What Is a Tensor? The mathematical point of view. - Physics Forums
2024年9月30日 · A four-dimensional version (tensor) could be viewed as the tensor product of two matrices, which themselves are tensor products of two vectors and always sums of them. Let us consider now arbitrary ##2 \times 2## matrices ##M## and order their entries such that we can consider them as vectors because ##\mathbb{M}(2,2)## is a vector space.
What, Exactly, Is a Tensor? - Mathematics Stack Exchange
2014年1月30日 · 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, …
An Introduction to Tensors - Mathematics Stack Exchange
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.
terminology - What is the history of the term "tensor"?
Voigt used tensors for a description of stress and strain on crystals in 1898 [14], and the term tensor first appeared with its modern meaning in his work.$^\dagger$ Tensor comes from the Latin tendere, which means "to stretch." In mathematics, Ricci applied tensors to differential geometry during the 1880s and 1890s.
abstract algebra - What exactly is a tensor product? - Mathematics ...
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 $\textbf{uv ...
differential geometry - What is the definition of tensor contraction ...
Most of this formalism is quite general and in particular there are contexts where you have tensor ...
What is a rank-1 tensor? What is the meaning of rank in this context?
That's what it seems like from studying Eq. (3.1). Does the term "rank" here have anything to do with the number of linearly independent columns or is it just describing the order of the tensor, i.e., the number of dimensions of the tensor? For example, a rank-2 tensor would then be a matrix and a rank-0 tensor would be a scalar?
What does "$(n,m)$-tensor" mean? - Mathematics Stack Exchange
Assuming your meaning of tensor is the first one, an $(n,m)$ tensor means an element of $(V^{\otimes n}) ...
What does the dot product of a tensor and a vector represent?
The meaning of the dot product of two vectors has been well explained below: What does the dot product of two vectors represent? What is physical interpretation of dot product? [duplicate] But, what is the meaning of the dot product of a tensor and a vector, if there is any?
Meaning of a $(0,2)$ tensor field being "nondegenerate"
2020年9月14日 · A $(0,2)$ tensor field can be thought of as a bilinear form on the module of vector fields over smooth functions, thus we can get a notion of nondegeneracy from just taking the original definition and generalizing to modules.