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Questions tagged [tensors]

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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What's contravariant about contravariance

I think, that the covariant basis varies with the coordinate system, and the contra varies opposite to it. Geometrically, is the thing that's varying against the coordinate system just the length/...
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How do I solve this partial differential equation involving tensors?

So I have the following differential equation to solve: $$\frac{\alpha ^2}{m^2c^2\sqrt{r^4\sin \left(\theta \right)}}\cdot \partial \:_x\left(A^{xy}\cdot \:\partial \:_y\:\psi \:\left(ct,\:r,\:\...
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tensor power method

Just as we can use the matrix power method to find eigenvalues/eigenvectors of matrices in an iterative way, we can analogously find the eigenvalues/eigenvectors of tensors in a similar way. My ...
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1answer
17 views

Understanding of the Kronecker delta

From my understanding $e_a\cdotp e_b$ = $\delta_{ab}$ (for $a,b = 1,2,3$) equals $1$ when $a=b$ and $0$ when $a \neq b $ where $e_a$ and $e_b$ are vectors with entries 1 in the $a$'th and $b$'th row, ...
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Biharmonic Operation on a Tensor [on hold]

This is a practice question given to us and I have no idea how to solve this question. Q. Use two distinct definitions of your choice of dot product between a first order tensor and a third order ...
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28 views

Christoffel symbols of the second kind transformation law

We want to show that the Christoffel symbols of the second kind transform like a connection. the Christoffel symbols of the second kind are given by: $$\begin{Bmatrix}a \\ bc\end{Bmatrix} = \frac{1}{...
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37 views

Form the 4th dimensional 2nd order T to find its determinant and eigenvalues.

A 4th dimensional 2nd order tensor is given as $\overline {\overline T}=2\vec e_1\vec f_1+2\vec e_2\vec f_2+\vec e_1\vec f_2+\vec e_2\vec f_1+\epsilon:\vec e_1\vec e_2$ where $\vec e_i.\vec f_j=\...
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Higher order covariant derivative notation

The covariant derivative of a tensor can be noted equivalently as $$\nabla_b X^a \qquad \mathrm{and} \qquad X^a_{\; ;b}$$ If we want to express a higher order covariant derivative $$\nabla_c \...
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Integration over cylinder surface of a fourth order tensor [on hold]

I would get the results of the following integration and the procedure of it. Thanks in advance.
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23 views

Is the covariant basis covariant?

Not sure where my reasoning is going wrong in the following (and also not sure if this is common notation/phrasing): Since $e_i=\frac{\partial \vec{R}(z')}{\partial z^i}$ if the coordinate system z ...
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How can I find the component of this 3D tensor in a 2D space?

Good day Dear community, I'm really new in this field, so I truly appreciate your help and advises. I have this math problem: Consider the two mutually perpendicular unit vectors: $$i_a=3/5 i_1-4/...
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Deriving the Transformation of the Coefficients of Surfaces of Second Degree as a Tensor. Per Einstein

See pages 12 and 13 of Einstein's https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_1 This question involves rectangular Cartesian coordinates in $\mathbb{R}^{3}$ related by orthogonal ...
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1answer
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Higher Order Multivariable Taylor Expansions

The quadratic multivariable Taylor approximation of a function $f(x, y)$ around a point $(a, b)$ is given by $f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) + \frac{1}{2}f_{xx}(a, b)(x - a)^2 + f_{xy}(...
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1answer
32 views

Bianchi identity to Einstein tensor

The Bianchi identity is $\triangledown_{[aR_{bc}]_d^e}=0$. Contraction of the Bianchi identity leads to an important equation satisfied by $R_{ab}$. We find $$\triangledown_a R_{bcd}^a + \...
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1answer
36 views

Deriving parallel transport equations

From Wald's General Relativity page 37, It is easiest to compute the change in $v^a$ when we return to $p$ by letting $\omega_a$ be an arbitray dual vector field and finding the change in the ...
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Smallest $i$ for Einstein summation $ij,ik,il→jkl$ to have a (under-determined) solution given arbitrary $jkl$ term

I'm wondering what is the smallest $i$ for Einstein summation $ij,ik,il→jkl$ to have a (under-determined) solution given arbitrary jkl term (and solve for $ij$, $ik$...
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Is there a specific mathematical terminology for matrix/tensor with holes(missing elements)?

Consider a matrix-with-hole like below: $$ \begin{bmatrix} None & x \\ y & 0 \end{bmatrix} $$ We can define two variants of means for the above object: one is to calculate ...
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1answer
68 views

equality of two simple tensors in $R=k[x,y]$

Consider $R=k[x,y]$, where $k$ is a field. Consider the $R$-module $M=\langle x,y\rangle$. I would like to see that $x \otimes y \neq y \otimes x$ in $M \otimes_R M$. My try to prove it: Let $F$ ...
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Symmetric algebra notation

For a vector space $V$ we have its tensor algebra $TV = \bigoplus _{k=0}^\infty \otimes ^k V$, its exterior algebra $\Lambda V = \bigoplus _{k=0}^\infty \Lambda ^k V$, and its symmetric algebra $S V = ...
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35 views

If $A^iB_i$ is called a contraction, what is $A^{ij}B_{ij}$ called?

I have a tensor $A^{ijk}$ and a tensor $B_{ijk}$, and I'd like to contract all the indices between them, resulting in the scalar $k$: $$k = A^{ijk}B_{ijk}$$ Is there a name for this sort of "multi-...
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1answer
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what does it mean to be symmetric for tensors and Kronecker delta symbols and help explain this answer to me

i understand how to change 2 tensors into Kronecker delta symbols but unsure how they managed to transform back to just one. If someone could add all the steps to get to the answer that would be ...
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1answer
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Is the metric tensor relative to a reference coordinate system?

I am fairly new to this topic, and I don't know anything about tensors, but I have to know what a metric tensor is and this is how I have been thinking about it: 1)A metric tensor is a mathematical ...
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2answers
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Basis vectors as a differentiation of position vector

Basis vectors are defined as $\vec {E_i}=\vec{E_i}(x_1,x_2,x_3)=\frac{\partial\vec{r}}{\partial x^i}$ i=1,2,3. In spherical coordinate system $x^1=r, x^2=\theta, x^3=\phi$ position vector $\vec {R}=r ...
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Prove that $ E^{ijk}=\frac{1}{g} E_{ijk}$

Prove that the components of Contravariant Levi-Civita tensor can be given in terms of the components of the Covariant Levi-Civita tensor as follows: $$ E^{ijk}=\frac{1}{g} E_{ijk},$$ where $g$ is ...
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1answer
19 views

is it possible to describe a set of matrix-vector products equivalently in terms of a 3D array and a matrix?

Is it possible to define the matrix-vector products \begin{align} A_i x_i = b_i; \ \forall i = 1,\cdots,N \ , \end{align} where $A_i \in \mathbb{R}^{M \times K}$, $x_i \in \mathbb{R}^{K \times 1}$, ...
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62 views

Interesting theorem in differential geometry

I was reading a while back in differential geometry, and noticed a theorem which states something along the lines of that for any given surface there always exists a point where the metric tensor $g_{...
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1answer
46 views

Explanation of the metric tensor

Before anything, I just want to say that I'm studying mathematics/physics in a different language (Serbian) so some of the English terms that I use might be a bit wonky. Just ask if a term makes no ...
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1answer
28 views

Why is the tensor product of two vector spaces a vector space?

We defined the tensor product of $v$ and $w$ to just mean a "symbol" $e_{vw}$, then considered the subspace spanned by all these symbols, and finally we quotient out by relations to make the tensor ...
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1answer
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The tensor $\epsilon_{ijk}$ is related to determinants?

My textbook says the following in an appendix on tensor notation: The tensor $\epsilon_{rst} = \begin{cases} 0 \qquad & \text{unless $r, s,$ and $t$ are distinct} \\ +1 \qquad & \text{if $...
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proof that the covariant or contravariant basis is a basis

Does anyone know how to prove that given a coordinate system Z, the covariant (contravariant) basis is in effect a basis for Z ?
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Transformation Matrices: $\hat{\mathbf{e}}_j = \sum_{i} H_j^i \mathbf{e}_i$ and $\hat{\mathbf{\mathrm{x}}} = H^{-1}\mathbf{\mathrm{x}}$?

My textbook says the following in an appendix on tensor notation: Consider a change of coordinate axes in which the basis vectors $\mathbf{e}_i$ are replaced by a new basis set $\hat{\mathbf{e}}_j$,...
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How to prove Penrose “Bianchi symmetry” with non-zero torsion tensor using abstract indexing?

I want to prove $R_{[αβγ]}^{\ \ \ \ \ \ \ δ} + ∇_{[α}T_{βγ]}^{\ \ \ \ δ} + T_{[αβ}^{\ \ \ \ ρ}\ T_{γ]ρ}^{\ \ \ \ δ} = 0$ EDIT: A brief discussion of the solution found by Matt is at the bottom ...
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Derivative of a scalar function with respect to a contravariant vector

So in the steps below, a scalar function, $f(x^2)$, is partial differentiated with respect to a contravarient vector: Where η is the metric tensor used for special relativity. I understand all the ...
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What is a basis for the space of multilinear maps from $V_1 \times \dots \times V_k \to W$?

I know that a basis for the space $L(V_1, \dots, V_k; \mathbb R)$ is $$\{\varepsilon^{i_1} \otimes \cdots \otimes \varepsilon^{i_k}\mid 1 \le \varepsilon^{i_j} \le \dim(V_j)\}$$ where $\varepsilon^{...
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components of antisymmetric tensor unchanged under rotations proof

In "classical theory of fields" Landau states that the components of a completely antisymmetric tensor of a rank equal to the number of dimensions of the space remain unchanged after a rotation. I ...
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1answer
20 views

Tensor coordinate transformations: cosine matrix vs rotation matrix

I have a tensor $\varepsilon(\vec r)$ and want to get $\varepsilon(\vec r')$. After some googling I've got confused, should cosine matrix to be used or coordinate transformation matrix? Some sources ...
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multilinear rank of a tensor

let $A$ be a tensor in $R^{n_1\times...\times n_d}$ if we use the truncated higher order singular value decomposition THOSVD to approximate this tensor to a tensor $B$ of multilinear rank $(s_1,...,...
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Tensor operations

I'm trying to do some scientific programming, but I have limited math experience. My supervisor has given me the following equation I need to solve for $d$: $$ Y = Xe^{-b\cdot d}. $$ where $X$ and ...
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1answer
48 views

Are all bivectors in three dimensions simple?

I want to show that all bivectors in three dimensions are simple. If I understand correctly, a bivector is simply an element from the two-fold exterior product $\bigwedge^2V$ of a vector space $V$, ...
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1answer
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How do you derive the row and column of an element in a tensor unfolding?

Tensor Unfolding defined with example: I'm trying to understand tensors and their unfoldings into matrices. I'm hitting a roadblock at the notation used for rows and columns mapped to the tensor ...
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Tensor product property

Can someone confirm if the below statement is correct? $\textbf{a}(\textbf{b}\bullet\textbf{c})=(\textbf{a}\otimes\textbf{b})\textbf{c}$ = $(\textbf{b}\bullet\textbf{c})\textbf{a}$
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Transform 4th order tensor from Cartesian to convected coordinates

Given a 4th order tensor and its Cartesian components, and a convected (non-orthogonal) covariant basis, how can I calculate variously the covariant and contravariant components of the tensor? ...
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49 views

Musical Isomorphisms

I'm studying from Fecko's Differential Geometry and Lie Groups for Physicists, and in the part introducing metric tensors, Fecko introduces the musical isomorphisms between the tangent and cotangent ...
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27 views

Tensors, Co- and Contravariance

Recently I have started to learn about Tensor-Calculus and in my head I made some connections/conclusions which I want to know whether I'm right or wrong about. As an example consider a $m\times n$ ...
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15 views

What is the type of a square matrix containing square matrices-of same dimension- of numbers?

Is it a $(4,0)$-tensor, or a $(2,2)$-tensor or something else?
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1answer
24 views

What's the inverse of the gradient of a gradient?

I am wondering if, for a given functio $f(\mathbf{x})$, there exists a tensor $\mathbf{A}$ such that $$\nabla \nabla f \cdot \mathbf{A} = \mathbf{I}$$ To make it clearer, in index notation (with ...
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1answer
25 views

Expressing the coordinate dependent and indepent forms of the $(0,1)$ tensor in different coordinate systems

I'm studying general relativity and and learning up on tensors through a lecture series. It says that $\omega_\mu$ represents a 1-form in the $x^\mu$ coordinate system. The coordinate independent 1-...
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Proving the type of the derivative of a tensor

Take the tensor $A_{ij}$ be the components of a tensor of type $(0,2)$. Prove that $\frac{\partial A_{ij}}{\partial x^r}$ form components of a tensor of type $(0,3)$. My main problem is I am not ...
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Show that the derivative of a second order tensor gives a third order tensor

Let $U_{i,j}$ be a second order tensor. Show that $\frac{\partial U_{i,j}}{\partial x_{k}}$ is a third order tensor. I know how to prove that the gradient of a scalar field (which is a tensor of ...
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1answer
30 views

Coordinate systems and derivative operators

Let $\psi$ be a coordinate system and let $\{\partial / \partial x^{\mu}\}$ and $dx^{\mu}$ be the associated coordinate bases. For any smooth tensor field $T^{a_1 \dots a_k}_{b_1 \dots b_l}$ we take ...