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This can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property as explained here and here. A type (p, q) tensor is defined in this context as an element of the tensor product of vector spaces, [9] [10]
A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b.Analogous to vectors, it can be written as a linear combination of the tensor basis e x ⊗ e x ≡ e xx, e x ⊗ e y ≡ e xy, ..., e z ⊗ e z ≡ e zz (the right-hand side of each identity is only an abbreviation, nothing more):
Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity, [8] in the mechanics of curved shells, [6] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials [9] [10] and in many other fields.
Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of metric. For example, an event in spacetime may be represented as a position four-vector , with coherent derived unit of meters: it includes a position Euclidean vector and a timelike component, t ⋅ c 0 (involving the speed ...
The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the divergence theorem. The divergence can also be defined on a Riemannian manifold, that is, a manifold with a Riemannian metric that measures the length of vectors.
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T • (V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product.It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property ...
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics.These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.
Similarly, every second rank tensor (such as the stress and the strain tensors) has three independent invariant quantities associated with it. One set of such invariants are the principal stresses of the stress tensor, which are just the eigenvalues of the stress tensor. Their direction vectors are the principal directions or eigenvectors.