Tensor Conventions for Contravariant and Covariant Notation

[quasar987]

If a tensor acts on V x V* x V, do we say that it is

i. 1 time contravariant, 1 time covariant, then 1 time contravariant

ii. 2 times contravariant, 1 time covariant

iii. 1 time covariantt, 2 times contravariant

vi. there is no convention

 

[dextercioby]

The tensor is neithzer covariant nor contravariant, it's an invariant geometrical object. However, in the canonical basis of the tensor product of spaces under discussion its components are

$$T=T^{i}{}_{j}{}^{k} e_{i}\otimes e^{j}\otimes e_{k}$$

So which one is it ?i, ii, iii or iv ?

 

[tacman]

for this

The correct answer would be ii. 2 times contravariant, 1 time covariant. This is because the notation for tensors follows the order of the spaces they act on in a specific order, with contravariant spaces written first and covariant spaces written last. In this case, the tensor acts on two contravariant spaces and one covariant space, so the notation would be 2 times contravariant, 1 time covariant. This follows the standard convention for tensor notation.