Questions tagged [tensors]
For questions about tensors, tensor computation and specific tensors (e.g.curvature tensor, stress tensor). Tensor calculus is a technique that can be regarded as a follow-up on linear algebra. It is a generalisation of classical linear algebra. In classical linear algebra one deals with vectors and matrices.
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Simplification of Levi-Civita Symbol
I need to simpify the following:
$\begin{equation*}
\varepsilon_{ijk}\dfrac{\partial^2 \phi}{\partial x_i \partial x_j}
\end{equation*}$
For the Levi-Civita symbol $\varepsilon$. I tried expanding ...
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Can I see an example of transforming a tensor from polar to cartesian coordinates?
I have been learning about tensors, and I understand about creating the Jacobian matrix to obtain the coordinate transformation for infinitesimals, so that we have
$$d\bar{x}^j=\frac{\partial \bar{x}^...
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How do normal coordinates take effect in the proof of evolution equations?
Though the title includes "(geometric) evolution equations", my question is really more of how to use normal coordinates to help us prove an identity involving components of a tensor. My ...
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covariant basis for spherical coordinate system
In this Wikipedia page the third covariant basis vector in the spherical coordinate system (when using the physics $r, \theta, \varphi$ convention) is:
$$ e_{\varphi} = R \cos \theta (-\sin \varphi \...
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Tensor Index Manipulation
I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem.
I have to prove that $\partial_{\mu} g^...
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Continuity of Linear Map on Tensor Product Spaces with Different Norm Properties
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and extending this to a map $\phi^k: V^{\otimes k} \rightarrow U^{\otimes k}$ defined by
$$
\phi^k(v_1 \otimes \...
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Algebra equation for 3 rank tensor
Suppose I work in $4$ dimensions. I have an algebraic equation in the following form, which contains a 3 rank tensor $X ^{\alpha \lambda \mu }$
\begin{equation}
X ^{\alpha \lambda \mu }\eta ^{\beta \...
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Understanding the Definition of a Rank-1 Tensor
A tensor is nothing but a multidimensional array. We can think of an $n-mode$ tensor as a structure whose each element has to be referred with the help of $n$ indices or $n$ axes.
Now, while reading ...
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Understanding the order of tensor application as a linear map.
I have a question regarding the application of one tensor to another, $Q(D)$. Let's start with the simplest example, considering the bilinear form $G=G_{ij}(\epsilon^{i}\otimes \epsilon^{j})$ acting ...
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Tensor fields and scalar function pullbacks
For convenience, $(p,q)$ tensor fields on a differentiable manifold $M$ is defined to be the entire scalar field.
On the other hand, in my textbook, the pullback of the $(p,q)$ tensor $T(x)$ at $x \...
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Derivation of momentum balance equation from covariant derivative of energy-momentum tensor
I am trying to derive the ($\beta = 0$, $\alpha = 1$) and ($\beta = 1$, $\alpha = 1$) components of the covarinat derivative of the energy-momentum tensor which is, $\nabla_{\alpha}T_{\beta}^{\alpha}=...
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What is the difference between the tensor product of a covector and a vector and a covector acting on a vector as a 1-form?
I believe I'm getting quite confused between the tensor product and the dual vector as a 1-form acting on a usual vector. Essentially, I'm struggling to distinguish the difference between $\epsilon^{i}...
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Chain rule applied on Jacobian
Say we want to find the acceleration vector in spherical coordinates and in cartesian basis.
By defining the position vectors in cartesian, $\mathbf{x}=(x,y,z)$, and in spherical coordinates, $\mathbf{...
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Proving vector field identity with tensor index notation [closed]
How do I show, in tensor notation, that $\nabla(\frac{1}{2}v^2)= (\nabla\vec{v})\cdot\vec{v}$, where $v^2=\vec{v}\cdot\vec{v}$?
So far, the conversion to tensor index notation I have is: $\frac{\...
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Does the Kronecker product for matrices satisfies the universal property of the tensor product (of modules)?
The Kronecker Product is defined here: https://en.m.wikipedia.org/wiki/Kronecker_product
It is sometimes also called 'tensor product'.
Hence, I would like to know whether it satisfies the universal ...
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Einstein summation notation and multi indices [closed]
Consider the Einstein summation notation $A^{ij}B_{ij}$. I know that this summation takes over a each pair of indices $i,j$ and there is no confusion if they are tensors and the indexing sets are ...
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Shape of a transform matrix that maps $\mathbb{R}^{a \times b}$ to $\mathbb{R}^{c \times d}$
I'm looking for the dimension of some tensor $T$ that transforms $a \times b$ matrices to $c \times d$ matrices. I initially thought it would have to be of a greater dimension, perhaps some shape like ...
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Tensor Calculus - Differentiation of a transformation.
I began reading about tensor calculus recently, and I've been having trouble understanding the differentiation of a transformation.
Let's say we are in an N-dimensional space with coordinates $x^1, x^...
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Need some help in understanding the orientations of fibres.
I was studying about "Tensor Decompositions and it's Applications" from a small handout, i.e Tensor Decompositions and Applications https://www.kolda.net/publication/TensorReview.pdf.
This ...
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Proving that $A^{[ab]}B_{ab} = A^{ab}B_{[ab]}$
On this problem sheet, I am trying to show that for tensors (given a vector space V and it's dual $V^{*}$) $$A:V^{*} \times V^{*} \to \mathbb{R} \\ B: V \times V \to \mathbb{R}$$
that $A^{[ab]}B_{ab} ...
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divergence of divergence of a matrix
looking recently at the higher dimensional fokker planck equation
i noticed it can be written pleasingly as
$$
\partial_t p = - \nabla\cdot(p \underline{\mu}) + \nabla\cdot(\nabla\cdot(p \underline{\...
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Proof of fundamental lemma of Riemannian geometry in the 1-form connection formalism
I'm trying to understand the proof of this version of the fundamental lemma of Riemannian geometry.
Let $\pi : \mathcal{F}_{on}(M) \rightarrow M $ the orthonormal frame bundle of an n-dimensional ...
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Euclidean first fundamental form as vector bundle valued, first order differential invariant
I'm self studying the book Cartan for beginners and i'm trying to solve the exercise 2.5.2. The setup is the following:
Let $\operatorname{ASO}(n+s)$ the set of rototranslation in the $n+s$ ...
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Are closed group of $GL(n)$ always determined by a set of tensors?
Let $\mathcal{T}$ be a set of tensors in $\mathbb{R}^n$, and let $G_{\mathcal{T}}$ be a subgroup of $GL(n)$ defined as
$$
G_{\mathcal{T}} = \{ g \in GL(n) \mid g \cdot T = T, \, \forall T \in \mathcal{...
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Proof of commutativity of trace and covariant derivative in orthonormal frame
Suppose we have a symmetric (0,2) tensor $k$. (The original setting is that $k$ is the scalar second fundamental form for a Riemannian hypersurface, but I don't think it matters in my specific ...
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Is it correct to exclude the nonlinear terms from the finite strains in linear elasticity, if their differentiation is needed in equilibrium equation?
It is clear that infinitesimal strains are obtained from finite strains (e.g. Green-Langragian or Eulerian-Almansi tensor) by removing nonlinear terms which smallness order is greater than that of ...
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Change of basis in tensor notation
When trying to deeply visualize the meaning of the entries of a basis transformation matrix, I realized that a (forward) change of basis matrix can be written in a tensor form like this:
$$Q=Q^i_j \; \...
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Tensor bundles are vector bundles
I would like to prove that every tensor bundle is a vector bundle. First some definitions.
A bundle of $(k,l)$-tensors on $M$ is defined as as their disjoint union, $T^k_lM:=\bigsqcup_{p\in M} T^k_l(...
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Notational question on matrices, particularly in the context of differential geometry.
I've seen the notation both in notes, and in for example Loring W. Tu:s book, the notation $a^j_i$ for a matrix-element in a matrix $$A = \Big\{a^i_j\Big\}_{1 \leq i,j \leq n}$$ What is the row and ...
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Efficient representations of the angular dependence of quadratic and quartic functions in 2d
Consider the set of arbitrary quadratic functions in two dimensions $F(x_1, x_2) = \sum_{ij} A_{ij} x_i x_j$, as well as the set of arbitrary quartic functions $G(x_1, x_2) = \sum_{ijkl} B_{ijkl} x_i ...
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Traceless tensor product: unclear definition
I do not understand here on the page $8$
that in the definition of traceless tensor product there are 3 indices $i,j,k$
in the rightmost term but in the preceding ...
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Dehn Invariants of two space-filling tetrahedra
Here are vertices for two space-filling tetrahedra.
$A = ((0, 0, 2),(0, 4, 2),(1, 2, 2),(2, 2, 0))$
$B = ((0, 3, 3),(1, 4, 2),(3, 2, 4),(1, 0, 2))$
Dihedral angles for A: {Pi/4, ArcCos[-(1/Sqrt[6])], ...
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Meaning of computing the tensor product of vectors
In a nutshell: a proposition in Abraham's Foundations of Mechanics states that, given a vector space $E$ with basis $e_1,\ldots,e_n$, the space $T^r_s(E)$ of tensors of type $(r,s)$, the dual basis $\...
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Trace of tensors in pseudo-Riemannian manifolds
I know that on a pseudo-Riemannian manifold $(M,g)$ a $(1,1)$ tensor field can be thought as an endomorphism of the tangent bundle. When one computes, for example, the trace of a $2$-covariant tensor, ...
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What formula can I use to translate $m$ indices into $n$ indices?
NOTE: The tensors are considered row-major. The sequence of elements is preserved.
Let $A$ be a $m$-dimensional tensor of size $m_1 \times m_2 \times m_3 \cdots$ and $B$ be a $n$-dimensional tensor of ...
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Why does the invariant 1-tensor integral gives 0 for any volume?
I was going through David Tong's vectors calculus notes. In Chapter 6.1.3 (Invariant Integrals) he gives the example
Here are some examples. First, suppose that we have a 3d integral over
the ...
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in $ds^2 = g_{\mu \nu} dx^\mu dx^\nu$, why is $dx^\mu$ represented as a row vector and $dx^\nu$ represented as a column vector? [closed]
Just like what's in the title, in the equation $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$, why are $dx^\mu$ and $dx^\nu$ represented differently?
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Coordinate expression for the codifferential of a $p$-form
I have been having difficulty obtaining the component expression of the codifferential of a $p$-form found on Wikipedia and in the book Riemannian Geometry and Geometric Analysis by Jürgen Jost. Let ...
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Ambiguity in raising/lowering the index of a tensor
I'm reading Sean Carroll's GR book, where it's written:
The metric and inverse metric can be used to raise or lower indices on tensors. That is, given a tensor $T^{\alpha\beta}_{\ \ \ \ \ \gamma\...
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Geometric meaning of the rising and the lowering of indices
I'm an undergraduate student and currently I'm approaching tensorial calculus. I was wondering: is there some geometric meaning to the operation of rising/lowering indices (and then if there was any ...
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Permutational invariants of degree-$m$ supersymmetric tensors
Let $A$ be a super-symmetric $(n \times.....\times n)$ $m$-fold tensor, so that:
$$
A\left(i_{1},\dots,i_{m}\right) = A\left(j_{1},\dots,j_{m}\right),
$$
when $\left(j_{1},\dots,j_{m}\right)$ is any ...
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How exactly does tensor contraction work with higher rank tensors?
I'm an undergraduate student doing my best to learn about tensor networks, but I'm stuck on Equation (2) of this paper "A Practical Introduction to Tensor Networks: Matrix Product States and ...
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Generalizing isomorphism between a $(1,1)$ tensor and linear map
(Throughout $\alpha,\beta$, etc. (Greek letters) denote covectors and $u,v$, etc. denote vectors)
I'm reading Sean Carroll's relativity text, with this being written:
Although we have defined tensors ...
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How is the arithmetic multiplication related to tensor products?
Mathematical objects are supposed to "model things", sometimes, more mathematics. In this sense, i think is "natural" to assume that tensors, being such universal objects as they ...
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Equation for Ricci tensor induced on a metric hypersurface derived from first Gauss-Codazzi equation
The first Gauss-Codazzi equation for a metric hypersurface $(S, h_{ab})$ of $(M, g_{ab})$ is:
$$\mathcal{R}^{a}_{\ bcd} = -2 \pi^{a}_{\ [c}\pi_{d]b} + h^a_{\ m}h^n_{\ b}h^p_{\ c}h^r_{\ d}R^m_{\ \ npr}$...
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Connection laplacian and abstract index notation
Let $(M,g)$ be a (pseudo-)Riemannian manifold. I am having some struggles to relate two different approaches, one based on the abstract index notation, and the other one based on global, coordinate-...
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Derivative of block matrix using einstein notation
Let $Y = [A \quad XB \quad C]$, where $A,B,C,X$ are all matrices with appropriate size. What is the derivative of $Y$ w.r.t. $X$?
The part that confuses me is that $\frac{\partial A}{\partial X}$ ...
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Mathematical relation between metric tensor and transformation matrix
In the context of non-orthogonal bases, I am trying to understand the mathematical relation between the metric tensor and the transformation matrix between the orthogonal and non-orthogonal basis.
I ...
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Locally constant tensors is the zero tensor on closed manifolds
Suppose I have $M^2$ a closed (i.e. compact with no boundary) Riemannian manifold, and
$$T: \text{Sym}^2(TM) \to \mathbb{R}$$
Coordinatize an open set $U \subseteq M^2$ with $\{(x,y)\}$. Suppose I ...
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Compute the tensor product $\mathbb{Z}\otimes_{\mathbb{Z}[x]}\mathbb{Z}$
Question:
For each $n\in\mathbb{Z}$, define the ring homomorphism
$$\phi_n:\mathbb{Z}[x]\to\mathbb{Z}$$
by $\phi_n(f)=f(n)$. This gives a $\mathbb{Z}[x]$-module structure on $\mathbb{Z}$, i.e,
$$f\...
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