Tensors

A tensor is defined as the linear map of the cross product of the cotangent space and the tangent space to the n-dimensional real space. A tensor takes r co-vectors and s vectors and gives us a number, $T({\omega }_{(1)},...,{\omega }_{(r)},V_{(1)},...,V_{(s)})$denoted as a (r,s) tensor field, and these (r,s) tensors form a vector space. A tensor is characterised by its components, $T_{{\nu }_1...{\nu }_s}^{{\mu }_1...{\mu }_r}=T(d{x}^{{\mu }_1},...,d{x}^{{\mu }_r},{\partial }_{{\nu }_1},...,{\partial }_{{\nu }_s})$ Under coordinate transformation, a tensor transforms in the same way as a vector in the upper indices and as a co-vector in the lower indices.

• Note that $T_{\nu }^{\mu }=V^{\mu }{\omega }_{\nu }$is an example of a Tensor Equation, an equation between the components of tensors of the same type, which is covariant under coordinate transformation.

**Rules for tensor equations:

• Needs to relate tensors of the same type.
• The indices can appear once (free index) or twice (contracted index) in any given term.
  • Free indices tell us what type of tensor it is: these indices need to appear in every term and in the same position (up or down).
  • Contracted indices represent sums and hence must be one up and one down in terms where it appears.
• Indices can be renamed to help with simplification.
• The order of tensors does not matter but the order of indices does.