Covariance and Gauge Invariance in Continuum Physics : Application to Mechanics Gravitation and Electromagnetism This book presents a Lagrangian approach model to formulate various fields of continuum physics ranging from gradient continuum elasticity to relativistic gravito electromagnetism. It extends the classical theories based on Riemann geometry to Riemann Cartan geometry and then describes non homogeneous continuum and spacetime with torsion in Einstein Cartan relativistic gravitation. It investigates two aspects of invariance of the Lagrangian: covariance of formulation following the method of Lovelock and Rund and gauge invariance where the active diffeomorphism invariance is considered by using local Poincaré gauge theory according to the Utiyama method. Further it develops various extensions of strain gradient continuum elasticity relativistic gravitation and electromagnetism when the torsion field of the Riemann Cartan continuum is not equal to zero. Lastly it derives heterogeneous wave propagation equations within twisted and curved manifolds and proposes a relation between electromagnetic potential and torsion tensor General introduction Basic concepts on manifolds spacetimes and calculus of variations Covariance of Lagrangian density function Gauge invariance for gravitation and gradient continuum Topics in continuum mechanics and gravitation Topics in gravitation and electromagnetism General conclusion Annexes