Central to the computation of stresses and strains is the Jacobian matrix which relates the
initial and deformed configuration:(1)
(2)

$$d{x}_{i}=\frac{\partial {x}_{i}}{\partial {X}_{j}}d{X}_{j}={D}_{j}{x}_{i}d{X}_{j}={F}_{ij}d{X}_{j}$$

$${D}_{j}=\frac{\partial}{\partial {X}_{j}}$$

The transformation is fully described by the elements of the Jacobian matrix
$F$
:(3)

$${F}_{ij}\equiv {D}_{j}{x}_{i}$$

So that Equation 1 can be written in matrix
notation:(4)

$$dx=FdX$$

The Jacobian, or determinant of the Jacobian matrix, measures the relation between the initial
volume
$d\Omega $
and the volume in the initial configuration containing the same
points:(5)

$$d\Omega =|F|d{\Omega}^{0}$$

Physically, the value of the Jacobian cannot take the zero value without cancelling the volume
of a set of material points. So the Jacobian must not become negative whatever the final
configuration. This property insures the existence and uniqueness of the inverse
transformation:(6)

$$dX={F}^{-1}dx$$

At a regular point whereby definition of the field
$u(X)$
is differentiable, the vicinity transformation is defined
by:(7)

$${F}_{ij}={D}_{j}{x}_{i}={D}_{j}\left({X}_{i}+{u}_{i}(X,t)\right)={\delta}_{ij}+{D}_{j}{u}_{i}$$

or in matrix form:(8)

$$F=1+A$$

So, the Jacobian matrix
$F$
can be obtained from the matrix of gradients of
displacements:(9)

$$A\equiv {D}_{j}{u}_{i}$$