Search results
Results from the WOW.Com Content Network
The tensor product of V and its dual space is isomorphic to the space of linear maps from V to V: a dyadic tensor vf is simply the linear map sending any w in V to f(w)v. When V is Euclidean n-space, we can use the inner product to identify the dual space with V itself, making a dyadic tensor an elementary tensor product of two vectors in ...
The divergence of a second-order tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero.
For example, in a fixed basis, a standard linear map that maps a vector to a vector, is represented by a matrix (a 2-dimensional array), and therefore is a 2nd-order tensor. A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. Scalars are simple numbers and are thus 0th-order tensors.
The map C defines the contraction operation on a tensor of type (1, 1), which is an element of . Note that the result is a scalar (an element of k ). In finite dimensions , using the natural isomorphism between V ⊗ V ∗ {\displaystyle V\otimes V^{*}} and the space of linear maps from V to V , [ 1 ] one obtains a basis-free definition of the ...
Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, [ 6 ] Naghdi, [ 7 ] Simmonds, [ 2 ] Green and Zerna, [ 5 ] Basar and Weichert, [ 8 ] and Ciarlet.
Non-degeneracy shows that the partial evaluation map is injective, or equivalently that the kernel of the map is trivial. In finite dimension, the dual space V ∗ {\displaystyle V^{*}} has equal dimension to V {\displaystyle V} , so non-degeneracy is enough to conclude the map is a linear isomorphism.
In machine learning, the term tensor informally refers to two different concepts (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector ...
It can be shown that the stress tensor is a contravariant second order tensor, which is a statement of how it transforms under a change of the coordinate system. From an x i -system to an x i ' -system, the components σ ij in the initial system are transformed into the components σ ij ' in the new system according to the tensor transformation ...