(maxima.info)Functions and Variables for atensor

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27.2 Functions and Variables for atensor
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 -- Function: init_atensor
          init_atensor (<alg_type>, <opt_dims>)
          init_atensor (<alg_type>)

     Initializes the 'atensor' package with the specified algebra type.
     <alg_type> can be one of the following:

     'universal': The universal algebra has no commutation rules.

     'grassmann': The Grassman algebra is defined by the commutation
     relation 'u.v+v.u=0'.

     'clifford': The Clifford algebra is defined by the commutation
     relation 'u.v+v.u=-2*sf(u,v)' where 'sf' is a symmetric
     scalar-valued function.  For this algebra, <opt_dims> can be up to
     three nonnegative integers, representing the number of positive,
     degenerate, and negative dimensions of the algebra, respectively.
     If any <opt_dims> values are supplied, 'atensor' will configure the
     values of 'adim' and 'aform' appropriately.  Otherwise, 'adim' will
     default to 0 and 'aform' will not be defined.

     'symmetric': The symmetric algebra is defined by the commutation
     relation 'u.v-v.u=0'.

     'symplectic': The symplectic algebra is defined by the commutation
     relation 'u.v-v.u=2*af(u,v)' where 'af' is an antisymmetric
     scalar-valued function.  For the symplectic algebra, <opt_dims> can
     be up to two nonnegative integers, representing the nondegenerate
     and degenerate dimensions, respectively.  If any <opt_dims> values
     are supplied, 'atensor' will configure the values of 'adim' and
     'aform' appropriately.  Otherwise, 'adim' will default to 0 and
     'aform' will not be defined.

     'lie_envelop': The algebra of the Lie envelope is defined by the
     commutation relation 'u.v-v.u=2*av(u,v)' where 'av' is an
     antisymmetric function.

     The 'init_atensor' function also recognizes several predefined
     algebra types:

     'complex' implements the algebra of complex numbers as the Clifford
     algebra Cl(0,1).  The call 'init_atensor(complex)' is equivalent to
     'init_atensor(clifford,0,0,1)'.

     'quaternion' implements the algebra of quaternions.  The call
     'init_atensor (quaternion)' is equivalent to 'init_atensor
     (clifford,0,0,2)'.

     'pauli' implements the algebra of Pauli-spinors as the
     Clifford-algebra Cl(3,0).  A call to 'init_atensor(pauli)' is
     equivalent to 'init_atensor(clifford,3)'.

     'dirac' implements the algebra of Dirac-spinors as the
     Clifford-algebra Cl(3,1).  A call to 'init_atensor(dirac)' is
     equivalent to 'init_atensor(clifford,3,0,1)'.

 -- Function: atensimp (<expr>)

     Simplifies an algebraic tensor expression <expr> according to the
     rules configured by a call to 'init_atensor'.  Simplification
     includes recursive application of commutation relations and
     resolving calls to 'sf', 'af', and 'av' where applicable.  A
     safeguard is used to ensure that the function always terminates,
     even for complex expressions.

 -- Function: alg_type
     The algebra type.  Valid values are 'universal', 'grassmann',
     'clifford', 'symmetric', 'symplectic' and 'lie_envelop'.

 -- Variable: adim
     Default value: 0

     The dimensionality of the algebra.  'atensor' uses the value of
     'adim' to determine if an indexed object is a valid base vector.
     See 'abasep'.

 -- Variable: aform
     Default value: 'ident(3)'

     Default values for the bilinear forms 'sf', 'af', and 'av'.  The
     default is the identity matrix 'ident(3)'.

 -- Variable: asymbol
     Default value: 'v'

     The symbol for base vectors.

 -- Function: sf (<u>, <v>)

     A symmetric scalar function that is used in commutation relations.
     The default implementation checks if both arguments are base
     vectors using 'abasep' and if that is the case, substitutes the
     corresponding value from the matrix 'aform'.

 -- Function: af (<u>, <v>)

     An antisymmetric scalar function that is used in commutation
     relations.  The default implementation checks if both arguments are
     base vectors using 'abasep' and if that is the case, substitutes
     the corresponding value from the matrix 'aform'.

 -- Function: av (<u>, <v>)

     An antisymmetric function that is used in commutation relations.
     The default implementation checks if both arguments are base
     vectors using 'abasep' and if that is the case, substitutes the
     corresponding value from the matrix 'aform'.

     For instance:

          (%i1) load("atensor");
          (%o1)       /share/tensor/atensor.mac
          (%i2) adim:3;
          (%o2)                                  3
          (%i3) aform:matrix([0,3,-2],[-3,0,1],[2,-1,0]);
                                         [  0    3   - 2 ]
                                         [               ]
          (%o3)                          [ - 3   0    1  ]
                                         [               ]
                                         [  2   - 1   0  ]
          (%i4) asymbol:x;
          (%o4)                                  x
          (%i5) av(x[1],x[2]);
          (%o5)                                 x
                                                 3

 -- Function: abasep (<v>)

     Checks if its argument is an 'atensor' base vector.  That is, if it
     is an indexed symbol, with the symbol being the same as the value
     of 'asymbol', and the index having a numeric value between 1 and
     'adim'.