The deformation gradient is a transformation matrix.
Symbol: "F".
Define:
X: Stretched-space matrix
B: Rest-space matrix
F: Deformation gradient
In an n-dimensional "rest" space (sometimes "material" space), the columns of "B" are the n basis vectors which span that space. B is an n-by-n matrix.
In an n-dimensional "stretched" space (sometimes "world" space), the columns of "X" are the n basis vectors which span that space. X is an n-by-n matrix.
Define F (the deformation gradient):
F = X*(B^-1)
For example:
The deformation gradient transforms a vector from rest-space to stretched-space.
F*B = X
The inverse of the deformation gradient transforms a vector from stretched-space to rest-space.
(F^-1)*X = B
