Jan 26, 2025 · The tensor product of two 1 dimensional vector spaces is 1 dimensional so it is smaller not bigger than the direct sum. The tensor product tof two 2 dimensional vector spaces is 4 dimensional so this is the the same size as the direct sum not bigger.This is correct but missing the relevant point: that the presentation contains a false statement.

A tensor field of type $(0, 0)$ is a smooth function. A tensor field of type $(1, 0)$ is a vector field. A tensor field of type $(0, 1)$ is a differential $1$-form. A tensor field of type $(1, 1)$ is a morphism of vector fields. A tensor field of type $(0, 2)$ which is symmetric and nondegenerate is a metric tensor.

Jan 30, 2014 · The complete stress tensor, $\sigma$, tells us the total force a surface with unit area facing any direction will experience. Once we fix the direction, we get the traction vector from the stress tensor, or, I do not mean literally though, …

Feb 5, 2015 · Tensor : Multidimensional array :: Linear transformation : Matrix. The short of it is, tensors and multidimensional arrays are different types of object; the first is a type of function, the second is a data structure suitable for representing a tensor in a coordinate system.

A part of the tensor history must come from tenses (past present future) and how Aristotle defined time as the measure of change / motion / movement. So really descriptions of changes of the state (or unchanging stillness) whether static or dynamic. EDIT: Tensor:= Tense + or (REF-1, includes William Rowan Hamilton algebraic origin)

Jun 6, 2013 · The components of a rank-2 tensor can be written in a matrix. The tensor is not that matrix, because different types of tensors can correspond to the same matrix. The differences between those tensor types are uncovered by the basis transformations (hence the physicist's definition: "A tensor is what transforms like a tensor").

This is a beginner's question on what exactly is a tensor product, in laymen's term, for a beginner who has just learned basic group theory and basic ring theory. I do understand from wikipedia that in some cases, the tensor product is an outer product, which takes two vectors, say $\textbf{u}$ and $\textbf{v}$, and outputs a matrix $\textbf{uv ...

A tensor field has to do with the notion of a tensor varying from point to point . A scalar is a tensor of order or rank zero , and a scalar field is a tensor field of order zero . A vector is a tensor of order or rank one , and a vector field is a tensor field of order one . …

Jun 1, 2018 · Instead, it is generally a linear combination of such objects. A better way of viewing a (rank 2) tensor is as a linear map from a vector space to itself, i.e., it maps a vector to another vector. In the case of the stress tensor, it is a linear transformation from directed surface normals to forces, both of which are vectors.

The tensor product $\color{darkorange}{\otimes}$ in $(2)$ is a tensor product not of (co)vectors, but of (co)vector spaces. The result of that tensor product describes not one tensor, but the set of all tensors of a given type. The tensors are then elements of the corresponding set.