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
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2answers
57 views

Exact meaning of “Not every matrix is a tensor”.

I've recently begun reading about tensors and am trying to understand the second order variety in the context of euclidean $\mathbb{R}^n$ with orthonormal basis {$e_1, e_2,\ldots, e_n$}. This seems ...
0
votes
2answers
38 views

Tensor Product over a ring

Given Two Fields $F,K$, and two vector spaces $V,W$ over $F$, what does tensor product $$V\otimes_{K} W$$ mean? I am not certain whether this is defined in general. I came across it in cases wheh ...
2
votes
1answer
41 views

The relation between principal curvature and curvature tensor?

To me, there are two systems of curvature of a surface, one is consist of 'principal curvature, mean curvature, Guass curvature, normal curvature' while the other is consist of 'curvature tensor'. I ...
0
votes
1answer
38 views

Another matrix of a given operator $A \otimes A$

Let $V$ be a $4$-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $(e_{1}, ...
1
vote
1answer
18 views

When is a symmetric 2-tensor field globally diagonalizable?

Suppose that $\mathbb{R}^n$ has a Riemannian metric $g$. Let $h$ be a smooth symmetric 2-tensor field on $\mathbb{R}^n$. At any point $p \in \mathbb{R}^n$, there is a basis of $T_p \mathbb{R}^n$ in ...
0
votes
1answer
14 views

Symmetrizing and Anti-Symmetrizing Tensors

Given any Tensor, we can obtain a symmetric tensor through symmetrising operator. by $T_{uv} \rightarrow T_{(uv)}=\frac{1}{n!}(T_{uv}+T_{vu})$ where $n$ is the order of the tensor, and you have to ...
1
vote
0answers
46 views

What exactly do the terms of $(f \circ g)'''$ mean?

Say, $g: X\to Y$ and $f: Y\to Z$ are smooth. One can find $(f \circ g)'''$ by using the Faà di Bruno's formula: $$(f \circ g)''' =(f'''\circ g)(g')^3 + 3(f''\circ g)g'g'' + (f'\circ g)g'''$$ But my ...
1
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0answers
18 views

Hessian matrix of $g\circ f$

Say, $f:\mathbb R^n\to\mathbb R^k$ and $g:\mathbb R^k\to\mathbb R$ are both $C^2$. I'd like to express the Hessian matrix of $g\circ f$ $$\left( \frac{\partial^2(g\circ f)}{\partial x_i \partial ...
1
vote
1answer
27 views

Matrix of a given operator $A \otimes A$

Let $V$ be a 3-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $e_{1}$, $e_{2}$, $e_{3}$ ...
0
votes
0answers
18 views

Divergence of a Tensor Examples?

Would anyone have a reference to a book where explicit examples of things like taking the divergence of a tensor (page 4) are given? Stupid computational examples like the one given here (bottom of ...
0
votes
0answers
10 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 ...
3
votes
2answers
166 views

Green's first identity

Good morning/evening to everybody. I'm interested in proving this proposition from the Green's first identity, which reads that, for any sufficiently differentiable vector field $\mathbf{\Gamma}$ and ...
1
vote
0answers
15 views

Symbolic cancellation in tensor notation of derivative

Start with this: $\frac{\partial f}{\partial x'^i} = \frac{\partial f}{\partial x^j} \frac{\partial x^j}{\partial x'^i}$ I think(?) the $\partial x^j$s cancel and this simplifies to ...
1
vote
0answers
23 views

How is the multiplication between a multidimensional tensor with a matrix defined?

I am thinking this calculation in the following way but I am wondering if it is correct. Can anybody explain to me please? For example, I have a 3-way tensor $T^{u×i×t}$. How do I multiply this ...
0
votes
1answer
58 views

Curl, $\vec\nabla \times (\hat{a}\times \vec{b})$

EDIT: FIXED TYPOS & Deleted most of my wrong work pointed out by others. Calculate the curl of $f(\vec r,t)$ where the function is given by $$ f(\vec r,t)=- (\hat{a}\times \vec{b}) \frac{e^{i(c ...
3
votes
1answer
85 views

The Curvature Tensor

I present three different ways I've seen the Riemann curvature written: $R(X,Y)Z=D_XD_YZ-D_YD_XZ-D_{[X,Y]}Z$ $R(e_c,e_d)e_b=D_{e_c}D_{e_d}e_b-D_{e_d}D_{e_c}e_b-D_{[e_c,e_d]}e_b$. $R^{\rho}_{\space ...
1
vote
0answers
23 views

Is there a name for this type of tensor rank?

Let $A\in\mathbb{R}^{n_1\times n_2\times n_3 \times n_4}$ be a tensor. Suppose that $k$ is the minimum integer there exist matrices $X_1,\ldots,X_j\in\mathbb{R}^{i_1\times i_2}$ and ...
1
vote
1answer
24 views

Proving that the Moment Tensor is super-symmetric

The Carathéodory theorem in the image bellow is the one about convex hull, isn't it? Would you please explain why can the tensor F be rewritten as that sum? From that representation the author ...
1
vote
1answer
32 views

Trace of tensor product vs Tensor contraction

I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality: $$ \text{Tr}(T_1\otimes ...
0
votes
0answers
18 views

Tensor notation and the “Zero-Value Theorem”

In the following picture: taken from Martin Sadd's book on elasticity, I am having trouble understanding the "zero-value theorem". I can't understand why this theorem is true. For example, when ...
6
votes
1answer
55 views

Tensor fields and vector bundles

Let $M$ be a differentiable manifold, $TM$ and $T^*M$ a tangent and cotangent bundle of $M$ and let $\Gamma (TM),\ \Gamma (T^*M)$ be spaces of smooth sections of $TM$ and $T^*M$. Let $T_s^r (M)$ ...
0
votes
0answers
40 views

tensor notation surprise

I'm trying to study tensors from several textbooks. One early example completely confuses me: Islam, Tensors and their Applications, in the "Preliminaries" chapter, gives this example (page 3, using ...
2
votes
0answers
64 views

Covariant vectors and Dual spaces

There are contravariant vectors and their duals called covariant vectors. But the duals are defined only once the contravectors are set up because they are the maps from the space of contravariant ...
0
votes
1answer
18 views

Is tensor product commutative on orthonormal basis?

In general the tensor product $\varphi\otimes\psi$ is not commutative, but I was thinking that if I have tensor product on two orthonormal bases of Hilbert spaces are they commutative i.e is ...
2
votes
2answers
39 views

Cartesian & Tensor Product

What is the difference between a cartesian product and tensor product of two vector spaces $V_1$ and $V_2$ defined over same field $F$ ?
0
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0answers
17 views

how to make geometric figure with tensor, are tensors covariant or contravariant

Heloł dear Colegues! Im curious how to make some geometric figure ie. cube, simple hull of ship like half of fish or half of cylinder, with tensor, or in more mathematical language: how to make ...
2
votes
0answers
69 views

Prove the Curvature Tensor is a Tensor

For an affine connection $\nabla$, prove the curvature R $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$ with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is ...
0
votes
1answer
22 views

derivation of rank of tensor from the product of two tensors

If $A^p$ is a first rank tensor and $A^pK^{qrs}$ is a 4th rank tensor we have to prove that $K^{qrs}$ is a tensor of rank three?.we can check here clearly that $A^p$ is first rank tensor and $K^{qrs}$ ...
2
votes
1answer
56 views

using the kronecker product and vec operators to write the following least squares problem in standard matrix form

I have a least squares problem with the following form: $$ \min_\mathbf{X} ~ \left\| \sum_{i=1}^n \mathbf{u}_i^\top \mathbf{X} \mathbf{v}_i - b_i \right\|^2 $$ where $\{\mathbf{u}_i\}_{i=1}^n$ and ...
1
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0answers
22 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 ...
0
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0answers
8 views

degrees of freedom (df) of a third order tensor

Does it make sense to simply unfold the tensor into a matrix and apply the df metric used for matrices? That is, a $\ n_1\times n_2 $ matrix of rank $\ r$ has df =$\ n_1r + (n_2 - r)r$ and so, a ...
0
votes
0answers
23 views

Rotation of axes transformation as definition of vectors

Given a three-axes coordinate system ${1,2,3} $ by the right-hand rule, and a new coordinate system ${1',2',3'}$ , I know that one can define a vector $\vec{x}$ to be something that obeys the ...
1
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1answer
38 views

Tensors and Transformations

In Griffiths E&M book, he says that a second rank tensor transforms with two factors of some transformational tensor on each of its nine components-I'm not sure why that is. I thought a second ...
0
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0answers
34 views

The trace of a wedge product of matrices

I'm trying understand a computation on Besse's book (p. 371). I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form relative to the direct sum ...
0
votes
1answer
41 views

Is it true to say that every tensor is an element of a monoid?

If we consider that: by definition, a tensor is an element of the tensor product of two algebraic structures the most abstract algrebraic structure on which the tensor product is defined are ...
0
votes
2answers
37 views

Is every tensor an element of a vector space?

As, the tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$, is it true to say that every tensor is an element of a vector space ? (if we do not consider ...
1
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0answers
18 views

Number of isotropic tensors of rank N

Could someone please help me out with this problem? I would like to know what the number of distinct isotropic tensors of rank N is. I am new to the field and got confused by apparently contradictory ...
0
votes
0answers
17 views

Alternative operator is a homomorphism?

Let $V$ be a real vector space of dimension $n$, for a (real valued) tensor $f$ of order $r$, define the alternative operator $A$ by $$(Af)(v_1,\cdots, v_r)=\frac{1}{r!}\sum_{\sigma\in ...
1
vote
1answer
53 views

Partial derivative with respect to $ \left( \frac{dx^m}{ds} \right) $

I don't understand how the following $$ 2g_{ml} \frac{dx^l}{ds} $$ partial derivative was obtained below. It is supposedly the partial derivative of the value between the parenthesis. $$L = ...
2
votes
1answer
79 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} ...
0
votes
1answer
29 views

Why must metric tensor be invertible?

The metric can be written as a matrix, but why must this matrix be invertible? At the points where the matrix is singular, why is the metric not defined?
3
votes
1answer
51 views

Transforming a matrix from cartesian to spherical coordinates

Consider a variable matrix $$\left[\begin{array}{ccc}a_{11}(x,y,z) \quad a_{12}(x,y,z) \quad a_{13}(x,y,z)\\ a_{21}(x,y,z) \quad a_{22}(x,y,z) \quad a_{23}(x,y,z)\\ ...
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0answers
45 views

Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...
2
votes
1answer
37 views

Calculating a certain tensor

Consider the $\mathbb{Z}$ module $\mathbb{Z}/n\mathbb{Z}$. What is $\mathbb{Z}/n\mathbb{Z} \otimes_{\mathbb Z} \mathbb{Z}/n\mathbb{Z}$?
1
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1answer
59 views

Tensor Einstein summation notation

I have two tensors $A^i$ and $B_j$ with components $(2,3,4)$ and $(1,2,3)$ respectively. What is the difference between $A^i B_i$ and $A^i B_j$? Is it just: $A^i B_i = 2+6+12 = 20$ $A^i B_j =$ $ ...
4
votes
1answer
48 views

Symmetry of the Riemannian curvature tensor

The Riemannian curvature tensor, in local coordinates, $R_{ijkl}$, has the following symmetries: $$R_{ijkl}+R_{jikl}=0;$$ $$R_{ijkl}+R_{ijlk}=0;$$ $$R_{ijkl}=R_{klij};$$ ...
14
votes
5answers
259 views

What, Exactly, Is a Tensor?

I've repeatedly read things that reference tensors, and despite reading the wiki page and other answers here on stackexchange I still don't know what a tensor is. I'm fine with hearing things in the ...
0
votes
1answer
25 views

Levi-Civita symbol identity: $\epsilon_{ijs}x_s L_i = -\epsilon_{ijs}x_i L_s$

I have the following identity I want to use but don't know whether it's correct or not (and if it is, why so): $$\epsilon_{ijs}x_s L_i = -\epsilon_{ijs}x_i L_s$$ Is this correct? How to arrive at ...
1
vote
0answers
25 views

Prove $\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A$

Is there any simple way to prove the following: $$\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A. \tag{$1$} $$
3
votes
2answers
75 views

What is the value of the 2-form $X(u(Y))-Y(u(X))-u([X,Y])$, is it $2du(X,Y)$?

Let $X,Y$ be two vector fields on a Riemannian manifold $(M,g,\nabla)$ where $\nabla $ is the symmetric metric connection on $M$ and let $u$ be a 1-form. I want to find the value or a simple form or ...