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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1answer
103 views

Tensors and General Relativity

I've recently begun self studying general relativity, using mostly the material found in Robert Wald's "General Relativity", and almost right out of the gate one encounters the notion of a tensor. In ...
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1answer
50 views

What is a high rank tensor?

Can someone please give me a good example of a rank 3, 2x2x2 or 3x3x3 tensor? Where are these forms arise from? Is a 4x3x3 tensor say, a pressure tensor on a 3D space in 4-dimensions? Are there any ...
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0answers
25 views

Misleading tensor notation for Jacobian inverse?

In Schutz, Geometrical Methods of Mathematical Physics, is written a Jacobian coordinate transform $\Lambda$, $$ \Lambda^i_j = \frac{\partial x^i}{\partial y^j} $$ The inverse matrix is written $$ ...
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1answer
23 views

jacobian times its inverse - should be identity

Here's an easy one. A Jacobian is $\frac{dx^i}{dy^j}$. The inverse is $\frac{dy^j}{dx^k}$. So, in tensor notation, $\frac{dx^i}{dy^j} \frac{dy^j}{dx^k} = \frac{dx^i}{dx^k} = \delta^i_k$ Now ...
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0answers
23 views

meaning of tensor “component”

In Schutz, Geometrical Methods of Mathematical Physics, it is written The components of a tensor are its values when it takes basis vectors and one-forms as arguments. It then gives an abstract ...
0
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1answer
64 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 ...
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0answers
21 views

How to change the parametric equations of a hypersurface in $V_N$ to another form…

This exercise was given in the first pages of Synge & Schild Tensor Calculus. The parametric equations of a hypersurface in $V_N$ are $x^1=a\cos{u}$, $x^2 = a\sin{u^1}\cos{u^2}$, $x^3 = ...
0
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1answer
52 views

Levi-Civita symbol

Is the Levi-Civita symbol a tensor? R. A. Sharipov afirm (In "Quick Introduction to Tensor Analysis", page 30) that "...the Levi-Civita symbol is NOT a tensor..." ...
0
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1answer
34 views

Time evolution of Laplacian

While reading monograph on the Ricci flow, I came accross a fact (at least I think it is a fact), which is not proved explicitly in that book. Assume a smooth 1-parameter family of Riemannian metrics ...
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0answers
31 views

On reconciling different definitions of the $\nabla$ operator in curvilinear coordinates

Note: This questions was originally asked in iMechanica. The main confusion appears to be on whether Christoffel symbols should appear in the divergence of a field expressed in curvilinear ...
4
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1answer
114 views

Tensor notation (practicing)

I'm praticing tensor notation, and I want to prove this way that given vectors $A,B,C,D$ then $(A \times B) \times (C \times D) = \det(A,C,D)B - \det(B,C,D)A$, where $\det$ means the triple product. ...
5
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1answer
63 views

The derivative of the determinant of a Kronecker product

For an invertible matrix $A$, we have the identity \begin{align} \dfrac{\partial \det A}{\partial A} = \det A (A^{-1})^T \end{align} where the $T$ denotes the transpose operation. How does this ...
2
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0answers
19 views

Index notation interpretation

I'm having some confusion with index notation and how it works with contravariance/covariance. $(v_{new})^i=\frac{\partial (x_{new})^i}{\partial (x_{old})^j}(v_{old})^j$ $(v_{new})^i=J^i_{\ ...
0
votes
1answer
17 views

Rotation operator for a point in a coordinate system linearly derived from Cartesian coordinates

For some experimental and practical reason, I have created a new coordinate system in the form $$x^\prime_i=T_{ij}x_j$$ where $T_{ij}$ isn't a square matrix. $x_i$ is standard Cartesian coordinates, ...
5
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2answers
69 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
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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
votes
1answer
55 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
47 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
20 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
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1answer
25 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
50 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 ...
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0answers
22 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
34 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
40 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 ...
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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 ...
3
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2answers
199 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 ...
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0answers
31 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
101 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
101 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 ...
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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
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1answer
28 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
222 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 ...
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0answers
19 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
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1answer
80 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
70 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
24 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
46 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
28 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 ...
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0answers
91 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 ...
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1answer
31 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
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1answer
97 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 ...
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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 ...
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0answers
16 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 ...
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0answers
42 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
53 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
54 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 ...
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2answers
51 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 ...