Differential geometry applied to continuum mechanics

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Differential geometry serves as a foundational framework for discussing continuum mechanics with precise mathematical terminology. The discussion begins with linear geometry in affine point spaces, leading into modern differential geometry on manifolds, covering topics such as topology, tensor algebra, bundles, tensor fields, exterior algebra, and both differential and integral calculus. These tools are then applied to fundamental aspects of continuum mechanics. The kinematics of a material body and mass balance are articulated using geometric terminology, while the principles of objectivity and material frame indifference in constitutive equations are explored. A clear distinction is made between the Lagrangian and Eulerian formulations. Additionally, a generalized Arbitrary Lagrangian-Eulerian (ALE) formulation on differentiable manifolds is presented, introducing a grid manifold that ensures a coherent description of the relationships among the material body, ambient space, and reference domain in the ALE framework. The goal is to compile essential formulas and key results—some with complete proofs—commonly utilized in the field. Practical point arguments and changes within equations are clearly indicated, and component and direct tensor notation are employed as necessary, avoiding a one-dimensional approach to the subject.