A representation theorem for volume-preserving transformations

A general solution is presented for the partial differential equation ∂u/∂x=k(x), where u and x are n-vector fields, ∂u/∂x denotes the Jacobian of the transformation x→u and k(x) is a scalar-valued function. The solution for the case k(x)=1 is of special interest because it furnishes a representation theorem for volume-preserving transformations in an n-dimensional space. Such a representation for the case n=2 was obtained by Gauss. The solution for n=3, presented here, furnishes a representation for isochoric (volume-preserving) finite deformations, which are important in the mechanics of highly deformable incompressible solid materials.