What Is a Tensor? The mathematical point of view. - Physics Forums 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 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, the stress tensor collapses to the traction vector
terminology - What is the history of the term tensor? - Mathematics . . . 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)
What are the Differences Between a Matrix and a Tensor? 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")
manifolds - Difference Between Tensor and Tensor field? - Mathematics . . . 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 Some additional mathematical details
Is Tensor Quantity a Combination of Vector and Scalar? - Physics Forums A tensor is a general quantity A scalar has magnitude with 0 direction, hence a rank 0 tensor A vector is a magnitude acting along a line, or 1 dimension, i e tensor of rank 1 A dyadic has magnitude(s), acting in various planes, i e 2 dimensions, making it a 2nd rank tensor These quantities are often expressed using matrices
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
What is a Rank 3 Tensor and Why Does It Matter? - Physics Forums A rank 3 tensor inputs three generalized vectors (i e either a vector or their dual vector), and spits out a scalar One can also think of it as inputting 2 generalized vectors (or a rank 2 tensor), and outputting a vector, or inputting 1 generalized vector, and outputing 2 vectors (or a rank 2 tensor)