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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
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")
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 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)
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
Tensors: raising and lowering indices - Physics Forums Note these are the same tensor expressed in different bases You can also make a (0,2) tensor by lowering both indices of X^{\mu \sigma} The (0,2), (1,1), and original (2,0) tensors are all different objects labeled by the same letter - it's the index placement that differentiates them edit: hah, it took me too long to type this - you beat me
Understanding tensor contraction - Physics Forums And tensor field is defined as a map "from real space to some tensor space" Number (5) and (6) are tensor spaces but somehow trivial - these are only (1,1) tensor spaces (well related to spaces [itex]V[ itex] and [itex]W[ itex] because of course it could be that [itex]W = V^* \otimes V^* \otimes V^*[ itex])
Taking the Trace of a Tensor Product - Physics Forums An antisymmetric tensor must be tracefree, but not vice versa For example, the LHS of Einstein's field equations is R ij - (1 2)R g ij , where R ij is the Ricci curvature tensor, which is symmetric , with 10 independent parameters, and R is its trace (a scalar, obviously only 1 parameter) … the tracefree part of R ij is R ij - (1 4)R g ij