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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2answers
45 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
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1answer
56 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
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1answer
50 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
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1answer
21 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 ...
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1answer
29 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 ...
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0answers
51 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 ...
2
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0answers
27 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
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1answer
36 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
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0answers
52 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 ...
1
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0answers
20 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
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2answers
225 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
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0answers
30 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
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0answers
35 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
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2answers
111 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
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1answer
105 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
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0answers
25 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
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1answer
31 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
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1answer
417 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
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0answers
21 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
86 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 $T_s^r (M)$ ...
1
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1answer
76 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
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0answers
73 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
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1answer
25 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
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2answers
48 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 $F$ ?
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0answers
29 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
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0answers
96 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 ...
1
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1answer
32 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
111 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
30 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 ...
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0answers
18 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
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0answers
48 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 ...
2
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1answer
49 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 ...
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0answers
56 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 ...
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1answer
55 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
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2answers
52 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 ...
5
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0answers
83 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
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0answers
19 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
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1answer
58 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
123 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
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1answer
46 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
109 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
51 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
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1answer
39 views

Calculating a certain tensor

Consider the $\mathbb{Z}$ module $\mathbb{Z}/n\mathbb{Z}$. What is $\mathbb{Z}/n\mathbb{Z} \otimes_{\mathbb Z} \mathbb{Z}/n\mathbb{Z}$?
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1answer
84 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
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1answer
64 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
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5answers
394 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
28 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
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0answers
26 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
79 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 ...
1
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0answers
49 views

Question about tensor product of homomorphisms

I've come to think about this problem when reading a proof in Commutative Algebra by N. Bourbaki. Say, let $R$ be a commutative ring, given 3 $R-$modules $A$, $B$, $C$, and the $R$-homomorphism $f:B ...