A continuum body is always embedded in three-dimensional Euclidean space, however its kinematics may be described in either rectilinear (Cartesian) or curvilinear coordinates. Expressed in curvilinear coordinates the differential geometry of flat Euclidean space is captured by the Christoffel symbols that take the role of a symmetric and integrable metric connection with associated zero curvature tensor. The position of a physical point together with the distortion, the double-distortion, and the triple-distortion are essential quantities to describe the continuum kinematics. Thus their representation is carefully elaborated. The kinematics of a continuum body are further characterized by an embedded non-Euclidean manifold. The embedded manifold is represented by a connection that additively decomposes into an integrable and a non-integrable contribution. The integrability conditions for the distortion and the double-distortion prove to be governed by the anholonomic object, the torsion, the curvature, and the non-metricity of the embedded manifold. To describe the deviation from integrability four defect density tensors, i.e. the primary and the secondary dislocation density tensors, the disclination density tensor, and the point-defect density tensor are introduced. Various types of continua with defects may be classified based on these defect density tensors.
The chapter also contains a comprehensive account on tensor calculus in Euclidean space in an extended supplement (which also introduces the symbolic notation used extensively throughout Part III).