  A coordinate independent derivation of the Eulerian and Lagrangian strain
tensors of finite deformation theory is given based on the parallel propagator,
the world function, and the displacement vector field as a three-point tensor.
The derivation explicitly shows that the Eulerian and Lagrangian strain tensors
are two-point tensors, each a function of both the spatial and material
coordinates. The Eulerian strain is a two-point tensor that transforms as a
second rank tensor under transformation of spatial coordinates and transforms
as a scalar under transformation of the material coordinates. The Lagrangian
strain is a two-point tensor that transforms as scalar under transformation of
spatial coordinates and transforms as a second rank tensor under transformation
of the material coordinates. These transformation properties are needed when
transforming the strain tensors from one frame of reference to another moving
frame.
