File:Cauchy strain and gradient tensor.svg

English: The infinitesimal strain tensor is the symmetric part of the gradient of the deformation tensor, ${\displaystyle {\underline {\nabla }}\,{\underline {u}}.}$ which is assumed constant with respect to ${\displaystyle x\equiv x_{1}}$ and ${\displaystyle y\equiv x_{2}.}$ The yellow highlights The blue colored square represents ${\displaystyle {\hat {e}}_{i}dx_{i}}$, where it is convenient let ${\displaystyle dx_{i}=1}$. This allows us to represent ${\displaystyle (\partial u_{j}/\partial x_{i})dx_{i}}$ as ${\displaystyle \partial _{i}u_{j},}$ These four expressions state how far the points (0,1) and (1,0) move in the x and y directions due to the strain in a coordinate system in which no displacement occurs at the origin (0,0).