################ (2023) Dr.W.G. Lindner,  Leichlingen DE
###    tensorBox        Tensor box 
################        for calculations with tensors

-- KRONECKER delta function d(i,j)
delta(i,j) = test( i=j, 1, 0) 

- KRONECKER delta 2-tensor dij
deltaij = zero(2,2)
   deltaij[1,1] = 1
   deltaij[2,2] = 1

-- 2D permutation tensor als FUNCTION
Eij(i,j) = j-i
Eij(1,2)
Eij(1,1)

-- 2D permutation tensor eij as TABLE
eij = zero(2,2)                           
for(i,1,2, for(j,1,2,  eij[i,j]=j-i )) 

-- 3D permutation tensor as FUNCTION
Eijk(i,j,k) = test(
   or( (i,j,k)==(1,2,3), (i,j,k)==(2,3,1), (i,j,k)==(3,1,2)), +1, 
   or( (i,j,k)==(3,2,1), (i,j,k)==(1,3,2), (i,j,k)==(2,1,3)), -1,
   or( i==j, j==k, k==i), 0 )

-- 3D permutation tensor as TABLE
eijk = zero(3,3,3)
  eijk[1,2,3] =  1
  eijk[2,3,1] =  1
  eijk[3,1,2] =  1
  eijk[3,2,1] = -1
  eijk[1,3,2] = -1
  eijk[2,1,3] = -1


-- 4D permutation tensor as FUNCTION
Eijkl(i,j,k,l) = (i-j)*(i-k)*(i-l)*(j-k)*(j-l)*(k-l)/12

-- 4D permutation tensor as TABLE
eijkl=zero(4,4,4,4)
for(i,1,4, for(j,1,4, for(k,1,4, for(l,1,4,
        eijkl[i,j,k,l] = (i-j)*(i-k)*(i-l)*(j-k)*(j-l)*(k-l)/12 ) )))

-- metric tensor g in 2D, an example
gdd = zero(2,2)
      gdd[1,1]= 4
      gdd[1,2]= 1
      gdd[2,1]= 1
      gdd[2,2]= 1

-- inverse guu to gdd
guu = inv(gdd)

-- volume element
Z = det(gdd)
V = sqrt(Z)

-- LEVI-CIVITA tensor epsilon in 2D
epsilon = V * eij

Epsilon = 1/V * eij

-- 2D/3D/nD CHRISTOFFEL tensor 2nd kind
-- specify dimension n = 2,3,..
n=3   
Gamma=zero(n,n,n)
for( k,1,n,
    for( l,1,n,
        for( m,1,n,
             Gamma[k,l,m] = 
                sum(i,1,2, guu[k,i]/2 * 
                      ( d(gdd[m,i],X[l]) + 
                        d(gdd[l,i],X[m]) - 
                        d(gdd[m,l],X[i]) )
                ))))