## Tagged Questions

0answers
44 views

### Null vector fields given Bondi metric

I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric $g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$ with \$d\Omeg …
0answers
61 views

### Geometric interpretation of tracing

Let $(M,g)$ be a Riemannian manifold. Is it true that for any symmetric 2-tensor $\alpha$ we have: $Trace_g(\alpha)=1/\omega_n\int_{S^{n-1}}\alpha(V,V)dvol(V)$ where $w_n$ is the v …
0answers
40 views

### covarient derivative of electromagnetic field tensor

I'm trying to prove the energy momentum tensor in curved spacetime for Electromagnetic field is Divergence-less directly(Without using general lie derivative method which can prove …
0answers
120 views

### How to find the tensor product of modules that we don’t know a basis for them?

Hi I know how to calculate some easy tensor products like $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}\cong_{\mathbb{Z}} \mathbb{Z}/(m,n)\mathbb{Z}$ or \$F[ …
1answer
159 views

### How many flavors should a notational system offer for rank-1 tensors?

The notation for tensors is like the plumbing in a very old Vermont farmhouse. It may once have been intentionally designed, but after that it just evolved. As an example, it seems …
11answers
3k views

### Why are matrices ubiquitous but hypermatrices rare?

I am puzzled by the amazing utility and therefore ubiquity of two-dimensional matrices in comparison to the relative paucity of multidimensional arrays of numbers, hypermatrices. O …
1answer
532 views

### Representation theory of (anti)self-dual tensors

I am using usual physics notations and I guess the physics motivations of this question are obvious. Let a basis of the $SO(n,m)$ Lie algebra be denoted by $S^{\mu \nu}$ and the …
3answers
2k views

### Geometrical meaning of the Ricci Tensor and its Symmetry

Let $M$ be a smooth, pseudo-Riemannian manifold with $\dim(M) \ge 2.$ Let $\nabla$ be any affine connection on $M$. No reason for it to be the Levi-Civita connection. All we assume …
0answers
210 views

### A property on the Green-St Venant strain tensor

Green-St Venant strain tensor is defined by $E(u)={1\over 2}[\nabla u+(\nabla u)^T+(\nabla u)^T\nabla u]$, where $\nabla u$ is the displacement gradient. Show that \$u\in H^1(\Om …
0answers
205 views

### Tensor products not left exact [closed]

Is there a simple example that shows that the functor $B\otimes_R(-)$ is not left exact, given a ring $R$ and a right $R$-module $B$?
2answers
830 views

### Who coined the name tensor and why?

Who coined the name "tensor" and why? What does the word "tensor" really mean, not the mathematical definition?
0answers
139 views

### Tensor analysis with alpha beta and i j coordinates. [closed]

The covariant differentiation or the levi civita connection represented by the Christoffel symbol, \Gamma_{\alpha,\beta}^{\gamma} \frac{dx^k}{dy^{\gamma}} =?= \Gamma_{i,j} \fra …
0answers
97 views

### Density of divergence in $L^2$ of vector bundles.

Let $V$ be a vector bundle over a smooth, complete Riemannian manifold $M$. In general, the manifold is not compact. Further, denote by $g$ the metric on $M$, the volume measure b …
2answers
414 views

### Tensor algebra question [closed]

1)Why embedding of ( not necessarily finite-dimensional) vector spaces $V\rightarrow W$ produces embedding of tensor algebras $T(V)\rightarrow T(W)$. I can prove it using Hamel ba …
0answers
220 views

### tensor/hypermatrix analogues of $GL(n,\mathbb{C})$?

Please excuse me if this question turns out to be incredibly silly for one reason or another. Are there tensor/hypermatrix analogues of $GL(n,\mathbb{C})$ that are interesting? Wh …

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