determinant tensorial de ordre 2

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

... ( a^{1}_{11}a^{2}_{22}+(-1)·a^{2}_{11}a^{1}_{22} )+...

... ( a^{1}_{12}a^{2}_{21}+(-1)·a^{2}_{12}a^{1}_{21} )

A^{k}_{ij}·a_{i}a_{j} = a^{k}

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

A^{1}_{11} = (1/a) & A^{2}_{22} = (1/b)

A^{2}_{11} = (b/a^{2}) & A^{1}_{22} = (a/b^{2})

A^{1}_{21} = (1/b) & A^{2}_{12} = (1/a)

A^{2}_{21} = (1/a) & A^{1}_{12} = (1/b)

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

A^{k}_{ij}·a_{i}a_{j} = ( a^{k} )^{n}

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

A^{1}_{11} = a^{(n+(-2))} & A^{2}_{22} = b^{(n+(-2))}

A^{2}_{11} = (b^{n}/a^{2}) & A^{1}_{22} = (a^{n}/b^{2})

A^{1}_{21} = (a^{(n+(-1))}/b) & A^{2}_{12} = (b^{(n+(-1))}/a)

A^{2}_{21} = (b^{(n+(-1))}/a) & A^{1}_{12} = (a^{(n+(-1))}/b)

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

A^{k}_{ij}·( a_{i} )^{p}·( a_{j} )^{q} = ( a^{k} )^{n}

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

A^{1}_{11} = (a^{n}/a^{(p+q)}) & A^{2}_{22} = (b^{n}/b^{(p+q)})

A^{2}_{11} = (b^{n}/a^{p+q}) & A^{1}_{22} = (a^{n}/b^{p+q})

A^{1}_{21} = (a^{(n+(-q))}/b^{p}) & A^{2}_{12} = (b^{(n+(-q))}/a^{p})

A^{2}_{21} = (b^{(n+(-p))}/a^{q}) & A^{1}_{12} = (a^{(n+(-p))}/b^{q})

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

det( b^{k}_{i} ) = b^{1}_{1}b^{2}_{2}+(-1)·b^{2}_{1}b^{1}_{2}

B^{k}_{i}·b_{i} = b^{k}

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

B^{1}_{1} = 1 & B^{2}_{2} = 1

B^{2}_{1} = (b/a) & B^{1}_{2} = (a/b)

det( B^{k}_{i} ) = 0

B^{k}_{i}·b_{i} = ( b^{k} )^{n}

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

B^{1}_{1} = a^{(n+(-1))} & B^{2}_{2} = b^{(n+(-1))}

B^{2}_{1} = (b^{n}/a) & B^{1}_{2} = (a^{n}/b)

det( B^{k}_{i} ) = 0

B^{k}_{i}·( b_{i} )^{m} = ( b^{k} )^{n}

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

B^{1}_{1} = a^{(n+(-m))} & B^{2}_{2} = b^{(n+(-m))}

B^{2}_{1} = (b^{n}/a^{m}) & B^{1}_{2} = (a^{n}/b^{m})

det( B^{k}_{i} ) = 0

Contracció Tensorial:

A^{j}_{ij}·a_{i}a_{j} = a^{j}

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

A^{1}_{11} = (1/a) & A^{2}_{22} = (1/b)

A^{1}_{21} = (1/b) & A^{2}_{12} = (1/a)

det( A^{j}_{ij} ) = 0

A^{j}_{ji}·a_{j}a_{i} = a^{j}

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

A^{1}_{11} = (1/a) & A^{2}_{22} = (1/b)

A^{1}_{12} = (1/b) & A^{2}_{21} = (1/a)

det( A^{j}_{ji} ) = 0

Teorema de contraccions tensorials:

Sum[ A^{k}_{ij} ] = Sum[ ( A^{j}_{ij}+A^{j}_{ii}+A^{j}_{ji} ) ]+(-2)·Sum[ A^{i}_{ii} ]

Prod[ A^{k}_{ij} ] = ( Prod[ ( A^{j}_{ij}·A^{j}_{ii}·A^{j}_{ji} ) ]/( Prod[ A^{i}_{ii} ] )^{2} )