2. Tensor Analysis

Tensor analysis is an essential tool for derivation of the basic laws of physics, whenever representation of the law in a different coordinates system is required. The shell structures, due to the different geometries and different shell theories, fall into this category. To write the kinematical relations of a shell geometry, either the reader should accept a given relation or he/she has to be able to derive the required relations. This is why this chapter is devoted to the tensor analysis. The chapter begins with simple definitions of tensor laws in the rectangular Cartesian coordinates and then follows to discuss the tensor laws in the general curvilinear coordinates. The covariant and contravariant tensors and their coordinate transformations are given. The covariant derivatives are replaced with the partial derivatives, giving the requirement for considering the Christoffel symbols, and base vectors take the role of unit vectors (but not unit in magnitude). The chapter comes to an end with the explanation of surface theory and the basic definitions of the Codazzi and Gauss conditions, which are replaced with the compatibility conditions of the theory of elasticity.