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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9 views

Is it possible to write the Hadamard product of two matrices in tensor notation?

Say I have two $4 \times 4$ matrices $(A^{\alpha \beta})$ and $(B^{\mu\nu})$ and want to compute the Hadamard (entry-wise) product. Is there an elegant way of writing this down in the common ...
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1answer
13 views

Schur's theorem in DoCarmo's “Riemannian Geometry”

The exercise 8 of chapter 4 of Do Carmo's "Riemannian Geometry" ask to prove the Schur's Theorem. I don't understand a step in the hint (the "hint" is essentially the proof of the theorem). Schur's ...
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1answer
22 views

Tensor Product of Spaces has Basis of Tensor Products

I am given the following definition of the Tensor Product of spaces Given two vector spaces $V,W$ a vector space S is a tensor product of $V,W$ if there exists a map $M$ $$ M: V \times W ...
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1answer
23 views

Exercise 5, chapter 4 in Do Carmo's “Riemannian Geometry”

This is probably a silly question and maybe in the end the answer is trivial but I can't see it. The problem is the following. Let $M$ be a Riemannian manifold, $\gamma \colon[0,l]\to M$ be a ...
3
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1answer
87 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 ...
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1answer
22 views

Where is my mistake in this exterior derivative of a $2$-form not being a tensor?

Using the invariant formula for the exterior derivative, one gets that for a $3$-form $\omega$ its derivative is evaluated on vector fields according to $$3 \ {\rm d} \omega (X, Y, Z) = X \ \omega ...
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24 views

Geometrical meaning of Tensor [on hold]

i really need a geometrical interpretation or analogy for tensors. Bear in mind that i am not a mathematician, so complicated explanations will not suffice. Thank you!
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0answers
33 views

About a particular definition of “tensor”

I came across this quiet new to me way of defining "tensors", That a tensor $A$ is a map of the form, $A : \mathbb{R}^{n \times m_1} \times \mathbb{R}^{n \times m_2} \times .. \times \mathbb{R}^{n ...
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0answers
16 views

Why are second-order tensor invariants different if calculated in dual basis?

Disclaimer: I'm an student in engineering, so please forgive me asking this stupid question. Consider the following tensor coordinates given: The covariant coordinates of the metric tensor ...
4
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0answers
68 views

Einstein notation

I'm confused about a specific issue that I have with the Einstein notation (for tensor fields on manifolds). I want to write the following thing: Let $X$ be a smooth manifold. Choosing local ...
0
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0answers
39 views

how to mathematically represent a matrix of vectors?

My problem is the following: I have a dataset in particular have $4$ dimensions, for didactic reasons I need to represent this dataset as a $m\times n$ matrix array such that the ($i$-th, $j$-th) ...
2
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2answers
40 views

Why are vectors considered to be rank (0,1) tensors and dual vectors considered to be rank (1,0) tensors?

Sean Carrol in his book of general relativity, he defines a tensor to be a multilinear map from a collection of dual vectors and vectors to $\mathbb{R}$: $T:T^*_p \times...\times T^*_p \times T_p ...
3
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1answer
32 views

Number of Different Elements of $S_{ijkl}$ with Some Symmetries

I am not good at combinatorics so I am asking this simple question to learn a little. In fact, this question is motivated by the symmetries happening for the stiffness and Eshelby tensors in the ...
1
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1answer
85 views

Tensor notation for 3-D matrix expression

I have the expression $y_i = \displaystyle\sum_j x_j \!\cdot\!a_{ij} \!\cdot\! \exp \Big(\sum_k b_{ijk} \!\cdot\! x_k \Big)$ which I want to shorten without introducing more notation than necessary. ...
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1answer
36 views

Tensor Product on Hilbert Spaces (well definedness)

Let $H_1$, $H_2$,...,$H_n$ be $n$ Hilbert Spaces. For each $\phi_i \in H_i$, Let $$ \phi_1 \otimes \phi_2 \otimes... \otimes \phi_n:= \text{Conjugate multilinear form which acts on $H_1 \times H_2.. ...
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1answer
63 views

different approaches of defining tensors

This Wikipedia article says that tensor can be defined as miltilinear maps or be defined using tensor products. Could anybody explain with a simple example why these two approaches give the same ...
0
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0answers
17 views

$A: \mathbb R^3 \to \mathbb R^3$ what is the representation of $A$ under coordinate change? (Tensor)

For our initial homework on Tensor Calculus we have to do the following: Consider a map $A: \mathbb R^3 \to \mathbb R^3$, where we use local coordinates $\mathbf{x}$ given by $x^1,x^2$ and $x^3$. ...
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5answers
10k views

An Introduction to Tensors

As a physics student, I've come across mathematical objects called tensors in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, ...
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1answer
2k views

Vorticity equation in index notation (curl of Navier-Stokes equation)

I am trying to derive the vorticity equation and I got stuck when trying to prove the following relation using index notation: $$ {\rm curl}((\textbf{u}\cdot\nabla)\mathbf{u}) = ...
0
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1answer
72 views

Computing the Inverse of a Fourth Order Tensor

Suppose that we have a fourth order tensor ${\bf{A}}$ $${\bf{A}}=A_{ijkl} {\bf{e}}_i \otimes {\bf{e}}_j \otimes {\bf{e}}_k \otimes {\bf{e}}_l$$ in the orthonormal basis ...
2
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0answers
68 views

What are the non-vanishing directions of $A_{a(b} B_{cd)} - B_{a(b} A_{cd)}$? ($A$ and $B$ symmetric tensors)

Question Consider two symmetric tensors $A_{ab}=A_{ba},\, B_{ab}=B_{ba}$ which are generally of full rank (non-degenerate, all eigenvalues non-zero but of general sign) and their tensor product under ...
2
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1answer
35 views

Why are contravariant vectors denoted with a superscript (and not a subscript)?

I wonder whether the choice of denoting contravariant/covariant vectors with a superscript/subscript is arbitrary (and could have been made the other way around), or whether there is a specific reason ...
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0answers
20 views

Infinitesimal volume element transforms like a scalar

Show that the infinitesimal volume element $d^3x$ transforms like a scalar Attempt: Let $R^{kh} = \frac {\partial \bar x^h}{\partial x^k}$ Since in general a coordinate transformation is $\bar x^h ...
3
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1answer
23 views

How to indicate the space of sesquilinear forms?

Is there a canonical way to indicate the space of sesquilinear forms? For bilinear forms I can use the (0,2) Tensor space, but since sesquilinear forms are not linear in one variable I don't know how ...
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0answers
111 views

Cauchy Equations and Navier Stokes

I'm attempting to take the Navier Stokes Equation and coming up with an expression that will allow me to numerically determine the velocity of non-Newtonian fluid flow. The text I'm using is Cengel ...
5
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1answer
42 views

Is this 2-tensor symmetric? It satisfies these conditions

I have some scalar field $p$ and a 2nd order tensor $\textbf{T}$ such that $div( \ \textbf{T} \ \textbf{grad}(p) \ )=0$ $\textbf{curl}(\ \textbf{T} \ \textbf{grad}(p) \ )=\textbf{0} $ $\textbf{T}$ ...
1
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1answer
27 views

Is there an invariant similar to the characteristic polynomial for (0,2) and (2,0) tensors?

The characteristic polynomial of a matrix - a (1,1) tensor - is its invariant (independent on basis transformation). Is there a similar invariant for (0,2) and (2,0) tensors? The characteristic ...
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0answers
19 views

Metric tensor for just one index

I'm novice in the territory of tensor calculus. I know the utilization of the metric tensor to transform the covariant basis to contravariant one, vice versa: $Z^i = Z^{ij}Z_j$ (Eq. 1) I am going ...
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1answer
23 views

Covariant Derivatives and Swapping Indices

Okay,there's a covariant derivative of a rank 2 tensor. Swapping any indices gives a different tensor. Can we associate any physical significance to the swapping? For example, if I have a velocity ...
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1answer
13 views

algebra operations

If $[x]=x^{i}\sigma^{i}$, Find an alternative form for the product $$ x^{i}n^{j}\sigma^{k}\epsilon^{ijk}=x^{i}(\vec{n} \times \vec{\sigma})^{i} $$ that has to be more compact than $$ ...
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34 views

What kind of operation is available here?

What kind of operation $\clubsuit$ is available here? $$ \underbrace{\left(v^{\mathrm T}A_{n\times n\times n}\right)}_{\in\mathbb R^{n\times n}}\underbrace{\left(v^{\mathrm T}B_{n\times n\times ...
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0answers
45 views

Well definedness of Lie bracket (coordinates approach)

For practice I wanted to define Lie bracet in terms of coordinates and show that this definition is independet of the choice of coordinates. So I started with the definition ...
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1answer
41 views

In the formula for Ricci curvature, do the terms $\Gamma^{\ell}_{ij}\Gamma^m_{\ell m} - \Gamma^m_{i\ell}\Gamma^{\ell}_{jm}$ cancel each other out?

$$\large R_{ij} = R^{\ell}_{i\ell j} = g^{\ell m}R_{i\ell jm} = g^{\ell m} R_{\ell imj} = \frac{\partial\Gamma^{\ell}_{ij}}{\partial x^{\ell}} - \frac{\partial ...
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1answer
53 views

Checking that a two-form transforms correctly under Lorentz transformations

This is exercise $7.22$ in Supergravity by Freedman and Van Proeyen, but I did not understand it and would appreciate if you clear it out. Given the below, I still don't get how, if we define the ...
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1answer
741 views

How to prove that the Kronecker delta is the unique isotropic tensor of order 2?

Is there a way to prove that the Kronecker delta $\delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = \lambda \delta_{ij}$ ...
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1answer
32 views

Tensor Contraction Invariance

On page 86 of Bishop and Goldberg's Tensor Analysis on Manifolds you are asked to show that contractions are invariants. Rather than doing it awkwardly for an (r,s)-tensor (r is the contravariant and ...
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1answer
41 views

Sectional curvature in a paraboloid is always positive.

I'm working on Lee's book ''Riemaniann Manifolds an Introduction to Curvature''. One exercise (11.1) is about to see that the paraboloid given by the equation $y=x_1^2+...+x_n^2$ has positive ...
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1answer
26 views

Tensors which are symmetric and antisymmetric in overlapping groups

Say I have the following tensor $T_{abc}$ such that $$ T_{(a[b)c]} $$ Ergo, it is symmetric in indices $a$ and $b$ and antisymmetric in $b$ and $c$. Keeping in mind the various properties that ...
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1answer
30 views

Multiplication of tensor and vector

How to do the multiplication of the multidimensional array $A_{n\times n\times n}$ and the vector $v_n$ (indices denote dimensions)? Can you kindly give suggestions or references? Thanks in advance.
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0answers
30 views

Tensors in a vector bundle with infinite-dimensional fibers

If $\xi = (\pi,E,M)$ is a vector bundle such that each fiber has finite-dimension, and $V=\Gamma(M,E)$ is the space of smooth sections, then there is an isomorphism between $(\otimes^r ...
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1answer
28 views

Treatments of Lie theory and Noether theorems in tensor notation

I am looking for a conceptual treatment of Lie theory and Noether theorems that uses tensors calculus rather than exterior calculus. I know that tensor calculus is not optimal for these subjects, ...
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0answers
39 views

Connection with zero torsion and curvature

I remember encountering the following theorem many, many time ago: if a connection $\nabla$ has $R = 0$ and $T = 0$ then... The problem is that I cannot remember the conclusion. Was it that the ...
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2answers
55 views

Repeated application of the gradient on a Riemannian manifold

While reading about Sobolev spaces on manifolds, I encountered the following notation regarding the norm in $H^k$: $\| \nabla ^k f \|_{L^2}$. There are two questions here: 1) What kind of object id ...
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1answer
23 views

Are Linear Transformations Always Second Order Tensors?

I've been reading a bit about tensors on Wikipedia (so correctness not guaranteed here) and I have a question. The order of a tensor $T$ is defined as $n+m$, where $n$ denotes the number of covariant ...
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17 views

Is every square matrix a tensor of 2nd order?

Is every square matrix a second order tensor? If not, what is an example of a square matrix, which is not a tensor? How can I prove that a matrix is in fact a tensor?
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20 views

#skew symmetric, symmetric and alternating multilinear map in a vector space over field of characteristic 2 [on hold]

can every skew symmetric multilinear map written as sum of symmetric and alternating multilinear map.(specially I mean in a field of characteristic 2)
11
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1answer
315 views

How can I determine the number of wedge products of $1$-forms needed to express a $k$-form as a sum of such?

This question was motivated by this related one: How "far" a differential form is from an exterior product . Let $\mathbb{V}$ be a vector space of dimension $n$ with underlying field ...
0
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1answer
31 views

How many components of an antisymmetric rank five tensor on $ \mathbb{R}^5 $ are independent?

How many components of the a rank five tensor on $\mathbb{R}^5$ which is antisymetric under exchange of any pair of indices are independent? If we write the tensor $E_{i_1i_2i_3i_4i_5}$ then ...
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0answers
44 views

Intuition Behind Dual Vectors

Recently I've been self-studying tensor analysis through the book titled "An Introduction to Tensors and Group Theory for Physicists," by Nadir Jeevanjee. I found it to be very intuitive, up until the ...
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1answer
35 views

What is considered to be the natural (injective) homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$?

Let $R$ be a ring and $I,J,L \unlhd R$ such that $J \subseteq I$. What is considered to be the natural homomorphism $\frac{IL+J}{IL} \rightarrow I \otimes \frac{R}{L}$ ? Remark: It must be ...