determinant tensorial de curvatura de ordre 2

det( a^{k}_{ijs} ) = ...

... ( a^{1}_{111}a^{2}_{222}+(-1)·a^{2}_{111}a^{1}_{222} )+...

... ( a^{1}_{122}a^{2}_{211}+(-1)·a^{2}_{122}a^{1}_{211} )+...

... ( a^{1}_{212}a^{2}_{121}+(-1)·a^{2}_{212}a^{1}_{121} )+...

... ( a^{1}_{221}a^{2}_{112}+(-1)·a^{2}_{221}a^{1}_{112} )

A^{k}_{ijs}·a_{i}a_{j}a_{s} = a^{k}

a_{1} = a & a_{2} = b

A^{1}_{111} = (1/a^{2}) & A^{2}_{222} = (1/b^{2})

A^{2}_{111} = (b/a^{3}) & A^{1}_{222} = (a/b^{3})

A^{1}_{211} = (1/(ab)) & A^{2}_{122} = (1/(ba))

A^{1}_{112} = (1/(ab)) & A^{2}_{221} = (1/(ba))

A^{1}_{121} = (1/(ab)) & A^{2}_{212} = (1/(ba))

A^{2}_{211} = (1/a^{2}) & A^{1}_{122} = (1/b^{2})

A^{2}_{121} = (1/a^{2}) & A^{1}_{212} = (1/b^{2}) 

A^{2}_{112} = (1/a^{2}) & A^{1}_{211} = (1/b^{2})

det( A^{k}_{ijs} ) = 0

Primera contracció tensorial:

A^{k}_{ijj}·a_{i}a_{j}a_{j} = a^{k}

a_{1} = a & a_{2} = b

A^{1}_{111} = (1/a^{2}) & A^{2}_{222} = (1/b^{2})

A^{2}_{111} = (b/a^{3}) & A^{1}_{222} = (a/b^{3})

A^{1}_{211} = (1/(ab)) & A^{2}_{122} = (1/(ba)) 

A^{2}_{211} = (1/a^{2}) & A^{1}_{122} = (1/b^{2}) 

det( A^{k}_{ijj} ) = 0

A^{k}_{jij}·a_{j}a_{i}a_{j} = a^{k}

a_{1} = a & a_{2} = b

A^{1}_{111} = (1/a^{2}) & A^{2}_{222} = (1/b^{2})

A^{2}_{111} = (b/a^{3}) & A^{1}_{222} = (a/b^{3})

A^{1}_{121} = (1/(ab)) & A^{2}_{212} = (1/(ba)) 

A^{2}_{121} = (1/a^{2}) & A^{1}_{212} = (1/b^{2})

det( A^{k}_{jij} ) = 0

A^{k}_{jji}·a_{j}a_{j}a_{i} = a^{k}

a_{1} = a & a_{2} = b

A^{1}_{111} = (1/a^{2}) & A^{2}_{222} = (1/b^{2})

A^{2}_{111} = (b/a^{3}) & A^{1}_{222} = (a/b^{3})

A^{1}_{112} = (1/(ab)) & A^{2}_{221} = (1/(ba)) 

A^{2}_{112} = (1/a^{2}) & A^{1}_{221} = (1/b^{2})

det( A^{k}_{jji} ) = 0

Teorema de contraccions tensorials:

Sum[ A^{k}_{ijs} ] = Sum[ ( A^{k}_{ijj}+A^{k}_{jij}+A^{k}_{jji} ) ]+(-2)·Sum[ A^{k}_{iii} ]

Prod[ A^{k}_{ijs} ] = ( Prod[ ( A^{k}_{ijj}·A^{k}_{jij}·A^{k}_{jji} ) ]/( Prod[ A^{k}_{iii} ] )^{2} )

Segona contraccció tensorial:

A^{j}_{jii}·a_{j}a_{i}a_{i} = a^{j}

a_{1} = a & a_{2} = b

A^{1}_{111} = (1/a^{2}) & A^{2}_{222} = (1/b^{2})

A^{2}_{211} = (1/a^{2}) & A^{1}_{122} = (1/b^{2})

det( A^{j}_{jii} ) = 0

A^{j}_{iji}·a_{i}a_{j}a_{i} = a^{j}

a_{1} = a & a_{2} = b

A^{1}_{111} = (1/a^{2}) & A^{2}_{222} = (1/b^{2})

A^{2}_{121} = (1/a^{2}) & A^{1}_{212} = (1/b^{2})

det( A^{j}_{iji} ) = 0

A^{j}_{iij}·a_{i}a_{i}a_{j} = a^{j}

a_{1} = a & a_{2} = b

A^{1}_{111} = (1/a^{2}) & A^{2}_{222} = (1/b^{2})

A^{2}_{112} = (1/a^{2}) & A^{1}_{221} = (1/b^{2})

det( A^{j}_{iij} ) = 0