Tagged Questions
3
votes
0answers
84 views
Tensors as mutlilinear maps
I am aware that many books on differential geometry define tensors as multilinear maps. Namely
$$
V\otimes W := L_2(V^*\times W^*,\Bbb F)
$$
I am also aware that this space is isomorphic to the tensor ...
0
votes
1answer
28 views
The operator in Tensor algebra.
Let $V$ be a vector space over a field $K$.
We define the $k^{th}$ tensor power of $V$:
$$T^kV = V \otimes V \otimes ... \otimes V$$
We contruct $T(V)$ as the direct sum of $T^kV$ for $k=0,1,2,...$
...
0
votes
0answers
26 views
what is the status of the theory of multilinear systems of equations?
What is the current status of the theory of multilinear systems of equations?
I have a particular interest for multilinear homogeneous systems of the form
$A_1 \otimes \cdots \otimes A_r) (x_1 ...
0
votes
1answer
17 views
Regarding confusion of basis tensors and the usage of tensors.
Let us for example give a tensor example of following: $X = X^i \partial_i$. According to mny knowledge, in this case $\partial_i$, basis, is treated as tensor (otherwise, $X$ as tensor won't be ...
0
votes
0answers
43 views
Tensor Products, various defintions
I came across a definition for the tensor product which differs from the standard definition. This book defined the tensor product of vector spaces $V$ and $W$ as the space $L(V,W,\Bbb K)$ of bilinear ...
5
votes
0answers
75 views
Tensor product of algebra
Can we find define a norm on tensor product $C(X) \otimes C(Y)$ such that the norm completion of $C(X)\otimes C(Y)=C(X\times Y)$
And can we define a norm on tensor product $L^1(X)\otimes L^1(Y)$ such ...
0
votes
0answers
79 views
Tensor calculus solution-why?
The text I read says that $\displaystyle\frac{\partial^2
x^\alpha}{\partial x^\delta
\partial x^\gamma}\frac{\partial x^\delta}{\partial
x^\beta} = 0$ leads to the solution $x^\alpha =
...
2
votes
1answer
87 views
Tensors in math and physics
I know how tensor product f two modules is defined in communtative algebra. But there is also a concept of tensors used in physics. Are these two concepts related? If yes, can someone explain me ...
0
votes
0answers
73 views
contraction with the metric tensor
What does mean "$\wedge_0^4V $ is the space of 4-vectors whose contraction with the metric tensor of the space $V$ vanishes" how can we formulate this set?
this means $i_gT=0$ for tensor $T$?
4
votes
2answers
127 views
Confusion when dealing with tensors.
I don't understand how tensors work, can someone please explain?
In particular, in the context of Electromagnetism, the dual of the field tensor $F$ is $$(*F)^{\mu\nu}:={1\over ...
4
votes
0answers
90 views
Mnemonic device for relationships between Hom and Tensor
Probably this is a stupid question, but nevertheless...
Let $A$, $B$, $C$ and $D$ be rings, and $M$, $N$ and $K$ be appropriate bimodules between them. There are extremely well-known canonical ...
2
votes
1answer
168 views
Tensor product and Kronecker Product
Is there any difference between tensor product and Kronecker Product?
1
vote
0answers
90 views
why do you need tensors of rank $>2$?
Question from someone just starting to study tensors (sorry if it's silly):
So I understand (maybe?) that tensors are basically about coordinate transformations (and things that are invariant under ...
4
votes
1answer
160 views
How do I do “calculations” with tensors?
I just started to read about tensor products and tensors and I understand that a tensor product $V \otimes W$ is a space used to replace bilinear maps $V \times W \to U$ with linear maps $V \otimes W ...
1
vote
2answers
216 views
What is the difference between tensors and tensor products?
The tensor product $S\otimes_R T$ of $S$ and $N$ over $R$ is a module. A multilinear form $L:V^r \to R$ is called an $r$-tensor on $V$.
On the one hand a tensor is a function sending elements of ...
5
votes
2answers
218 views
Unique symmetric covariant $k$-tensor satisfying $(\operatorname{Sym} T)(A,…,A)=T(A,…,A)$ for all $A \in V$
Let $T$ be a covariant $k$-tensor on a finite dimensional vector space $V$. I want to prove that the symmetrization of $T$ is the unique symmetric $k$-tensor satisfying the following condition:
...
5
votes
2answers
186 views
$e_1\otimes e_2 \otimes e_3$ cannot be written as a sum of an alternating tensor and a symmetric tensor
Let $(e_1,e_2,e_3)$ be the standard dual basis for $(\mathbb{R}^3)^\ast$. How can I show that $e_1\otimes e_2 \otimes e_3$ cannot be written as a sum of an alternating (or antisymmetric) tensor and a ...
