Tensor

Article summary

\mathbf{T} describes the stress experienced by a material at a given point. For any unit vector \mathbf{v} , the product \mathbf{T} \cdot \mathbf{v} is a vector, denoted \mathbf{T}(\mathbf{v}) , that quantifies the force per area along the plane perpendicular to \mathbf{v} . This image shows, for cube faces perpendicular to \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 , the corresponding stress vectors \mathbf{T}(\mathbf{e}_1), \mathbf{T}(\mathbf{e}_2), \mathbf{T}(\mathbf{e}_3) along those faces. Because the stress tensor takes one vector as input and gives one vector as output, it is a second-order tensor. In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.…

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