Let \(U\), \(V\) be two vector spaces on the space field. Then \(U \otimes V\) is vector space with a bilinear form \(\otimes\), which maps a tuple \((u,v)\in U\times V\) to an element \(U\otimes V\) denoted by \(u\otimes v\). Such an element is called tensor.
This space can be seen as a quotient space of \(U\times V\), where elements which are satisfies the bilinearity relation are identified with each other. [@lee2013smooth_manifolds]
We call elements of \(T^k(V^*)=V^*\otimes \cdots \otimes V^*\) (\(V^*\) appears \(k\) time) covariant tensors on \(V\) of rank \(k\) (or just covariant \(k\)-tensors) and elements of \(T^k(V)=V\otimes \cdots \otimes V\) are called contravariant tensors on \(V\) of rank \(k\). The space \(T^{(k,l)}(V)=T^k(V)\otimes T^l(V^*)\) contains mixed tensors on \(V\) of type \((k,l)\).
- Covariant 1-tensors are covectors and contravariant 1-tensors are vectors.
- The name covariant and contravariant is derived from tangent vectors and covectors .
- Tensors may be identified with multilinear maps.