (2*m*r^11+(-3*e^2-20*m^2)*r^10+(32*e^2*m+80*m^3)*r^9+(-5*e^4-140*e^2*m^2-160*m^4)*r^8+(42*e^4*m+312*e^2*m^3+160*m^5)*r^7+(-e^6-136*e^4*m^2-352*e^2*m^4-64*m^6)*r^6+(4*e^6*m+200*e^4*m^3+160*e^2*m^5)*r^5+(5*e^8-4*e^6*m^2-112*e^4*m^4)*r^4-24*e^8*m*r^3+(4*e^10+29*e^8*m^2)*r^2-10*e^10*m*r+e^12)/(r^4*(e^2-2*m*r+r^2)^5), R[2, 1, 1, 2] = -4*(m-(1/2)*r)^2*r^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[2, 1, 2, 1] = r^2*(e^2-m*r)*(2*m-r)^2/(e^2-2*m*r+r^2)^3, R[2, 3, 2, 3] = r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[2, 3, 3, 2] = -r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[2, 4, 2, 4] = -4*(m-(1/2)*r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[2, 4, 4, 2] = (2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[3, 1, 1, 3] = -4*(m-(1/2)*r)^2*sin(theta)^2*r^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[3, 1, 3, 1] = sin(theta)^2*r^2*(2*m-r)^2*(e^2-m*r)/(e^2-2*m*r+r^2)^3, R[3, 2, 2, 3] = -r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[3, 2, 3, 2] = r^2*sin(theta)^2*(e^2-4*m^2+2*m*r)/(e^2-2*m*r+r^2), R[3, 4, 3, 4] = -4*(m-(1/2)*r)^2*sin(theta)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[3, 4, 4, 3] = sin(theta)^2*(2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), R[4, 1, 1, 4] = (2*m*r^11+(-3*e^2-20*m^2)*r^10+(40*e^2*m+80*m^3)*r^9+(-13*e^4-196*e^2*m^2-160*m^4)*r^8+(114*e^4*m+456*e^2*m^3+160*m^5)*r^7+(-17*e^6-368*e^4*m^2-512*e^2*m^4-64*m^6)*r^6+(108*e^6*m+520*e^4*m^3+224*e^2*m^5)*r^5+(-11*e^8-228*e^6*m^2-272*e^4*m^4)*r^4+(48*e^8*m+160*e^6*m^3)*r^3+(-4*e^10-53*e^8*m^2)*r^2+10*e^10*m*r-e^12)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), R[4, 1, 4, 1] = (-2*m*r^11+(3*e^2+20*m^2)*r^10+(-40*e^2*m-80*m^3)*r^9+(13*e^4+196*e^2*m^2+160*m^4)*r^8+(-114*e^4*m-456*e^2*m^3-160*m^5)*r^7+(17*e^6+368*e^4*m^2+512*e^2*m^4+64*m^6)*r^6+(-108*e^6*m-520*e^4*m^3-224*e^2*m^5)*r^5+(11*e^8+228*e^6*m^2+272*e^4*m^4)*r^4+(-48*e^8*m-160*e^6*m^3)*r^3+(4*e^10+53*e^8*m^2)*r^2-10*e^10*m*r+e^12)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), R[4, 2, 2, 4] = (e^2-2*m*r+r^2)*(e^2-m*r)/r^4, R[4, 2, 4, 2] = -(e^2-2*m*r+r^2)*(e^2-m*r)/r^4, R[4, 3, 3, 4] = (e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4, R[4, 3, 4, 3] = -(e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4}$

(9)

To get the components of Riemann for a different metric, naturally, you need to indicate the different metric. You get the idea. Define G and get the component of R using TensorArray.

You may prefer to explore these components, as in

(10)

Or get all the non-zero ones in a sort of more compact form

${(1, 2, 1, 2) = (4*e^6*m-2*e^6*r-8*e^4*m^3-8*e^4*m^2*r+14*e^4*m*r^2-4*e^4*r^3+16*e^2*m^4*r-24*e^2*m^2*r^3+16*e^2*m*r^4-3*e^2*r^5-16*m^5*r^2+32*m^4*r^3-24*m^3*r^4+8*m^2*r^5-m*r^6)*r^3/(e^2-2*m*r+r^2)^5, (1, 2, 2, 1) = -(4*e^6*m-2*e^6*r-8*e^4*m^3-8*e^4*m^2*r+14*e^4*m*r^2-4*e^4*r^3+16*e^2*m^4*r-24*e^2*m^2*r^3+16*e^2*m*r^4-3*e^2*r^5-16*m^5*r^2+32*m^4*r^3-24*m^3*r^4+8*m^2*r^5-m*r^6)*r^3/(e^2-2*m*r+r^2)^5, (1, 3, 1, 3) = r^2*(4*sin(theta)^2*csc(theta)^2*e^6*m^2+sin(theta)^2*csc(theta)^2*e^6*r^2+3*sin(theta)^2*csc(theta)^2*e^4*r^4+3*sin(theta)^2*csc(theta)^2*e^2*r^6-32*sin(theta)^2*csc(theta)^2*m^5*r^3+80*sin(theta)^2*csc(theta)^2*m^4*r^4-80*sin(theta)^2*csc(theta)^2*m^3*r^5+40*sin(theta)^2*csc(theta)^2*m^2*r^6-10*sin(theta)^2*csc(theta)^2*m*r^7-4*e^6*m^2-3*e^6*r^2-7*e^4*r^4-6*e^2*r^6+16*m^5*r^3-48*m^4*r^4+56*m^3*r^5-32*m^2*r^6+9*m*r^7-r^8-4*sin(theta)^2*csc(theta)^2*e^6*m*r-24*sin(theta)^2*csc(theta)^2*e^4*m^3*r+36*sin(theta)^2*csc(theta)^2*e^4*m^2*r^2-18*sin(theta)^2*csc(theta)^2*e^4*m*r^3+48*sin(theta)^2*csc(theta)^2*e^2*m^4*r^2-96*sin(theta)^2*csc(theta)^2*e^2*m^3*r^3+72*sin(theta)^2*csc(theta)^2*e^2*m^2*r^4-24*sin(theta)^2*csc(theta)^2*e^2*m*r^5+8*e^6*m*r+16*e^4*m^3*r-44*e^4*m^2*r^2+32*e^4*m*r^3-32*e^2*m^4*r^2+96*e^2*m^3*r^3-96*e^2*m^2*r^4+40*e^2*m*r^5+sin(theta)^2*csc(theta)^2*r^8)*sin(theta)^2/(e^2-2*m*r+r^2)^5, (1, 3, 2, 3) = r^3*(sin(theta)^2*csc(theta)^2-1)*cos(theta)*sin(theta)*(2*m-r)^2/(e^2-2*m*r+r^2)^2, (1, 3, 3, 1) = -r^2*(4*sin(theta)^2*csc(theta)^2*e^6*m^2+sin(theta)^2*csc(theta)^2*e^6*r^2+3*sin(theta)^2*csc(theta)^2*e^4*r^4+3*sin(theta)^2*csc(theta)^2*e^2*r^6-32*sin(theta)^2*csc(theta)^2*m^5*r^3+80*sin(theta)^2*csc(theta)^2*m^4*r^4-80*sin(theta)^2*csc(theta)^2*m^3*r^5+40*sin(theta)^2*csc(theta)^2*m^2*r^6-10*sin(theta)^2*csc(theta)^2*m*r^7-4*e^6*m^2-3*e^6*r^2-7*e^4*r^4-6*e^2*r^6+16*m^5*r^3-48*m^4*r^4+56*m^3*r^5-32*m^2*r^6+9*m*r^7-r^8-4*sin(theta)^2*csc(theta)^2*e^6*m*r-24*sin(theta)^2*csc(theta)^2*e^4*m^3*r+36*sin(theta)^2*csc(theta)^2*e^4*m^2*r^2-18*sin(theta)^2*csc(theta)^2*e^4*m*r^3+48*sin(theta)^2*csc(theta)^2*e^2*m^4*r^2-96*sin(theta)^2*csc(theta)^2*e^2*m^3*r^3+72*sin(theta)^2*csc(theta)^2*e^2*m^2*r^4-24*sin(theta)^2*csc(theta)^2*e^2*m*r^5+8*e^6*m*r+16*e^4*m^3*r-44*e^4*m^2*r^2+32*e^4*m*r^3-32*e^2*m^4*r^2+96*e^2*m^3*r^3-96*e^2*m^2*r^4+40*e^2*m*r^5+sin(theta)^2*csc(theta)^2*r^8)*sin(theta)^2/(e^2-2*m*r+r^2)^5, (1, 3, 3, 2) = -r^3*(sin(theta)^2*csc(theta)^2-1)*cos(theta)*sin(theta)*(2*m-r)^2/(e^2-2*m*r+r^2)^2, (1, 4, 1, 4) = -(e^12-10*e^10*m*r+4*e^10*r^2+29*e^8*m^2*r^2-24*e^8*m*r^3+5*e^8*r^4-4*e^6*m^2*r^4+4*e^6*m*r^5-e^6*r^6-112*e^4*m^4*r^4+200*e^4*m^3*r^5-136*e^4*m^2*r^6+42*e^4*m*r^7-5*e^4*r^8+160*e^2*m^5*r^5-352*e^2*m^4*r^6+312*e^2*m^3*r^7-140*e^2*m^2*r^8+32*e^2*m*r^9-3*e^2*r^10-64*m^6*r^6+160*m^5*r^7-160*m^4*r^8+80*m^3*r^9-20*m^2*r^10+2*m*r^11)/(r^4*(e^2-2*m*r+r^2)^5), (1, 4, 4, 1) = (e^12-10*e^10*m*r+4*e^10*r^2+29*e^8*m^2*r^2-24*e^8*m*r^3+5*e^8*r^4-4*e^6*m^2*r^4+4*e^6*m*r^5-e^6*r^6-112*e^4*m^4*r^4+200*e^4*m^3*r^5-136*e^4*m^2*r^6+42*e^4*m*r^7-5*e^4*r^8+160*e^2*m^5*r^5-352*e^2*m^4*r^6+312*e^2*m^3*r^7-140*e^2*m^2*r^8+32*e^2*m*r^9-3*e^2*r^10-64*m^6*r^6+160*m^5*r^7-160*m^4*r^8+80*m^3*r^9-20*m^2*r^10+2*m*r^11)/(r^4*(e^2-2*m*r+r^2)^5), (2, 1, 1, 2) = -r^2*(e^2-m*r)*(2*m-r)^2/(e^2-2*m*r+r^2)^3, (2, 1, 2, 1) = r^2*(e^2-m*r)*(2*m-r)^2/(e^2-2*m*r+r^2)^3, (2, 3, 1, 3) = r*(sin(theta)^2*csc(theta)^2-1)*sin(theta)*cos(theta), (2, 3, 2, 3) = (-r^2*(2*m-r)^2*sin(theta)^2+(e^2-2*m*r+r^2)*r^2*sin(theta)^2*cos(theta)^2*csc(theta)^2+(e^2-2*m*r+r^2)*r^2*(sin(theta)^2-cos(theta)^2))/(e^2-2*m*r+r^2), (2, 3, 3, 1) = -r*(sin(theta)^2*csc(theta)^2-1)*sin(theta)*cos(theta), (2, 3, 3, 2) = (r^2*(2*m-r)^2*sin(theta)^2-(e^2-2*m*r+r^2)*r^2*sin(theta)^2*cos(theta)^2*csc(theta)^2-(e^2-2*m*r+r^2)*r^2*(sin(theta)^2-cos(theta)^2))/(e^2-2*m*r+r^2), (2, 4, 2, 4) = -(2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), (2, 4, 4, 2) = (2*m-r)^2*(e^2-m*r)/(r^2*(e^2-2*m*r+r^2)), (3, 1, 1, 3) = sin(theta)^2*(-(e^2-2*m*r+r^2)^3*csc(theta)^2*sin(theta)^2+(e^2-2*m*r+r^2)^3*sin(theta)^4*csc(theta)^4-(2*m-r)^2*r^2*(e^2-m*r)*csc(theta)^2*sin(theta)^2)/(e^2-2*m*r+r^2)^3, (3, 1, 2, 3) = r*(sin(theta)^2*cos(theta)*csc(theta)^2-2*sin(theta)*cot(theta)+cos(theta))*csc(theta)^2*sin(theta)^3, (3, 1, 3, 1) = sin(theta)^2*((e^2-2*m*r+r^2)^3*csc(theta)^2*sin(theta)^2-(e^2-2*m*r+r^2)^3*sin(theta)^4*csc(theta)^4+(2*m-r)^2*r^2*(e^2-m*r)*csc(theta)^2*sin(theta)^2)/(e^2-2*m*r+r^2)^3, (3, 1, 3, 2) = -r*(sin(theta)^2*cos(theta)*csc(theta)^2-2*sin(theta)*cot(theta)+cos(theta))*csc(theta)^2*sin(theta)^3, (3, 2, 1, 3) = r*(sin(theta)^2*csc(theta)^2-1)*sin(theta)^3*cos(theta)*csc(theta)^2, (3, 2, 2, 3) = sin(theta)^2*(r^2*(2*m-r)^2*csc(theta)^2*sin(theta)^2+(e^2-2*m*r+r^2)*r^2*sin(theta)^2*cos(theta)^2*csc(theta)^4-2*(e^2-2*m*r+r^2)*r^2*sin(theta)*cos(theta)*csc(theta)^2*cot(theta)-(e^2-2*m*r+r^2)*csc(theta)^2*r^2*(sin(theta)^2-cos(theta)^2))/(e^2-2*m*r+r^2), (3, 2, 3, 1) = -r*(sin(theta)^2*csc(theta)^2-1)*sin(theta)^3*cos(theta)*csc(theta)^2, (3, 2, 3, 2) = sin(theta)^2*(-r^2*(2*m-r)^2*csc(theta)^2*sin(theta)^2-(e^2-2*m*r+r^2)*r^2*sin(theta)^2*cos(theta)^2*csc(theta)^4+2*(e^2-2*m*r+r^2)*r^2*sin(theta)*cos(theta)*csc(theta)^2*cot(theta)+(e^2-2*m*r+r^2)*csc(theta)^2*r^2*(sin(theta)^2-cos(theta)^2))/(e^2-2*m*r+r^2), (3, 3, 1, 2) = 2*r*(sin(theta)*cot(theta)-cos(theta))*csc(theta)^2*sin(theta)^3, (3, 3, 2, 1) = -2*r*(sin(theta)*cot(theta)-cos(theta))*csc(theta)^2*sin(theta)^3, (3, 4, 3, 4) = -(2*m-r)^2*sin(theta)^4*(e^2-m*r)*csc(theta)^2/(r^2*(e^2-2*m*r+r^2)), (3, 4, 4, 3) = (2*m-r)^2*sin(theta)^4*(e^2-m*r)*csc(theta)^2/(r^2*(e^2-2*m*r+r^2)), (4, 1, 1, 4) = -(e^12-10*e^10*m*r+4*e^10*r^2+53*e^8*m^2*r^2-48*e^8*m*r^3+11*e^8*r^4-160*e^6*m^3*r^3+228*e^6*m^2*r^4-108*e^6*m*r^5+17*e^6*r^6+272*e^4*m^4*r^4-520*e^4*m^3*r^5+368*e^4*m^2*r^6-114*e^4*m*r^7+13*e^4*r^8-224*e^2*m^5*r^5+512*e^2*m^4*r^6-456*e^2*m^3*r^7+196*e^2*m^2*r^8-40*e^2*m*r^9+3*e^2*r^10+64*m^6*r^6-160*m^5*r^7+160*m^4*r^8-80*m^3*r^9+20*m^2*r^10-2*m*r^11)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), (4, 1, 4, 1) = (e^12-10*e^10*m*r+4*e^10*r^2+53*e^8*m^2*r^2-48*e^8*m*r^3+11*e^8*r^4-160*e^6*m^3*r^3+228*e^6*m^2*r^4-108*e^6*m*r^5+17*e^6*r^6+272*e^4*m^4*r^4-520*e^4*m^3*r^5+368*e^4*m^2*r^6-114*e^4*m*r^7+13*e^4*r^8-224*e^2*m^5*r^5+512*e^2*m^4*r^6-456*e^2*m^3*r^7+196*e^2*m^2*r^8-40*e^2*m*r^9+3*e^2*r^10+64*m^6*r^6-160*m^5*r^7+160*m^4*r^8-80*m^3*r^9+20*m^2*r^10-2*m*r^11)/(r^8*(e^2-2*m*r+r^2)*(2*m-r)^4), (4, 2, 2, 4) = (e^2-2*m*r+r^2)*(e^2-m*r)/r^4, (4, 2, 4, 2) = -(e^2-2*m*r+r^2)*(e^2-m*r)/r^4, (4, 3, 3, 4) = (e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4, (4, 3, 4, 3) = -(e^2-2*m*r+r^2)*sin(theta)^2*(e^2-m*r)/r^4}$

(11)

There are other ways of accomplishing the same, basically exploring that you can Define tensors represneting any tensorial equation you want. See ?Physics,Tensors, section 2.c amd 2.j.

Download Riemann_for_different_metrics.mw

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

Physics:-Redefine...

Answered: ecterrab 14540

April 12 2023

2 0

That command redefines the tetrad (either the one set, or any other one that you pass) according to the signature you want - check ?Physics,Redefine and the Examples section.

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

Fixed; Maplesoft Physics Updates v.1430...

Answered: ecterrab 14540

March 31 2023

2 0

This is fixed, and the fix is distributed to everybody using Maple 2023 within the Maplesoft Physics Updates v.1430 or newer.

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

Two different things...

Answered: ecterrab 14540

March 30 2023

1 2

Entering

subs(-e - r = s*(e + r), s = -1, %)

You get - (e + r) * cos(alpha/2-90), which is one thing. Is it possible to address this systematically? Yes, and I would vote for it. How? It would require a change in the so-called kernel, maybe in a new version of Maple; or maybe I find a way to resolve this at the so-called library level and distribute this within the Maplesoft Physics Updates.

The second issue is, as mentioned by @acer : there is a normalization for all mathematical functions; in the case of cot, you have cot(90 - z) -> -cot(z - 90). This can also be changed to be the reverse of that, but why would one do that? Your expression is very particular; for a different expression, the current normalization may look more convenient; besides that, too many things were coded over the years that may or not work as in the past if you change the normalization of a function. I wouldn't vote for such a change in the normalization of cot.

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

One way you can do it...

Answered: ecterrab 14540

March 29 2023

5 2

Add to your initialization file:

`simplify/trig/from_sincos/do/22` := eval(`simplify/trig/from_sincos/do/2`):

`simplify/trig/from_sincos/do/2` := () -> eval(`simplify/trig/from_sincos/do/22`(args), [csc = 1/sin, sec = 1/cos])

That suffices to achieve what you are asking. Now, changes like this would require testing to be sure that there are no other things in Maple 2023 that rely on the new csc and sec output by simplify; I suppose through using this or equivalent approaches you will discover.

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

[x = rhocos(phi(t)), y = rhosin(phi(t)...

Answered: ecterrab 14540

March 24 2023

3 1

You forgot to tell the system that phi is not constant? You indicate that entering not phi but phi(t). Use the transformation equations in the title to get the result you expect for non-constant phi(t). Independent of that, in your worksheet, I see just cos, instead of cos(phi) or cos(phi(t)).

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

for ~i in [ ~1, ~2, ~3, ~4] do...

Answered: ecterrab 14540

March 22 2023

5 1

Change your line

for i from 1 by 1 to 4 do

for ~i in [ ~1, ~2, ~3, ~4] do

This change may be sufficiently self-explanatory; if not, feel free to ask. To the side, these commands are related to your work, and you don't use them:

CompactDisplay (you don't need PDEtools, nor PDEtools:-declare)
SubstituteTensorIndices (as in SubstituteTensorIndices(i = Physics:-Library:-Contravariant(i), ...)
TensorArray (this does visually what you are doing in a difficult-to-read double loop
Setup(cosmologicalconstant = ...)
Gtaylor (instead of convert(series(....), polynom)

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

Maplesoft Physics Updates v.1423...

Answered: ecterrab 14540

March 17 2023

2 0

A correction for this is distributed within the Maplesoft Physics Updates v.1423.

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

works fine in Maple 2023....

Answered: ecterrab 14540

March 17 2023

1 5

For historical reasons only, Maple preferred the form rational in sin and cos. That has changed in Maple 2023:

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

Hi Jean-Michel Indeed the web page for t...

Answered: ecterrab 14540

March 17 2023

1 0

Hi Jean-Michel
Indeed the web page for the Maplesoft Physics Updates, a webpage that gets updated at every release, was not up-to-date :). It is now, and tells, as @Pascal4QM says, that the last update for Maple 2022 is v.1409.

Best

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

Fixed within the Maplesoft Physics Updat...

Answered: ecterrab 14540

March 11 2023

1 0

Hi
A fix for this one is distributed to everybody using Maple 2023 within the Maplesoft Physics Updates v.1414 or newer.

To install the Physics Updates in Maple 2023, use the MapleCloud toolbar to install the package - you need to do this once; then use Physics:-Version(latest) from the GUI in order to get next updates.

Note that the Maplesoft Physics Updates webpage reports v.1414 as being for Maple 2022, but that is not the case. This webpage is waiting for an update, probably this week. The last version of the Physics Updates for Maple 2022.2 is v.1409 and to install it in Maple 2022 input Physics:-Version(1409);.

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

PDEtools:-dpolyform...

Answered: ecterrab 14540

March 09 2023

1 0

Take a lookt at the help page of the command PDEtools:-dpolyform - it does precisely what you ask, in the most general way, for multivariate expressions and nonlinear equations.

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

Christoffel[nonzero]...

Answered: ecterrab 14540

March 08 2023

2 0

The title tells how. In general, it can be of help taking a quick look at the help page ?Physics,Tensors where mostly everything about Tensors is presented; or directly to the help page ?Christoffel; the nonzero keyword is mentioned there.

Besides Christoffel[nonzero], which will give you the all-covariant nonzero, with Christoffel[~, nonzero] you get the all-contravariant, and with - say - Christoffel[alpha, ~beta, gamma, nonzero] you get the corresponding ones.

Also fancy, you may want to try TensorArray(<..a tensorial expression here ...>, explore), where in your case the tensorial expression is Christoffel[... the covariant and/or contravariant indices ...]

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

Just a way to represent the norm of a ve...

Answered: ecterrab 14540

March 07 2023

3 3

Answering the question the way you put it, abs(F) always represents the absolute value of the complex variable F, regardless of whether Physics is loaded.

Now, about this Mechanics (Statics) section: in general, depending on the problem, you can choose to represent the objects with which you compute in one way or another. In this problem, the modulus of a vector - say A_ - which in Physics:-Vectors is represented by Norm(A_) - can also be represented as abs(A) (with A, not A_). Sometimes in textbooks, you see Norm(A_) represented by just the letter A, not |A| and not ||A_||.

And why using abs(A) (that displays |A|) instead of Norm(A_) (that displays ||A_||)? The technique used to solve this problem you are mentioning uses assignments. You see right at the beginning R_[B] := abs(R[B]) * _k. An assignment like this one would interrupt you with an "Error, recursive assignment" if on the right-hand side of that assignment, you use Norm(R_[B]). Using abs(R[b]) represents the object properly (within the context of this problem) avoids that recursive assignemnt, is short input, and places a visually open and close |...| which helps the readability of the output.

One could as well tackle this problem without using assignment, using equations instead, as in R_[B] = Norm(R_[B]) * _k, (note = instead of := ), then use substitutions to perform the computational steps. That is done in the majority of the other solved problems in this MaplePrimes presentation about Mechanics.

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

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ecterrab

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These are answers submitted by ecterrab

Yes, use the Physics package...

No and Yes....

Physics:-Redefine...

Fixed; Maplesoft Physics Updates v.1430...

Two different things...

One way you can do it...

[x = rhocos(phi(t)), y = rhosin(phi(t)...

for ~i in [ ~1, ~2, ~3, ~4] do...

Maplesoft Physics Updates v.1423...

works fine in Maple 2023....

Hi Jean-Michel Indeed the web page for t...

Fixed within the Maplesoft Physics Updat...

PDEtools:-dpolyform...

Christoffel[nonzero]...

Just a way to represent the norm of a ve...

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