Editing Tensor (section)

=== Tensor fields ===
{{Main|Tensor field}}
In many applications, especially in differential geometry and physics, it is natural to consider a tensor with components that are functions of the point in a space.  This was the setting of Ricci's original work.  In modern mathematical terminology such an object is called a [[tensor field]], often referred to simply as a tensor.<ref name="Kline" />

In this context, a [[coordinate basis]] is often chosen for the [[tangent space|tangent vector space]].  The transformation law may then be expressed in terms of [[partial derivative]]s of the coordinate functions,
:<math>\bar{x}^i\left(x^1, \ldots, x^n\right),</math>

defining a coordinate transformation,<ref name="Kline" />
:<math>
  \hat{T}^{i'_1\dots i'_p}_{j'_1\dots j'_q}\left(\bar{x}^1, \ldots, \bar{x}^n\right) =
  \frac{\partial \bar{x}^{i'_1}}{\partial x^{i_1}}
  \cdots
  \frac{\partial \bar{x}^{i'_p}}{\partial x^{i_p}}
  \frac{\partial x^{j_1}}{\partial \bar{x}^{j'_1}}
  \cdots
  \frac{\partial x^{j_q}}{\partial \bar{x}^{j'_q}}
  T^{i_1\dots i_p}_{j_1\dots j_q}\left(x^1, \ldots, x^n\right).
</math>