As in the one dimensional strain derivation, suppose that point P in a body shifts to point P after deformation.
The infinitesimal straindisplacement relationships can be summarized as,
where u is the displacement vector, x is coordinate, and the two indices i and j can range over the three coordinates {1, 2, 3} in three dimensional space.
Expanding the above equation for each coordinate direction gives,
where u, v, and w are the displacements in the x, y, and z directions respectively (i.e. they are the components of u).
