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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2
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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 ...
2
votes
1answer
223 views

Contraction of the second Bianchi identity

The second Bianchi identity is $${R^a}_{b[cd;e]}=0$$ And contracting it with respect to $a$ and $e$ we get $${R^a}_{b[cd;a]}=0 \Leftrightarrow $$ $${R^a}_{bcd;a}+R_{bc;d}-R_{bd;c}=0$$ What I don't ...
2
votes
1answer
385 views

Quotient theorem for tensors

Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
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{ ...
1
vote
0answers
179 views

Riemannian curvature and its application on covariant derivative of tensors

This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows $ \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ...
2
votes
1answer
47 views

Regarding the definition of covariant derivative and its use on basis vector fields

we find that for general vector fields ${\mathbf v}= v^ie_i$ and ${\mathbf u}= u^je_j$ we get :$\nabla_{\mathbf v} {\mathbf u} = \nabla_{v^i {\mathbf e}_i} u^j {\mathbf e}_j = v^i \nabla_{{\mathbf ...
1
vote
1answer
176 views

Gradient with respect to a matrix variable

I want to find the gradient of the function $\mathcal{F}_1$ with respect to the matrix $\mathbf{X}$ (differentiate with respect to $\mathbf{X}$): $$ \mathcal{F}_1 (\mathbf{X}; \mathbf{\lambda})= ...
1
vote
1answer
38 views

Regarding confusion of basis tensors and the usage of tensors.

Let us for example give a tensor example of following: $X = X^i \partial_i$. According to mny knowledge, in this case $\partial_i$, basis, is treated as tensor (otherwise, $X$ as tensor won't be ...
1
vote
1answer
38 views

Finding an “inverse” of a deviatoric tangent

I have have a material model, defining the deviatoric stress for a nonlinear fluid: $\boldsymbol{\sigma}_{\mathrm{dev}} = f(\dot{\boldsymbol{\varepsilon}}_{\mathrm{dev}})$ Now I wish to find the ...
1
vote
2answers
1k views

Proving the symmetry of the Ricci tensor?

Consider the Ricci tensor : $R_{\mu\nu}=\partial_{\rho}\Gamma_{\nu\mu}^{\rho} -\partial_{\nu}\Gamma_{\rho\mu}^{\rho} +\Gamma_{\rho\lambda}^{\rho}\Gamma_{\nu\mu}^{\lambda} ...
2
votes
1answer
42 views

If $T$ is a $k$-tensor and $S$ is an $l$-tensor, then $\text{Alt}(T \otimes S) = (-1)^{kl} \text{Alt}(S \otimes T)$

Could someone please help me with the following algebra question? I know it should be easy, but the textbook leaves the proof to the reader and I am having a hard time with it. Thank you in advance. ...
1
vote
0answers
66 views

What is this tensor called?

Is there a standard name for this "generalized identity" tensor $x^{i j k ...} = I(i = j = k = ...)$ where $I$ is the indicator function, ie the tensor is 1 when all indexes are equal & zero ...
0
votes
0answers
67 views

Tensor Products, various defintions

I came across a definition for the tensor product which differs from the standard definition. This book defined the tensor product of vector spaces $V$ and $W$ as the space $L(V,W,\Bbb K)$ of bilinear ...
0
votes
1answer
279 views

Dot product between two vectors or vector and 1-form?

When we take a dot product between two vectors of a vector space, we actually "act" by a 1-form (dual vector) on a vector. So why most books define the dot product between vectors? Of course with the ...
3
votes
2answers
122 views

Tensor notation and rules

I have a few questions about tensors: I appreciate that $g^{\alpha\beta}=g^{\beta\alpha}$ but when contracting say $T^{\sigma}_{\mbox{ }\;\mu\nu\rho}$ to $T_{\;\;\mu\nu}$, first of all can it be ...
0
votes
0answers
72 views

Differentiation of a vector (in index notation) with respect to its square

I have a formula which contains the derivative $\partial_{z^2}$(with respect to the square of a vector $z^\mu$). However, as the functions inside the formula might not directly depend on $z^2$ but on ...
0
votes
3answers
221 views

How to differentiate a differential form?

Please explain me the idea of differentiating differential forms (tensors). Example: compute d(xdy + ydx) The answer is known, we should have 0. What's the rule?
6
votes
1answer
148 views

Tensor product of algebra

Can we find define a norm on tensor product $C(X) \otimes C(Y)$ such that the norm completion of $C(X)\otimes C(Y)=C(X\times Y)$ And can we define a norm on tensor product $L^1(X)\otimes L^1(Y)$ such ...
1
vote
0answers
41 views

Tensors and 4-vectors

This may be a very trivial question, but help would be appreciated. It has to do with P. 55 in these notes I don't understand why the fact that $$U_a\dot U^a=0$$ applied to $$U_a\nabla_b T^{ab}=\rho ...
0
votes
1answer
2k views

How can I write a tensor in MATLAB?

My project is about tensors and I must write the program in MATLAB. How can I write a tensor in MATLAB? Is there anybody to help me? Can you explain me what is the code for import a tensor? thanks for ...
1
vote
0answers
310 views

How to derive covariant derivative and Lie derivative of tensors

1) As title says, how does one derive the following equation for covariant derivate of tensor: $A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} + \Gamma^{\alpha} {}_{\gamma\beta}A^\gamma$ where ...
1
vote
0answers
141 views

Multi-dimensional array decomposition

My question is about decomposing a muti-dimensional array into a product of matricies. To ask the question I will work towards the tensor, and then ask the question about the reverse process. Let ...
2
votes
1answer
152 views

Tensors in math and physics

I know how tensor product f two modules is defined in communtative algebra. But there is also a concept of tensors used in physics. Are these two concepts related? If yes, can someone explain me ...
1
vote
1answer
78 views

Tensor and conservation

I am reading a book "The Early Universe " by Kolb and Turner. On P.48, it says For $T^\mu\,_\nu=\operatorname{diag} (\rho, -p,-p,-p)$, the $\mu=0$ component of the conservation of stress energy ...
4
votes
1answer
184 views

What is the last index of a third-order tensor called?

In a third-order tensor I guess the first and second index would be called row and column respectively but is there a name for the third index?
3
votes
3answers
70 views

Definition of a tensor field

Could anybody explain to me the following: If $$T_{ij}=\nabla_i V_j-\nabla_j V_i$$ where $V_i$ is a covector field and $\nabla_i$ is the covariant derivative, then $T_{ij}$ is a tensor field. ...
0
votes
1answer
35 views

Does double antisymmetrisation always introduce a factor 2?

In theoretical electrodynamics, I came across terms with double antisymmetrisation, one with brackets, the other with a Levi-Civita-Tensor ($\epsilon$). The particular example was ...
4
votes
0answers
155 views

Use of graph theory to determine tensor contraction ordering

I am considering using a computer program to execute tensor contractions like the following: $\displaystyle \sum_{ij}^{o} \sum_{ab}^{v} \sum_{KL}^{X} B_{ia}^{K} B_{ia}^{L} B_{jb}^{K} B_{jb}^{L} $ ...
1
vote
2answers
101 views

Action of $\mathfrak{sl}({V})$ in tensor spaces

what is the natural action of $\mathfrak{sl}({V})$ in tensor spaces ?
3
votes
0answers
44 views

Equivalent definitions of tensors on finite dimensional spaces.

I have recently been studying various texts on differential geometry, and I am quite puzzled that various authors define the notion of a tensor quite differently. I have come across the following ...
2
votes
2answers
37 views

$4$-vectors and indices

I am reading through some material about $4$-vectors. And came across the following for which an explanation woud be greatly appreciated. The index for $\partial_\alpha$ can be raised giving ...
2
votes
2answers
64 views

Splitting a tensor

Is it possible to write $$\int d^3x \,\,\, x_i\,\,x_j\,\,\,f(\vec x)$$ where $f(\vec x)$ is some function of the position and the indices indicate which component, as a sum of a traceless tensor and ...
4
votes
2answers
193 views

Confusion when dealing with tensors.

I don't understand how tensors work, can someone please explain? In particular, in the context of Electromagnetism, the dual of the field tensor $F$ is $$(*F)^{\mu\nu}:={1\over ...
4
votes
2answers
150 views

Tensor Components

I would like to ask something On the Barrett Oneill's Semi-Riemann Geometry there is a definition of tensor component: Let $\xi=(x^1,\dots ,x^n)$ be a coordinate system on $\upsilon\subset M$. If $A ...
3
votes
1answer
172 views

Difference between tensor and tensor field?

I couldn't get the difference between tensor and tensor field. I'm learning from Barret O'neill's Semi-Riemann Geometry and here are the definitions: if $A:(V^*)^r \times V^s\to K$ transformation is ...
5
votes
1answer
209 views

A tensor calculus problem

If the relation $a_{ij}u^iu^j=0$ holds for all vectors $u^i$ such that $u^i\lambda_i=0$ where $\lambda_i$ is a given covariant vector, show that $$a_{ij}+a_{ji}=\lambda_iv_j+\lambda_j v_i$$ where ...
8
votes
2answers
609 views

Rotation invariant tensors

It is often claimed that the only tensors invariant under the orthogonal transformations (rotations) are the Kronecker delta $\delta_{ij}$, the Levi-Civita epsilon $\epsilon_{ijk}$ and various ...
4
votes
1answer
97 views

Connection between dual space V* and negation P^c

Notice the following similarity between the vector space dual and negation in propositional logic: $$ V^* \equiv V \rightarrow F $$ $$ P^c \equiv P \rightarrow \bot $$ Is there some general notion ...
0
votes
1answer
113 views

How does Kronecker $\delta_{ijkl}$ determined?

Let's have symbol $\delta_{ijkl}$. Is it equal to 1 for $i = j = k = l$ and $0$ in the other cases?
3
votes
1answer
498 views

What does cubic symmetry mean?

Is it an invariance under rotations around x-, y-, z-axis? Does this invariance separately include rotations around an arbitrary $(x$, $y$, $z)$ axis?
1
vote
1answer
73 views

Meaning of tensor expression

If $\epsilon$ is an alternating unit tensor and $\mu$ is any arbitrary tensor, then what does the expression $\epsilon : \mu = 0$ mean ? I came across this is some textbook I was reading. Is is ...
5
votes
3answers
561 views

Definition of a tensor for a manifold

While reading Nakahara's geometry, topology and physics. I came across the following definition of a tensor. A tensor $T$ of type $(p, q)$ is a multilinear map that maps $p$ dual vectors and $q$ ...
1
vote
1answer
220 views

Showing symmetry of the stress tensor by applying divergence theorem to $\int\int_{\delta V(t)} \vec{x}\times \vec{t} dS$

I'm currently working through the symmetry of the stress tensor, in relation to viscous flow. I am looking at this by examining the conservation of angular momentum equation for a material volume ...
2
votes
1answer
128 views

Question about stiffness tensor

Let's have a stiffness tensor $$ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $$ It has a 21 independent components for anisotropic body. How does body symmetry (cubic, hexagonal etc.) ...
3
votes
1answer
1k views

Christoffel symbol transformation law

It is known that the transformation rule when you change coordinate frames of the Christoffel symbol is: $$ \tilde \Gamma^{\mu}_{\nu\kappa} = {\partial \tilde x^\mu \over \partial x^\alpha} \left [ ...
1
vote
1answer
100 views

A basis for k-tensors

Let $V$ be a vector space of dimension $n$ with basis $e_1, \dots, e_n$. Let $a_1, \dots, a_n$ be the dual basis for $V^\ast$. Show that a basis for the space $L_k(V)$ of $k$-linear functions on $V$ ...
15
votes
3answers
3k views

What is a covector and what is it used for?

From what I understand, a covector is an object that takes a vector and returns a number. So given a vector $v \in V$ and a covector $\phi \in V^*$, you can act on $v$ with $\phi$ to get a real number ...
4
votes
0answers
38 views

Is $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in $R^n$?

Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors. Then define $H^{\alpha}_{ij,k}=e_k\langle ...
2
votes
2answers
379 views

Tensor invariants under isometries

UPD Equivallent formulation --- how do you find all the independent isometric invariants of a tensor? In what follows $V$ is a real inner product space. I want to understand how does one find all ...
6
votes
0answers
142 views

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 ...