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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rank 4 self-adjoint tensor?

In rank 2, and given an inner product, the self-adjoint matrix is easy to define: $$ \langle \mathbf{v}, A \mathbf{u} \rangle = \langle A \mathbf{v}, \mathbf{u} \rangle \tag{1} $$ If the inner product ...
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Conceptual confusion regarding raising and lowering index

In Pavel Grinfeld's tensor calculus course (eg: 18:51) , it is said that in a given coordinate system on $\mathbb{R}^3$, we can write: $$ e^i = g^{i \alpha} e_{\alpha}$$ Suppose we evaluate the above ...
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Obtaining an explicit formula for a connection applied to an almost complex structure

Let $(M,J)$ be an almost complex manifold and $\nabla$ be a connection on $TM$. I am trying to see how we can obtain an explicit formula for $\nabla_X J$. I know that the way to extend $\nabla$ to a ...
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2 Levi Civita Summation

I dont understand how i can get the following expression: $$\varepsilon_{ijk}\varepsilon_{ijk'}=2\delta_{k k'}$$ I try the following and end up with a $-2$ instead of a $2$: $$\varepsilon_{ijk}\...
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Relation between symmetric outer product decomposition and symmetric multilinear decomposition

Suppose tensor $\mathcal{A}$ is a symmetric real tensor of order $k$. Then, symmetric outer product decomposition of $\mathcal{A}$ is $$ \mathcal{A} = \sum_{i=1}^p \lambda_i v_i^{\bigotimes k}, $$ ...
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Gradient of Frobenius inner product involving a homogeneous function

Let $f:\mathbb{R}^{m\times n} \rightarrow \mathbb{R}^{m \times n}$ be a differentiable function. I want to take the gradient of the Frobenius inner product between $X$ and $f(X)$ with respect to $X$. ...
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Regarding the decomposition of a tensor as an antisymmetric and symmetric part

I have seen for two index tensor that the sum of symmetric and anti symmetric part is the total tensor again.. but is it true for higher index tensor? I thought it was but then I saw this diagram in ...
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How to reconcile two approaches to the Gaussian curvature

Gaussian curvature can be found as the ratio of determinants of the second to the first fundamental forms (from here): $$K =\frac{\det \mathrm {II}}{\det \mathrm I}=\det \left(\mathrm {I}^{-1}\rm {II}\...
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The connection between a Metric Tensor and a Metric in Topology

Does the Metric tensor define a metric on the topology of the manifold/space it's on, and/or does the metric on the topology of the space define a metric tensor, if you go through all the trouble of ...
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Factorial factor when finding odd/ even part of a tensor

Suppose we want antisymmetric part of product of component of three vector, we have: $$ \zeta^{[r} \eta^s \chi^{t]} = \frac16 (\zeta^r \eta^s \chi^t + \zeta^t \eta^r \chi^s - \zeta^r \eta^t \chi^s - \...
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Why is the following property true for the Schouten tensor?

Why is the following property true for the Schouten tensor $P_{ij}$: $$\nabla^k\nabla_iP_{kj}=\nabla^k\nabla_jP_{ki}$$ The Schouten tensor is defined as $$P_{ij}=\frac{1}{n-2}\left(\operatorname{Ric}_{...
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Proof that the covariant derivative is orthogonal to the tangent plane

Pavel Grinfeld in here writes: $$\vec S^\gamma \nabla_\alpha \vec S_\beta= \vec S^\gamma \cdot \frac{\partial \vec S_\beta}{\partial S^\alpha}=\Gamma^\omega_{\alpha \beta} \vec S_\omega\cdot \vec S^\...
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General form of rank $2$ tensor invaiant under certain rotations

I have the problem of determining the most general form of a rank 2 tensor $t_{ij}$ (in 3 dimensions) satisfying: $t_{ij}$ is invariant under any rotation about the $z$-axis $t_{ij}$ is invariant ...
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Derivative of the inverse of a symmetric matrix w.r.t itself

I'm trying take the derivative of a symmetrix matrix $\mathbf{C}$ with respect to itself. $$ \begin{equation} \frac{\partial \mathbf{C}^{-1}}{\partial \mathbf{C}} \end{equation} $$ Using the indicial ...
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Why is $\mathbb{R} \otimes \mathbb{S}^{1} $ infinite?

Intuitively I see why this is infinite, but there is no obvious way to apply the tensor relations to reduce every element into a finite set.I don't really have an idea on how to approach this problem (...
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Mechanics in the contraction $e^{ijk}e_{ijk}$

$e^{ijk}$ is the permutation symbol of $1,2,3$ assuming values $1$ and $-1$ depending on the sign of the permutation. Otherwise the value is zero. The result is $6,$ but I am not following the ...
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Decomposition of order 3 tensor symmetric along two dimensions

I have a 3rd order tensor $\mathbf{A}$ consisting of symmetric covariance matrices (with dimensions of space by space) stacked in time. I would like to compute the leading spatial features that ...
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Solving a matrix equation involving tensors

While doing my research, I came across the following matrix equation in $W \in \mathbb{R}^{n \times d}$ that I could not solve. $$ \sum_{i=0}^{t} X_{i} W Y_{i} + X'_{i}WY'_{i} = Z $$ where $X_{i}, X'...
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Initial step in proving that a partial derivative of a tensor is not a tensor

Exercise 91 in Pavel Grinfeld's book Introduction to Tensor Analysis and the Calculus of Moving Surfaces reads Similarly, for a contravariant tensor $T^i,$ derive the transformation rule for $∂ T^i ∕∂...
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Matrix of adjoint linear map is transposed matrix

I am stuck with some exercise and hope that someone can give me a hint: Let $V$ and $W$ be vector spaces and let $B_V = \big\{ v_1, \ldots, v_n \big\}$ and $B_w = \big\{ w_1, \ldots, w_n \big\}$ be ...
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Contravariant Vector Component Transformation from Polar to Cartesian

I am new to tensors and I just learned that the contravarient components of a vector transforms in the following way (using Einstein summation convention) $$A^{'i}=\frac {\partial x^{'i}}{\partial x^j}...
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Tensor and Gauss divergence theorem

I am trying to see whether, in spherical coordinates, $$\int \left( \boldsymbol{r} \times \boldsymbol{\nabla} \cdot \boldsymbol{T} \right) \cdot \boldsymbol{e}_z dV$$ where $T$ is a 2D symmetric ...
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What does the dot product of a tensor and a vector represent?

The dot product (or inner product) of a tensor T and a vector a produces a vector b = T . a: $$ b_i = T_{ij}a_j = \begin{pmatrix} T_{11} a_1 + T_{12} a_2 + T_{13} a_3\\ T_{21} a_1 + T_{22} a_2 + T_{23}...
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Differentiate the summation over an index of a rank 3 tensor.

$X^{i,j,\lambda}$ is a rank 3 tensor, $W^{i,j}$ is a matrix. Is it possible to analytically solve the following? $\frac{d}{dX}\left|| W - \sum_\lambda X \right||^{2}$ I know with matrices we can ...
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Are tensors constructed such that one forms "act" on some complex vector field?

I am a physics student, trying to learn differential geometry. I have some confusion understanding the motivation in constructing tensors (or tensor fields). On a differentiable manifold $\mathcal{M}$ ...
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What does R mean in this equation.

I have encountered an equation of (zeppelin = a tensor with the same perpendicular eigenvalues (l2=l3) ) that is simple but I don't understand the operation R. the equation is for implementing Zepplin ...
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Implicit definition of stress tensor

I'm reading Giacomini, Alessandro, and Luca Lussardi. "Quasi-static evolution for a model in strain gradient plasticity." SIAM journal on mathematical analysis 40.3 (2008): 1201-1245 and I ...
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Relating the Lie derivative to inner product of 2-tensors

Let $(M,g)$ be a Riemannian manifold. Let $f \in C^\infty(M)$, $X$ be a vector field and $h$ be a (symmetric) covariant $2$-tensor. Denote by $\langle \cdot, \cdot \rangle$ the inner product induced ...
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Which (contravariant) tensor does represent the rate of contravariant components with respect to covariant basis?

The rate of covariant basis with respect to contravariant components may be expressed as $$\frac{\partial \mathbf{Z}_{i}}{\partial Z^{j}} = \Gamma_{ij}^{k}\mathbf{Z}_{k}.$$ The inverse, i.e., $\frac{\...
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Prove invariant formula for exterior derivative

In Marsden "Foundation of Mechanics", it is said that the properties defining the exterior derivative are readily proven to be satisfied by the invariant formula $d\omega(u)(e_0,...e_k)=\...
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Why is covariant basis positive-definite?

I'm studying a (tensor calculus) book that says: Any covariant basis, interpreted as a matrix, is positive-definite. But I have stumbled upon seeing this claim. Let's assume a non-trivial vector $\...
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Why do we not need Christoffel to differentiate contravariant component of a vector?

OK, perhaps a very stupid question but I am going nuts thinking about it. In page-106 of Pavel Grinfeld's Tensor Calculus book, we consider a vector $V= V^i Z_i$ and we consider the derivative of this ...
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Demonstrate the inverse tranformation of coordinates (Cartesian Tensors)

I am starting to learn tensors but i'm already stuck in the first question of my problem set, which says: Show that the inverse transformation of $$g'_{i} = \sum_{j=1}^{3} a_{ij} g_{j} $$ is $$ g_{i} ...
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Solving tensor version of homogeneous Sylvester equation

I have a system that reads (in summation convention) $$ X_{j_1 j_2 ... j_N} R^{j_1}_{k_1} R^{j_2}_{k_2} ... R^{j_N}_{k_N} = X_{k_1 k_2 ... k_N} $$ where $N$ is a fixed value, $R$ a known matrix (...
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2 votes
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Shape operator as a tensor field defined in a submanifold

Definition $4.18$ in O'Neill's book Semi-Riemannian geometry, with Applications to Relativity states that given a unit vector field $U$ normal to a hypersurface $M\subset \overline M$, then, there is ...
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Is the covariant derivative a tensor?

While studying general relativity, the covariant derivative is constructed (in no rigorous manner) in order to make the derivative of a tensor transform like a tensor. Symbolically, $$ \nabla'_{\mu} V'...
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A "trivial" question on tensors that I'm not sure about

(Notation $L^k$ is the space of $k$-covariant tensors) A friend posed this question to me - he's been studying about tensors and this question is an exercise there, claimed to be "trivial". ...
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Steps to generalize the concept of tensors from Euclidean to non-Euclidean spaces

In the book by Pavel Grinfeld Introduction to Tensor Analysis and the Calculus of Moving Surfaces the "tensor property" is introduced on the grounds of defining the covariant basis $\left\{\...
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Wedge product and linear independence

The following question was from my assignment on smooth manifolds and I am struck on this problem Let V be a n-dimensional vector space and $w_1,...,w_k \in A^{1} (V)$. Show that {$w_1,...,w_k$} are ...
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Alternating 3-tensor and $Alt(f)(v_1,v_2,v_3)$ [closed]

This question was left as an exercise in the lecture on tensors( Course is on manifolds) and I am unable to solve it. Question: Let V be a vector space. If f is an alternating 3-tensor on V and $v_1,...
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Tensors and the gradient - how to see that the gradient if formed by covariant elements and basis.

In Introduction to Tensor Analysis and the Calculus of Moving Surfaces by Pavel Grinfeld the gradient is addressed as a problem to be solved because both the coefficients and the basis vectors are ...
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Any 2-tensor = Sum of a symmetric 2-tensor + an alternating 2-tensor

The following problem was asked in my assignment of linear algebra and I was not able to solve this. I am looking for hints here. Problem: Show that every 2-tensor can be uniquely written as sum of a ...
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non-degenerate 2 forms, dimension and wedge product

Consider the problem from my assignment on tensor forms: Problem: We say that a 2-form n on a vector space V is non-degenerate of for $u\in V$ , n(u,v)=0 for all v$\in V$ implies u=0. Let w be a non-...
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pullback map on k-tensors of a linear isomorphism and finding it's inverse

This question was asked in an assignment on k-tensor forms and I am having hard time solving this problem. I have been following my class notes which are based on the book of john lee. Question: Let V,...
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How do I simplify $\delta_{ij} \delta^{jk}$?

How do I simplify $\delta_{ij} \delta^{jk}$? I know that $\delta_{ij} \delta_{jk}=\delta_{ik}$, but what do I do if the there's a Kronecker Delta symbol with upper indices and one with lower indices?
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How can I find the tangent projection of a tensor field onto a surface orientation tensor?

Given a second order tensor that represents the orientation distribution of a set of vectors (specifically surface normals), $S_{ij} = \oint \hat n_i \hat n_j dA$, where $\hat n_i$ are surface ...
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Compute a transformation law for k-tensor field

Let $A$ be a smooth covariant $k$-tensor field on a smooth manifold $M$. In two overlapping charts $(U, x^i)$ and $(U, \tilde{x}^i)$ I can write $A$ using the these local coordinates as $A = A_{i_1 ......
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3 votes
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Why is this called the conformal tensor?

Suppose you have a Riemannian manifold $(M, g)$ and a diffeomorphism $f: M \longrightarrow M$. Define $E^f:= f^*g - g$. I can understand that $E^f$ defines a $(0, 2)$-tensor which in a sense keeps ...
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pullback map on k-tensors

This question was asked in an assignment of k-tensors and I struck on them. I am having a hard time solving questions related to pullback maps. I have followed my notes which are based on the textbook ...
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Checking Metric Tensor relations

I need to check some simple tensor relations for continuing my calculations. Please write me if you think anyone is incorrect. P.S.: I know this is very simple question, but i need to be assured about ...
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