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
74 views

Why is the curvature tensor defined to be anti-symmetric?

I'm reading Chern's Lectures on Differential Geometry. On page 117 of section 4-2, he points out that \begin{eqnarray} d\omega_i^j - \omega_i^h \wedge\omega_h^j &=& \frac{\partial\Gamma^j_{ik}}...
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1answer
64 views

Why does the Levi-Civita connection commute with pullbacks and pushforwards?

If $i: M \to N$ is an embedding of Riemannian manifolds, I am trying to prove that $\nabla i^* T = i^* \nabla T$ for any covariant tensor $T$ (I use the same letter for the two Levi-Civita connections)...
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1answer
63 views

Calculating Christoffel symbols using variational geodesic equation

Given the line element $$ds^2 = e^v dt^2 - e^{\lambda} dr^2 - r^2 d \theta^2 - r^2 \sin^2 \theta d \phi^2$$ we wish to compute the Christoffel symbols $\Gamma^{a}_{bc}$ using the geodesic equation. ...
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2answers
71 views

An example for a mathematical object that has two indexes, and is not a tensor

I understand how to check if something is a tensor (if it transforms like a tensor, it's a tensor). I have this example of a rank 2 tensor: $$\tau_{ij}=r_if_j - r_jf_i$$ and I understand that it is a ...
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1answer
53 views

Divergence theorem in curvilinear coordinates

Suppose I have a tensor \begin{gather} \stackrel{\leftrightarrow}{A} = \begin{bmatrix} a_{11}(\vec{r}) & a_{12}(\vec{r}) & a_{13}(\vec{r})\\ a_{21}(\vec{r}) & a_{22}(\vec{r}) & a_{23}(...
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1answer
41 views

Analyzing the Hessian of this function

The problem I'm looking at is, given a matrix $A$ of size $m\times n$ where $A_{ij}\ge 0$ minimize $f(x, y) = ||A - xy||^{2}_{F}$ where $x$ is a column vector of length $m$, $y$ is a row vector of ...
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0answers
32 views

Derivative of an eigenvalue with respect to tensor itself

I have a rank-2 tensor $\mathbf{C}=C_{\alpha\beta} \mathbf{A}^\alpha \otimes \mathbf{A}^\beta$ defined in a curvilinear coordinate system. I want to compute the derivative of an eigenvalue $\Lambda_i$ ...
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1answer
34 views

Why, given an object with rotational symmetry, is the axis of symmetry a principal axis?

When consulting textbooks and notes online about principle axes of inertia, I couldn't find a source which directly addressed the reasoning/proof behind following statement: "Given an object with a ...
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1answer
40 views

Divergence theorem with tensors

I'm unsure how to apply the divergence theorem correctly on a volume integral of following type of object: $\partial_{a} \partial_{b} \partial_{c} T^{a b c}$. The volume integral is over a $n$-...
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0answers
40 views

Generalizations of Positive Definiteness

What, if any, notions of positive definiteness can be extended to 3rd order tensors (and beyond)? The reason I ask is because the Hessian matrix of a convex function is positive semi-definite, but ...
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1answer
35 views

Components of Maxwell tensor under Lorentz boost transformation

The following is taken from exercise 12.4 in D'Inverno. We wish to compute the transformation properties of the electric field and magnetic induction under a Lorentz boost. Given the following boost ...
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1answer
24 views

Kronecker delta equation simplification

I am trying to simplify a tensor equation with Kronecker delta $$ A_{ij} \big ( \delta_{ik}\delta_{jm} -\frac{1}{3}\delta_{ij}\delta_{km} \big) $$ $A$ and $\delta$ are Cartesian tensors. I know ...
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0answers
17 views

Identity in continuum mechanics

For a problem in the textbook I am reading, I need to prove that $\int_Vw_{i,j}v_jdV = \int_Sw_iv_jn_jdS$, where $S$ is the boundary of the volume $V$, $v_i$ is the velocity vector field of a ...
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2answers
59 views

Can we get a tensor out of summation of one vector with its transpose?

I don't know much about tensor calculus and here is something I'm trying to figure out. $$T=\mu({\nabla}\vec{V}+{\nabla}\vec{V}^T)$$ T is viscous stress tensor and $\vec{V}$ is the velocity vector. ...
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0answers
17 views

Relation between torsion and curvature in Tensor calculus

In the context of tensor calculus by using Serret-Frenet formula or otherwise how to prove that $\tau^2=\displaystyle\frac{r'''^2}{k^2}-k^2-(\frac{k'}{k})^2$ where $\tau$ and $k$ represent ...
1
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1answer
34 views

Vector norm in tensor notation

Just starting out with tensor notation. I'm trying to express a vector sum (norm) in tensor notation and its derivative. Suppose we have $s = \sum_k (a_k - b_k)^2$ What would be the natural way to ...
1
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1answer
43 views

Tensor equation of a line

So, tensors appear to have no good description on the internet that actually starts from the basics. I've put together so far that, philosophically, tensors form a way to do your math in frames but ...
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1answer
19 views

Describe the state of stress in a deformed elastic body with a given Cauchy Stress Tensor

How can I describe the state of stress in a deformed elastic body if the corresponding Cauchy stress tensor $T$ is constant and given by $T = −pI$, where $p > 0$ ??
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1answer
31 views

An identity involving the derivative of a scalar function with respect to a tensor-valued variable [duplicate]

Let B be a tensor-valued variable, taking values from the set of second order tensors on the vector space naturally associated with Euclidean 3-space. It is given that B is invertible. I am looking ...
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0answers
31 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 $\...
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1answer
48 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 ...
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0answers
35 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 \...
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0answers
60 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 $...
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1answer
29 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 ...
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0answers
25 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
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1answer
43 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
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1answer
81 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
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2answers
66 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 ...
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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\\ &...
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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
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1answer
37 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
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1answer
40 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
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1answer
33 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
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1answer
65 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 ...
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1answer
36 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,...
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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 \...
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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 $\...
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0answers
92 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 ...
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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
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2answers
48 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 \...
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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 $x^3$. ...
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1answer
45 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
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 ...
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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
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1answer
69 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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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 =...
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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 ...
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1answer
240 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{...
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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} $ $\textbf{T}$ ...
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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 ...