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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5
votes
3answers
745 views

Multiplying 3D matrix

I was wondering if it is possible to multiply a 3D matrix (say a cube $n\times n\times n$) to a matrix of dimension $n\times 1$? If yes, then how. Maybe you can suggest some resources which I can read ...
4
votes
1answer
265 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
1
vote
1answer
78 views

Represent in a matrix form: $\sum_{ijl}(F_{il}-F_{jl})^2W_{ij}$

How can I represent this in a matrix form: $\sum_{ijl}(F_{il}-F_{jl})^2W_{ij}$ where all the entries are real and $W$ is a known(constant) matrix and $F$ is a rectangular matrix. When I say matrix ...
1
vote
1answer
130 views

Is the third order term of a Taylor approximation a covariant tensor of rank 3 restricted to (h,h,h)?

Recently reading Peter Lax's linear algebra text, he had a very concise way of writing the Taylor approximation up to second order: $f(x+h) = f(x) + l(h) + \frac{1}{2}q(h) + \|h\|^2 \epsilon(\|h\|)$, ...
3
votes
1answer
465 views

Basis of vector fields on manifold

For a typical real manifold of $n$ dimensions, the basis of a tangent vector space $T$ at point $p$ is $$\frac{\partial}{\partial x_i},\ldots,\frac{\partial}{\partial x_j}$$ So is the general basis of ...
2
votes
1answer
695 views

What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and ...
1
vote
1answer
115 views

Tensors Inner Product

If I had two tensors of rank $3$ and wanted their inner product, what would it be? Also, how could I represent the process with indices and please explain that? Could someone demonstrate this with ...
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. ...
0
votes
1answer
64 views

How do I compute the Laplacian of a function in terms of a given (general) coordinate transformation?

Consider a coordinate transformation $\boldsymbol{x} = \boldsymbol{x}(\boldsymbol{\xi})$ (with Jacobian $\partial \boldsymbol{x}/\partial \boldsymbol{\xi})$, the scalar function $f(\boldsymbol{x}) = ...
0
votes
1answer
33 views

Handling more than three indices/super indices, tensorial calculus

I need to code an equation such as the following one : $$ \frac{\partial u^j}{\partial q^i} = \frac{\partial \mathrm A^j_{pl}}{\partial q^i}\dot q^p \dot q^l + \frac{\partial \mathrm B^{jl}}{\partial ...
1
vote
1answer
34 views

Metric spaces and curvature

Suppose that it is given that the Riemann curvature tensor in a metric space of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the ...
0
votes
1answer
44 views

The operator in Tensor algebra.

Let $V$ be a vector space over a field $K$. We define the $k^{th}$ tensor power of $V$: $$T^kV = V \otimes V \otimes ... \otimes V$$ We contruct $T(V)$ as the direct sum of $T^kV$ for $k=0,1,2,...$ ...
1
vote
1answer
317 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 ...
1
vote
1answer
86 views

3-d Matrices (as in, $A=[a_{ijk}]_{ijk}$) [duplicate]

In programming I've come across the idea of arrays containing arrays containing arrays etc., and as it's pretty intuitive to think of an array of arrays as a matrix, it seems like a reasonable idea to ...
4
votes
1answer
49 views

Is a $1^{\mathrm{st}}$ rank tensor identical to a vector?

As far as I know, we define vectors as elements of a vector space, then there is an isomorphism (by choosing a basis) from the vector space to tuples of components in some field, $\mathbb{F}$ say. ...
1
vote
0answers
69 views

What is the needed background to study tensors?

What is the needed background to study tensors? I do not plan to study it but, I want to know the background! It's a matter of curiosity . Thanks.
1
vote
0answers
70 views

To show that something is a four-vector

I hope this question is not too inane... it would be really helpful for me to have this cleared up. I want to know what I need to show to demonstrate that something is a four-vector. I have checked ...
3
votes
1answer
605 views

“Inverse” of tensor product

I am trying to figure out something. I have a 4-tensor $\phi_{i \, j \, k \, \ell}$ and I know that $\phi = A \otimes B$, being $A$ and $B$ two matrices. With indices, I know this: $\phi_{i \, j \, k ...
2
votes
1answer
73 views

Tensor compact/matrix form.

I have got this tensor $S_{ij} = \frac{1}{2}(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i})$ Anyway I solve it for my problem and get $$ S_{ij} = \left( \begin{array}{ccc} 0 ...
2
votes
1answer
134 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
231 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
418 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
259 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
182 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
49 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
179 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
40 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
40 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
69 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
299 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
125 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
73 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
229 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
151 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
312 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
144 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
156 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
199 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
71 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 ...