# 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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### How to express the covariant derivative on $TM$ by the covariant derivative on $M\times M$?

Let $(M,g,\Gamma)$ be a Riemannian manifold with the Levi-Civita connection. $M\times M$ and $TM$ have natural metric structrures inherited from $(M,g,\Gamma)$. Let $\phi:TM \rightarrow M \times M$ be ...
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### What is the definition of the nuclear norm (aka trace norm, Ky-Fan n-norm) of a tensor?

What is the direct definition for the trace norm of a tensor? By direct I mean without matricization. Edit 1: By tensor, I mean a multi-way array, a generalization of vectors and matrices. (The ...
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### Is there a method of knowing which of the Christoffel symbols of second kind survive and vanish for a given metric?

I'm trying to solve a problem and the given $g_{ij}$ metric is $$\left[\begin{array}{cc}1&0&0\\0&(x^1)^2&0\\0&0&(x^1 \sin x^2)^2\end{array}\right]$$ The non-zero Christoffel ...
1answer
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### Symmetric Rank-1 Decomposition for Density Matrices

Let $(H,\langle\cdot,\cdot\rangle)$ be an $n$-dimensional complex Hilbert space. For concreteness, you can just take $H=\mathbb{C}^n$ with standard inner product. Note that we will use the physicist's ...
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### Integration by parts for covariant tensor fields

Let $(M,g)$ be a compact Riemannian manifold with boundary, and suppose $N$ is the outward unit normal vector field along $\partial M$. I am trying to prove the following integration by parts formula: ...
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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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### Trace in Riemannian geometry

The first time I met the definition of the trace of the Ricci curvature $Ric$ on a Riemannian manifold (M^n,g), it was formulated thanks to local orthonormal coordinates: if $(x_{1}, \cdots, x_{n})$ ...
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### Eigendecomposition and eigenvalue scaling for tensor networks

I've recently been interested in studying tensors and networks of tensors. Consider the following quantity: $$Tr(A^N)$$ Where $A$ is a square matrix and $N$ is large. This is extremely easy - all we ...
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### Examples of geometric structures on real manifold that lead to almost complex structure

I have a $M^{2n}$ real manifold. I am interested, if there are any well known geometric structures on manifold that lead in some natural manner to almost complex structure on $M.$ I read on wiki ...
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### Decomposition of order-$n$ tensors

If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$. The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is ...
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### Elastic wave equation in curvilinear coordinates: how do you perform a coordinate change?

The essence of this question is that I don't know how to convert an equation from Cartesian coordinates into curvilinear coordinates, and would like to know how, preferably using the language of co/...
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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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### Mean Gaussian Curvature using non-unit vectors.

Pg.248 of "Textbook in Tensor Calculus and Differential Geometry" by Prasun Nayak. Let us suppose that $\lambda_{h|}^i$ is not a unit vector and therefore, the mean curvature $M_h$ in this ...
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### Curvature of a circle using tensors

Using the tensor methods in polar coordinates, find the curvature of the circle: $x^1=b$ $x^2=t$ with $t$ a free parameter and $b$ a constant. I know how to obtain the curvature of any parametric ...
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### Exterior power of symmetric product of GL(2,R) tensor representations

A paper I am reading uses an identification of $GL(2,\mathbb{R})$ representations: \begin{align} \Lambda^2(\text{Sym}^3\mathbb{R}^2) = (\text{Sym}^4\mathbb{R}^2 \otimes \Lambda^2\mathbb{R}^2 ) \...
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### Is there a coordinate-free proof of this Lie derivative identity?

Wikipedia mentions (here and here) that the Lie derivative has the following appealing commutator: $$[\mathcal{L}_X,\iota_Y]=\iota_{[X,Y]}$$ The only way I know to demonstrate this identity relies on ...
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### Linear independence of tensor product basis $\{ v_i \otimes w_j\}$ for $\{v_i\}$ and $\{w_j\}$ linearly independent.

Show that the set $\{v_i \otimes w_j\}$ is a linear independent subset of $V\otimes W$ when $\{v_i\}$ and $\{w_j\}$ are independent subsets of V and W respectively. I want to find an error in a proof ...
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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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### An svd-like tensor decomposition splitting a tensor into two lower-dimensional tensors and a singular vector

I have also posted this question on mathoverflow, but it seems there are a lot of questions related to SVDs here and a tag "tensor-decomposition", so I will give it a shot. I am looking for ...
1answer
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### Are $(1, 0)$ tensors always vectors? (resolved)

An $(r, s)$ tensor $T$ is defined to be an element of the tensor product of a vector space and its dual: $$T \in T^r_sV := V^{\otimes r}\otimes V^{* \otimes s}.$$ However, when $V$ is finite ...
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### Naturality/when diagrams commute in Commutative Algebra

As I work my way through commutative algebra (Atiyah and Macdonald), I am often confronted with the following sort of scenario. One has an exact sequence $$0 \to M' \to M \to M'' \to 0$$ of modules ...
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