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In classical mechanics, there are two objectivities: The ``isometricobjectivity'' which concerns the constitutive laws of materials once expressedin a reference frame, and the ``covariant objectivity'' which concerns theuniversal laws of physics required to be observer independent. Covariantobjectivity is the main topic in this manuscript. We therefore first introducemotions and Eulerian variables and functions, then we deduce Lagrangianvariables and functions, then define the function ``deformation gradient'',from which we deduce simple definitions of the Lie derivatives. Then we detailthe construction of the velocity addition formula and recall the proof of theobjective covariance of the Lie derivatives. And constituvive laws built fromLie derivatives are introduced.

Objectivity in continuum mechanics, an introduction. Motions, Eulerian  and Lagrangian variables and functions, deformation gradient, Lie  derivatives, velocity-addition formula, Coriolis