# Vicinity Transformation

Central to the computation of stresses and strains is the Jacobian matrix which relates the initial and deformed configuration:(1)
$d{x}_{i}=\frac{\partial {x}_{i}}{\partial {X}_{j}}d{X}_{j}={D}_{j}{x}_{i}d{X}_{j}={F}_{ij}d{X}_{j}$
(2)
${D}_{j}=\frac{\partial }{\partial {X}_{j}}$
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 $u\left(X\right)$ is differentiable, the vicinity transformation is defined by:(7)
${F}_{ij}={D}_{j}{x}_{i}={D}_{j}\left({X}_{i}+{u}_{i}\left(X,t\right)\right)={\delta }_{ij}+{D}_{j}{u}_{i}$
or in matrix form:(8)
$F=1+A$
So, the Jacobian matrix $F$ can be obtained from the matrix of gradients of displacements:(9)
$A\equiv {D}_{j}{u}_{i}$