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Since modern continuum mechanics is mainly characterized by the strong influence of microstructure, Fractional Continuum Mechanics has been a promising research field, satisfying both experimental and theoretical demands. The geometry of the fractional differential is corrected and the geometry of the tangent spaces of a manifold is clarified providing the bases of the missing Fractional Differential Geometry. The Fractional Vector Calculus is revisited along with the basic field theorems of Green, Stokes and Gauss. New concepts of the differential forms, such as fractional gradient, divergence and rotation are introduced. Application of the Fractional Vector Calculus to Continuum Mechanics is presented. The Fractional right and left Cauchy-Green deformation tensors and Green (Lagrange) and Euler-Almanssi strain tensors are exhibited. The change of volume and the surface due to deformation (configuration change) of a deformable body are also discussed. Fractional stress tensors are also introduced. Further the Fractional Continuum Mechanics principles yielding the fractional continuity and motion equations are also derived.

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