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Underlying spacetime is the Weitzenböck spacetime, which has a quadruplet of parallel vector fields as the fundamental structure. These parallel vector fields give rise to the metric tensor as a by-product. All physical laws are expressed by equations that are covariant or form invariant under the group of general coordinate transformations.
The argument of methods symmetrize() and antisymmetrize() in tensor classes is now directly a sequence of index positions (and no longer a single list/tuple encapsulating such a sequence). Method self_contract() of tensor classes renamed trace(). The code for tensor contractions has been optimized; moreover multiple tensor contractions are now ...
Covariant differentiation of tensors was given a geometric interpretation by Levi-Civita (1917) who introduced the notion of parallel transport on surfaces. His discovery prompted Weyl and Cartan to introduce various notions of connection, including in particular that of affine connection.
An example for recurrent tensors are parallel tensors which are defined by = with respect to some connection .. If we take a pseudo-Riemannian manifold (,) then the metric g is a parallel and therefore recurrent tensor with respect to its Levi-Civita connection, which is defined via
In continuum mechanics, a compatible deformation (or strain) tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of the conditions under which such a displacement field can be guaranteed.
for any two points x, y on C, parallel transport is an affine transformation from A x to A y; parallel transport is defined infinitesimally in the sense that it is differentiable at any point on C and depends only on the tangent vector to C at that point; the derivative of the parallel transport at x determines a linear isomorphism from T x M ...
The system of n(n + 1)/2 one-forms (ω i, ω j i (i<j)) gives an absolute parallelism of F(n), since the coordinate differentials can each be expressed in terms of them. Under the action of the Euclidean group, these forms transform as follows.
The dual tree hypercomplex wavelet transform (HWT) developed in [9] consists of a standard DWT tensor and 2 m -1 wavelets obtained from combining the 1-D Hilbert transform of these wavelets along the n-coordinates. In particular a 2-D HWT consists of the standard 2-D separable DWT tensor and three additional components: H x {ψ(x) h ψ(y) h ...