Editing Tensor field (section)

== Notation ==

The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces.  Thus, the tangent bundle ''TM'' = ''T''(''M'') might sometimes be written as
: <math>T_0^1(M)=T(M) =TM </math>
to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold ''M''. This should not be confused with the very similar looking notation
: <math>T_0^1(V)</math>;

in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined for each point in the manifold ''M''.

Curly (script) letters are sometimes used to denote the set of [[smooth function|infinitely-differentiable]] tensor fields on ''M''. Thus,
: <math>\mathcal{T}^m_n(M)</math>
are the sections of the (''m'',''n'') tensor bundle on ''M'' that are infinitely-differentiable. A tensor field is an element of this set.