# Questions tagged [tensors]

For questions about tensor, tensor computation and specific tensors (e.g.curvature tensor, stress tensor). Tensor calculus is a technique that can be regarded as a follow-up on linear algebra. It is a generalisation of classical linear algebra. In classical linear algebra one deals with vectors and matrices.

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### Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
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### What is the difference between tensor calculus and exterior derivative type concepts?

I am trying to clarify terms in order to help me figure out what I'd like to study. I understand that $p$-forms and $p$-vectors are used with things like wedge products, exterior algebras, and a ...
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### 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 ...
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### Transformation law for the Ricci curvature

In 4 dimensions, for a conformal change of metric $g=e^{2u}g_0$ the Ricci curvature tensor $\operatorname{Ric}$ satisfies the transformation law \begin{equation}\tag{1} \operatorname{Ric}_g = \...
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### Intuition of divergence and curl

There is the well known expression for the divergence of a vector field $V$ as the limit of smaller and smaller surfaces of the flux of a surface. However it occurred to me that there is another way ...
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### Constant tensors and covariant derivatives

I seem to have trouble with an elementary computation, and figure it may help others if faced with a similar situation. The basic question is as follows: if I have a tensor field $T$ on some ...
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### Extending Tensor Fields defined on Manifolds to Ambient Space

I am currently reading about tensor fields on manifolds, and I came across two comments that sound contradictory to me. The first comment is made in the book by James Munkres "Analysis on Manifolds", ...
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### Is there a specific mathematical terminology for matrix/tensor with holes(missing elements)?

Consider a matrix-with-hole like below: $$\begin{bmatrix} None & x \\ y & 0 \end{bmatrix}$$ We can define two variants of means for the above object: one is to calculate ...
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### Find $T\left(\frac{\partial}{\partial x^k}, dx^l\right)$ for a tensor of type ${1}\choose{1}$ on $T_p(\mathbb{R}^n)$

So say I have the tensor of type ${1}\choose{1}$, with $T \in T_p(\mathbb{R}^n) \otimes T_p^*(\mathbb{R}^n)$ where $$T = T^{a}_{b} \frac{\partial}{\partial x^a} \otimes dx^b$$ summed over all $a, b$. ...
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### Historical motivation for the tensor definition

I would like to understand more about the tensors product/tensors and apart from the definition, I think it would be useful to understand first historical use or motivation of a tensors. For example ...
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### Apparent (minor) error in Cauchy's article on pressure or tension in a solid body

In his article De la pression ou tension dans un corps solide [On the pressure or tension in a solid body], Cauchy introduces a theory that allows to define Cauchy stress tensor. It looks like he ...
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### Need help calculating entropy using huge tensors

I am no expert in tensor algebra. I am stuck with computing the following tensor product (you will recognize an entropy-like equation): $H(\mathbf{T}) = \mathbf{T}^t \cdot \log_2(\mathbf{T})$ where \$...