# Vicinity Transformation

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

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

So that Equation 1 can be written in matrix notation:

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:

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:

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

or in matrix form:

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