What Is a Tensor? The mathematical point of view. - Physics Forums
2024年9月30日 · 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.
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.
Are there any differences between tensors and multidimensional …
2015年2月5日 · 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.
What are the Differences Between a Matrix and a Tensor?
2013年6月6日 · 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").
terminology - What is the history of the term "tensor"?
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)
abstract-algebra proof-verification modules tensor-products
By universal property of tensor products, this factors through $\mathbb{Z}_m\otimes\mathbb{Z}_n$. Using ...
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 ...
Adjointness of Hom and Tensor - Mathematics Stack Exchange
The tensor product's construction as a quotient of a free module, while sometimes called a definition, is more accurately called a construction. That the result of the construction satisfies the univesal property of tensor products follows easilly from the univesal properties of free groups and quotients. $\endgroup$
Why is a linear transformation a $(1,1)$ tensor?
2015年1月18日 · $\begingroup$ After tangling a bit on the Wikipedia's article talk page I've discovered that the regulars there have the following agenda/beliefs: (1) tensors are never defined over infinite-dimensional vector spaces and (2) tensors are never defined over different vector spaces (even if finite).