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For example, the Ricci tensor is a non-metric contraction of the Riemann curvature tensor, and the scalar curvature is the unique metric contraction of the Ricci tensor. One can also view contraction of a tensor field in the context of modules over an appropriate ring of functions on the manifold [ 5 ] or the context of sheaves of modules over ...
Note: the magnitude of the axial vector, , is the sole invariant of the skew part of , whereas these distinct three invariants characterize (in a sense) "alignment" between the symmetric and skew parts of .
The tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2) -tensor; an inner product is an example of a (0, 2) -tensor, but not all (0, 2) -tensors are ...
Totally antisymmetric tensors include: Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric). The electromagnetic tensor, in electromagnetism. The Riemannian volume form on a pseudo-Riemannian manifold.
Examples: is a model for 3-dimensional space. The metric is equivalent to the standard dot product., =, equivalent to dimensional real space as an inner product space with =. In Euclidean space, raising and lowering is not necessary due to vectors and covector components being the same.
For example, if a few points have non-trivial stabilisers, then the categorical quotient will not exist among schemes, but it will exist as a stack. In the same way, moduli spaces of curves, vector bundles, or other geometric objects are often best defined as stacks instead of schemes.
The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor. A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain the correct order. Dyadic tensor A dyadic tensor is a tensor of order two, and may be represented as a square matrix.
The minimum number r for which such a decomposition is possible is the (symmetric) rank of T. The vectors appearing in this minimal expression are the principal axes of the tensor, and generally have an important physical meaning. For example, the principal axes of the inertia tensor define the Poinsot's ellipsoid representing the moment of ...