# Tagged Questions

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### Tensor Algebra and Isomorphism

Consider the standard tensor algebra over a vector space: $$T(V)=\bigoplus_{n\in N}V^{\otimes n}$$, I am trying to define a linear map from $T(V)\times T(V)\rightarrow T(V)$. Up to this time, I have a ...
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### Confusing question on tensors

A tensor of rank $4$ satisfies $T_{ijkl}=T_{jilk}=-T_{jikl}$ and $T_{ijij}=0$. I need to show that: $$T_{ijkl}=-\varepsilon_{ijp}\varepsilon_{klq}T_{rqrp}$$ Could someone offer a hint? I have tried ...
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### Eigenvalues of a rank 2 tensor defined by an integral

I've been given the question: "Consider the tensor: $$C_{ij}=\int_{V}{x_ix_j|\mathbf {x}|^2 + x_ix_j(\mathbf {x.n})^2} dV$$ where V is the volume of a sphere radius R centred on the origin. What ...
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### Isomorphism between $T^k_{l+1}(V)$ and $\mathcal{L}\left( (V^*)^k \times (V)^l\right)$.

V is a real $n$-dimensional vector space. I'm using the notation as in John Lee's Riemannian Manifolds: an Introduction to Curvature for the set of $\binom{k}{l}$ tensors and the set of multilinear ...
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### The divergence of the Weyl tensor

First, the Weyl tensor is given by $$W_{ijkl}=R_{ijkl}-\frac{1}{n-2}(g_{ik}A_{jl}-g_{il}A_{jk}-g_{jk}A_{il}+g_{jl}A_{ik})$$ where, $A_{ij}$ is the Schouten tensor, given by ...
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### notation question (bilinear form)

So I have to proof the following: for a given Isomorphism $\phi : V\rightarrow V^*$ where $V^*$ is the dual space of $V$ show that $s_{\phi}(v,w)=\phi(v)(w)$ defines a non degenerate bilinear form. ...
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### How to solve a tensor differential equation?

Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$ where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor. The original Problem How does ...
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### What is mathematics (in physics) of this tensor equation?

I want to know what is $C^{a}C_{a}$ if $C^{a}=A^{a}+B^{a}$ and $C_{a}=A_{a}+B_{a}$? a. this one $A^{a}A_{a}+B^{a}B_{a}$ or b. this one $(A^{a}+B^{a})(A_{a}B_{a})$
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### Prove that a tensor field is of type (1,2)

Let $J\in\operatorname{end}(TM)=\Gamma(TM\otimes T^*M)$ with $J^2=-\operatorname{id}$ and for $X,Y\in TM$, let $$N(X,Y):=[JX,JY]-J\big([JX,Y]-[X,JY]\big)-[X,Y].$$ Prove that $N$ is a tensor field of ...
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### Matrix representing $\Lambda^k$(A)
Let V , W be finite dimensional vector spaces over R. Let A : V->W be a linear map. Choose bases of V and W and the corresponding bases of $\Lambda^k$(V ) and of $\Lambda^k$(W). How to show that the ...
Given is a second-order tensor $T$, and three arbitrary vectors, $u$, $v$ and $w$, defined in Euclidean point space $\mathcal{E}$. Prove that the determinant of the tensor $T$ \$\det T=\frac{Tu.(Tv ...