1
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0answers
20 views

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 ...
0
votes
1answer
63 views

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 ...
1
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0answers
19 views

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 ...
1
vote
0answers
29 views

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 ...
2
votes
1answer
101 views

Inverse of a matrix

I am looking for a way to derive that the inverse of a matrix using Levi-Civita. I know that the final result looks like this for a $3 \times 3$ matrix: $$(A^{-1})_{ij} = \frac{1}{2!}\frac{1}{\det A} ...
1
vote
0answers
42 views

Derivative of a tensor

I have a rank-2 tensor given by $$ P_{\alpha \beta} = p\delta_{\alpha \beta} + (u_1^2, u_1u_2 ; u_1u_2, u_2^2) $$ whose derivative with respect to $x_{\alpha}$ I would like to find. According to my ...
3
votes
1answer
61 views

Help with notation in second order tensor.

I have seen recently this notation: $$F=F_{ij}\,e_i\otimes e_j $$ Where $F_{ij}=\frac{\partial x_i}{\partial X_j}$ is the tensor matrix: $$ F=\left[ {\begin{array}{cc} ...
4
votes
1answer
230 views

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 ...
2
votes
3answers
56 views

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. ...
1
vote
1answer
300 views

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 ...
0
votes
0answers
55 views

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 $(A^{a}+B^{a})(A_{a}B_{a})$
2
votes
1answer
129 views

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 ...
3
votes
1answer
255 views

Parallel Transport along a curve

We had this homework assignment for our geometry course, and we couldn't figure it out, any ideas on how to do this: Consider the Poincare model of Lobachevsky plane, $H^2=\left\lbrace{ ...
0
votes
0answers
126 views

Derivative of a vector with respect to a matrix

I am at an impasse. I don't know if homework is allowed on here or not, so if it isn't, someone delete this. Given: $H_{\gamma} = C_{\beta \beta} v_{\gamma} + C_{\beta \varepsilon} C_{\varepsilon ...
1
vote
1answer
90 views

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 ...
1
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2answers
948 views

Prove the determinant of a tensor is invariant

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 ...