Before we state the next difficulty we should give a definition; a tensor density of weight W transforms like an ordinary tensor, except that in addition the W th power of the Jacobian,
12.
The bundle of densities cannot seriously be defined'at a point'; and therefore a limitation of the contemporary mathematical treatment of tensors is that tensor densities are defined in a roundabout fashion.
13.
The Cotton tensor can be regarded as a vector valued 2-form, and for " n " = 3 one can use the Hodge star operator to convert this into a second order trace free tensor density
14.
The hyperdeterminant can be written in a more compact form using the Einstein convention for summing over indices and the Levi-Civita symbol which is an alternating tensor density with components ? ij specified by ? 00 = ? 11 = 0, ? 01 = " ? 10 = 1:
15.
A " tensor density " is the special case where " L " is the bundle of " densities on a manifold ", namely the determinant bundle of the cotangent bundle . ( To be strictly accurate, one should also apply the absolute value to the transition functions this makes little difference for an orientable manifold . ) For a more traditional explanation see the tensor density article.
16.
A " tensor density " is the special case where " L " is the bundle of " densities on a manifold ", namely the determinant bundle of the cotangent bundle . ( To be strictly accurate, one should also apply the absolute value to the transition functions this makes little difference for an orientable manifold . ) For a more traditional explanation see the tensor density article.
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