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 ...

learn more… | top users | synonyms

0
votes
0answers
32 views

Definition of the Hessian on Rieamannian manifolds

I can't understand the following definition given by Petersen in his Riemannian Geometry book. Let $M$ be a Riemannian manifold and let $f \colon M \to \mathbb{R}$ be a smooth function. Let $\...
2
votes
1answer
49 views

Definition of tensors

I'm studying the book "Riemannian Geometry" by Petersen and since I'm new to the subject, I'm helping myself also with the more introductory DoCarmos's book. I'm a bit confused about the definition ...
1
vote
0answers
36 views

Linear Algebra: Inverting an induced operator.

Question: Given an invertible linear map $U:V\to V$, consider the induced map $\tilde{U}:\Lambda^k(V)\to \Lambda^k(V)$ given by $$\tilde{U}(v_1\wedge \cdots\wedge v_k):=\sum_{j=1}^kv_1\wedge \cdots \...
0
votes
0answers
62 views

$\nabla\times(\nabla\times \boldsymbol{A})$ using Levi-Civita

I want to prove that $\nabla\times(\nabla\times \boldsymbol{A}) = \nabla(\nabla\cdot\boldsymbol{A}) - \nabla^2\boldsymbol{A}$ using the Levi Civita. This is solved considering the $i$-th component of $...
0
votes
1answer
30 views

What does this colon in tensor notation mean?

I was reading a paper earlier an found the following: "The tensors satisfy orthogonality $$ <S_{:,j,:,:}|S_{:,j',:,:}> =0 $$ if $j \neq j' $. Here $<S_{:,j,:,:}|S_{:,j',:,:}>$ is the ...
0
votes
0answers
26 views

Divergence of outer product in polar coordinates

Right now I am trying to solve Euler's conservation equations for circular domain. Due to several factors, I am restricted to polar coordinates. I can't manage to correctly calculate divergence of ...
4
votes
1answer
44 views

Covariant Derivative Clarification

In my notes I have the following when taking the divergence, $\partial_\mu$ of $\partial_\alpha\varphi^\alpha g^{\mu\nu}$ $$ \partial_\mu \partial_\alpha \varphi^\alpha g^{\mu\nu} = \partial_\nu \...
3
votes
1answer
85 views

Doing Symbolic Computations With Tensors And Differential Operators

Motivation Consider the following expression $${\varepsilon}= \frac{1}{2} \left( \nabla \otimes u + \nabla \otimes u^\text{T} \right) \tag{1}$$ where $u:\mathbb{R^3} \to \mathbb{R^3}$ is a vector ...
3
votes
2answers
69 views

Covariant derivative identity

I am trying to prove the following identity for contravariant vectors $X$ and $Y$ (this appears in exercise 6.7 of D'Inverno): $\nabla_{X}(fY) = (Xf)Y + f \nabla_X Y$. I have a way of proving it but ...
1
vote
0answers
32 views

Riemann curvature tensor for geometry surfaces on $\mathbb{R}^3$

Let $M\subseteq \mathbb{R}^3$ be a regular surface and $p\in M$. The Riemann curvature tensor is defined by: $$\begin{array}{rcll} R_p:&T_pM\times T_pM\times T_pM&\longrightarrow &T_pM\\ &...
1
vote
1answer
24 views

Sum of skew symmetric and symmetric parts of tensors

Denoting the skew-symmetrisation and symmetrisation of a $(0,p)$-tensor $X_{a_1 \ldots a_p}$ by $X_{[a_1 \ldots a_p]}$ and $X_{(a_1 \ldots a_p)}$ respectively, is it always true that $X_{a_1 \ldots ...
3
votes
1answer
43 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 component,...
1
vote
1answer
46 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 ...
2
votes
1answer
37 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 \...
2
votes
1answer
71 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 ...
0
votes
1answer
37 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 (Y,...
1
vote
0answers
36 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 \...
1
vote
0answers
24 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
votes
0answers
95 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
votes
0answers
53 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
votes
2answers
49 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 \...
0
votes
0answers
18 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 $x^3$. ...
2
votes
1answer
46 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.. \...
3
votes
1answer
43 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
vote
1answer
74 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 ...
2
votes
1answer
78 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 ...
0
votes
0answers
30 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
votes
1answer
29 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 ...
0
votes
1answer
302 views

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 $\{{\bf{e}}_1,{\bf{e}}_2,{\bf{...
5
votes
1answer
45 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
vote
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 ...
2
votes
0answers
69 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 ...
0
votes
0answers
22 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 ...
-3
votes
1answer
18 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 $$ [[x],\vec{\...
0
votes
0answers
39 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 n}v\...
1
vote
0answers
54 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 $$[X,Y]:=X^i\partial_i(Y^j)...
1
vote
1answer
26 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 ...
1
vote
1answer
44 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 ...
0
votes
1answer
57 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 ...
0
votes
0answers
36 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 V)\otimes(\...
1
vote
1answer
35 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.
1
vote
0answers
41 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 ...
2
votes
1answer
33 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 ...
0
votes
0answers
25 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?
1
vote
0answers
79 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 ...
0
votes
1answer
37 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 ...
0
votes
0answers
26 views

Covariant derivative of curved space basis vector is not 0. Why?

https://www.youtube.com/watch?v=jQTm-YyKWs0&index=24&list=PLlXfTHzgMRULkodlIEqfgTS-H1AY_bNtq In this video at 24:30 the teacher writes down the expression for the covariant dervative of the ...
0
votes
1answer
52 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 there ...
0
votes
1answer
48 views

Derive second fundamental from metric

If M is immersed in $\mathbb{R}^2$ and the metric is given by $g=e^{x^2+y^2}(dx^2+dy^2)$. How to calculate Gaussian curvature $K_M$? First, I use the Gauss equation $K_{\mathbb{R}^2}-K_M=\frac{1}{2}|...
1
vote
1answer
60 views

How to extend the parallelepiped volume formula to higher dimensions?

The volume of a parallelepiped $(V)$ is given by the triple scalar product: $$V=\mathbf{c}\cdot{}(\mathbf{a}\times\mathbf{b})$$ where $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are the vectors ...