Coordinate invariance

For fixed coordinates, invariance of the action integral for gauge variations about SU(2) field solutions implies... 

The principle of coordinate invariance states that constitutive rules must be stated in such a way that they do not depend on the coordinate system. One way to insure that this principle is satisfied is to work with dyadic or invariant notation as is common practice in the more mathematical engineering disciplines. 

It is important to emphasize that the mathematical constraint imposed by coordinate invariance addresses only the selection of an allowable form of a constitutive equation, given the physical assumption, based on an educated guess, that there is a linear relationship between q and VO. Whether the resulting constitutive equation captures the behavior of any real material is really a question of whether the physical assumption of linearity is an adequate approximation. In fact, in the generalized Fourier heat conduction model, Eq. (2-65), there are several additional physical assumptions that must be satisfied, besides linearity between q and V0 ... 

The quantities cyku describing the material properties, are called elastic stiffness constants or stiffiiesses. The transformation properties of the cyki under a rotation of the coordinate system is uniquely determined by the requirement of a coordinate invariant formulation of Eq. (3.51). The rank of this material tensor is equal to the sum of the ranks of strain and stress tensor. Accordingly, cyki are the coordinates of a fourth rank tensor. The symmetry of strain and stress tensors reqitires the symmetry of the stiffness tensor with regard to an interchange of i and j as well as k and /... 

The mirror planes with - 1 characters do not leave the a2 coordinate invariant. One thus retains the Q subgroup as the kernel of the conrotatory path. Equation (2) may therefore be rewritten as ... 

In isotropic materials, the yield surface must not depend on the orientation of the load. Thus, the function / describing the yield surface can only contain those parts of the deviatoric stress tensor that do not change during coordinate transformations. This is already ensured if the principal stresses are used because the hydrostatic stress (Thyd is also coordinate invariant. 

Additionally, questions of the coordinate invariance of these paths have to be answered (cf. Refs. 24a,25a,9a and Chap.3). Starting with the Cartesian coordinate system, one may transform and describe the RP in any other coordinate systems by means of differential geometry. In this manner, a gradient path (e.g. the steepest descent) is invariant with respect to the choice of the coordinate system and may be regarded as a fundamental (mathematical) characteristic of a chemical reaction. 

Epstein ST (1975) Coordinate invariance, the differential force law, and the divergtaice of the stress-enCTgy tensor. J Chem Phys 63 3573-3574... 

---