Book review: Marcelo Epstein, The Geometrical Language of Continuum Mechanics

Intended mainly for continuum mechanicists, Epstein's book introduces modern geometry and some of its applications to theoretical continuum mechanics. Thus, examples for the mathematical objects introduced are chosen from the realm of mechanics. In particular, differentiable manifolds, tangent and cotangent bundles, Riemannian manifolds, Lie derivatives, Lie groups, Lie algebras, differential forms and integration theory are presented in the main part of the book. Once the reader's familiarity with continuum mechanics is used for the introduction of basic geometry, geometry is used in order to generalize notions of continuum mechanics. Integration of differential forms is used to formulate flux theory on manifolds devoid of a Riemannian structure. More specialized topics, namely, Whitney's geometric integration theory and Sikorski's differential spaces are used to relax smoothness assumptions for bodies and fields defined on them. Finally, an overview is given of the work that Epstein and co-workers carried out in recent years where the theory of inhomogeneity of constitutive relations is developed using the geometry of principal fiber bundles, G-structures and connections.