# Tagged Questions

Use this tag for questions about specific tensors (curvature tensor, stress tensor), or questions regarding tensor computations as they appear in multivariable calculus and differential/Riemannian geometry (specifically, when it is amenable to be treated as objects with multiple indices that ...

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### Tensors, indices and matrix notation - is there a common convention?

For a tensor named T with two indices, there are four possibilities: $T_{ij}$ , $T_i^{\ j}$, $T^i{\ _j}$ and $T^{ij}$. Is there a common convention as to how these tensors would be represented as ...
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### Christoffel Symbols in Flat Space-Time

Does it make sense to talk about Christoffel symbols in flat space time? Do they have non-zero values? I understand that the Christoffel symbols appear as an indication of curvature in space. So, are ...
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### Border rank of tensors

Can anyone help me find the rank and border rank of the following tensor: \begin{align} T=a_{11}\otimes b_{11}\otimes c_{11}+a_{12}\otimes b_{21}\otimes c_{11}+a_{11}\otimes b_{12}\otimes c_{12}\\ ...
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### Is every tensor the gradient of a vector?

I guess no, since every vector is not the gradient of a scalar - I assume ! Please confirm or guide in this regard .
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### Strassen's Laser Method Technique AND Tensors in matrix multiplication algorithms

I understand the first algorithm presented by Strassen in 1968, for fast matrix multiplication. This was the first improvement to the naive approach of multiplying matrices. Thereafter, he went on to ...
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### Ricci Contraction

Is Ricci Contraction different from ordinary Contraction or are they the same? I understand that in ordinary tensor contraction, you contract( equate two indices) one index of the tensor with respect ...
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### Connection between covariant and contravariant components o tensor

What is the general proof of the relation between covariant and contravariant components of a tensor using the metric tensor? $${g^{mr}g_{rn}=\delta^{m}_{n}}$$
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Question is to prove that $Alt(T)=0$ if $T$ is a symmetric tensor. We have $$Alt(T)=\sum_{\sigma}sgn(\sigma)T^{\sigma}$$ As $T$ is symmetric we have $T^{\sigma}=T$ for all $\sigma$. So, we have $$Alt(... 0answers 47 views ### Transformation laws for tensors on general manifolds I was interested in tensor products rather from the mathematical, abstract point of view, so topics like various tensor products in the context of, for example, Banach spaces, C^*-algebras and so on.... 0answers 54 views ### derivative of a tensor A with respect to transpose(A)*A? What is the derivative of \partial A/\partial ({A^T}A) ? Where A is a 3x3 tensor. (in index notation, I want to find explicit components of {D_{ijpq}} = \partial {A_{ij}}/\partial ({A_{kp}}{A_{... 1answer 74 views ### d \times d \times d tensor rank vs d \times d tensor rank I am trying to understand rank of a d \times d \times d tensor. The way that I understand the d \times d case is that a rank r, d \times d tensor is a tensor that can be written as the sum of ... 0answers 45 views ### Example of two modules M, N where the set of the m\otimes n is not a submodule If I remember my Algebra class correctly, this is in general not true because, for example, the addition isn't closed. So if I take: B:\mathbb{R}^2 \times \mathbb{R}^2\to \mathbb{R}^{2^2}\cong \... 1answer 45 views ### Solution to tensor/matrix equation I need to find a real, symmetric matrix, A, that satisfies: \sum_{i,j} c_i c_j A_{ki}A_{jl} = A_{lk} I believe this is an equation of the form: c^T B c = A, where c is \mathbb{R}^{N \times ... 0answers 34 views ### Does \mathfrak T^r(\Bbb R^m) count as an vector space? Here \mathfrak T^r (\Bbb R^m) denotes all the r-th tensors (multi-linear functions) acting upon the elements (u_1,\cdots,u_r) from the product space \displaystyle \prod^r \Bbb R^m. And the ... 1answer 59 views ### Understanding Symmetric tensor field I am reading an article in which author calls some basic tensor analysis result. He states in general we define on \mathbb R^N that$$ \mathcal T^k(\mathbb R^N):=\{\xi:\,\mathbb R^N\times\cdots\...
How to write a tensor in terms of a product of matrices? For example, I have $a \times b$ matrix $F$, and I want to create a 3D $a \times a \times a$ tensor $T$, where \$T_{i,j,k} = \sum_{m=1}^{b} F_{...