Editing Tensor field (section)

== Via coordinate transitions ==
Following {{harvtxt|Schouten|1951}} and {{harvtxt|McConnell|1957}}, the concept of a tensor relies on a concept of a reference frame (or [[coordinate system]]), which may be fixed (relative to some background reference frame), but in general may be allowed to vary within some class of transformations of these coordinate systems.<ref>The term "[[affinor]]" employed in the English translation of Schouten is no longer in use.</ref>

For example, coordinates belonging to the ''n''-dimensional [[real coordinate space]] <math>\R^n</math> may be subjected to arbitrary [[affine transformation]]s:
: <math>x^k\mapsto A^k_jx^j + a^k</math>
(with ''n''-dimensional indices, [[Einstein summation convention|summation implied]]).  A covariant vector, or covector, is a system of functions <math>v_k</math> that transforms under this affine transformation by the rule
: <math>v_k\mapsto v_iA^i_k.</math>
The list of Cartesian coordinate basis vectors <math>\mathbf e_k</math> transforms as a covector, since under the affine transformation <math>\mathbf e_k\mapsto A^i_k\mathbf e_i</math>.  A contravariant vector is a system of functions <math>v^k</math> of the coordinates that, under such an affine transformation undergoes a transformation
: <math>v^k\mapsto (A^{-1})^k_jv^j.</math>
This is precisely the requirement needed to ensure that the quantity <math>v^k\mathbf e_k</math> is an invariant object that does not depend on the coordinate system chosen.  More generally, the coordinates of a tensor of valence (''p'',''q'') have ''p'' upper indices and ''q'' lower indices, with the transformation law being
: <math>{T^{i_1\cdots i_p}}_{j_1\cdots j_q}\mapsto A_{i'_1}^{i_1}\cdots A_{i'_p}^{i_p}{T^{i'_1\cdots i'_p}}_{j'_1\cdots j'_q}(A^{-1})_{j_1}^{j'_1}\cdots (A^{-1})_{j_q}^{j'_q}.</math>

The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to be [[smooth function|smooth]] (or [[differentiable function|differentiable]], [[analytic function|analytic]], etc.).  A covector field is a function <math>v_k</math> of the coordinates that transforms by the [[Jacobian matrix|Jacobian]] of the transition functions (in the given class).  Likewise, a contravariant vector field <math>v^k</math> transforms by the inverse Jacobian.