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A scalar function that depends entirely on the principal invariants of a tensor is objective, i.e., independent of rotations of the coordinate system. This property is commonly used in formulating closed-form expressions for the strain energy density , or Helmholtz free energy , of a nonlinear material possessing isotropic symmetry.
The theory of optimizing compilers, the methodology of design by contract, and formal methods for determining program correctness, all rely heavily on invariants. Programmers often use assertions in their code to make invariants explicit. Some object oriented programming languages have a special syntax for specifying class invariants.
[1] are a set of tensor invariants that span the space of real, symmetric, second-order, 3-dimensional tensors and are isomorphic with respect to principal stress space. This right-handed orthogonal coordinate system is named in honor of the German scientist Dr. Walter Lode because of his seminal paper written in 1926 describing the effect of ...
Invariant-based programming [1] is a programming methodology where specifications and invariants are written before the actual program statements. Writing down the invariants during the programming process has a number of advantages: it requires the programmer to make their intentions about the program behavior explicit before actually implementing it, and invariants can be evaluated ...
We are, in classical language, looking at invariants of n-ary r-ics, where n is the dimension of V. (This is not the same as finding invariants of GL(V) on S(V); this is an uninteresting problem as the only such invariants are constants.) The case that was most studied was invariants of binary forms where n = 2.
a tensor of order k. Then T is a symmetric tensor if = for the braiding maps associated to every permutation σ on the symbols {1,2,...,k} (or equivalently for every transposition on these symbols). Given a basis {e i} of V, any symmetric tensor T of rank k can be written as
where is the Ricci curvature tensor and is the Ricci scalar curvature (obtained by taking successive traces of the Riemann tensor). The Ricci tensor vanishes in vacuum spacetimes (such as the Schwarzschild solution mentioned above), and hence there the Riemann tensor and the Weyl tensor coincide, as do their invariants.
The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci. [1] An equivalent definition of a tensor uses the representations of the general linear group. There is an action of the general linear group on the set of all ordered bases of an n-dimensional vector space.