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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36 views

Two forms of application of metric tensor to get differential length

I'm reading the monograph "Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity" by Joseph Kolecki, now retired, of NASA. I have a ...
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312 views

expanding tensor notation of Navier stokes equation

I'm trying to expand the variable density and viscosity Navier-stokes equation for incompressible flows but I've had no luck so far. The Navier-Stokes in tensor notation is: $$ \rho \dfrac{Du_i}{Dt} ...
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1answer
73 views

Decomposition of Tensor into a Product of Tensors

I was working on a text that made use of the assumption that, for some rotation matrix $a_{ij}$, and some tensor $U$, the tensor under rotation is represented by $U'_{\alpha}=a_{\alpha i}U_i$. It made ...
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136 views

Double dot product in Cylindrical Polar coordinates - Strain energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$ 2W = σ_{ij}ε_{ij} $$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
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1answer
82 views

Differentiation of scalar fields using tensor notation

I'm learning tensor calculus to understand differential geometry. Please verify if I've understood how to employ Einstein's sum convention and index notation correctly. Suppose that $\varphi := ...
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1answer
67 views

Interpretation of $(r,s)$ tensor

A tensor of type $(r,s)$ on a vector space $V$ is a $C$-valued function $T$ on $V×V×...×V×W×W×...×W$ (there are $r$ $V$'s and $s$ $W$'s in which $W$ is the dual space of $V$) which is linear in each ...
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109 views

Why abstract index notation should not be confused with the Ricci calculus?

Considering this answer, it is mentioned that the range of indices $a, b, c,\dots$ are seen as abstract and coordinate-free and linear operations can be represented with them; and the range of indices ...
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256 views

How to prove that the Kronecker delta is the unique isotropic tensor of order 2?

Is there a way to prove that the Kronecker delta $\delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = \lambda \delta_{ij}$ ...
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39 views

Is it necessary for a linear map to be an automorphism to allow polar decomposition?

Bowen and Wang's Introduction to Vectors and Tensors I (pg. 168) states a general form of the polar decomposition theorem as Every automorphism A has two unique multiplicative decompositions $$ ...
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2answers
87 views

$\nabla \cdot (\mathbf{B}\mathbf{B} - \frac{1}{2}B^2 \tilde{1})=(\nabla \cdot \mathbf{B})\mathbf{B} - \mathbf{B} \times (\nabla \times \mathbf{B})$

Does someone know how to show this identity? $\nabla \cdot (\mathbf{B}\mathbf{B} - \frac{1}{2}B^2 \tilde{1})=(\nabla \cdot \mathbf{B})\mathbf{B} - \mathbf{B} \times (\nabla \times \mathbf{B})$
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1answer
100 views

Using metric to raise and lower indices

Everything I read on tensors makes it clear that using the metric matrix $g_{ab}$ and its inverse $g^{ab}$ to respectively lower and raise indices of a tensor is very important. As far as I know (and ...
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3answers
32 views

Tensor tranformation between basis?

If I am the basis vector $e_i$ into another basis to get $e'_j$ I use: $$e'_j=S_{ij}e_i$$ My text book says that $S_{ij}$ is the ith component of the vector $e'_j$ with respect to the unprimed basis. ...
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2answers
108 views

Kronecker delta symmetry?

This is quite a simple question for the Kronecker delta does $$\delta_{ij}=\delta_{ji}$$ My textbook implies it does but does not actually state it and hence am looking for clarification.
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1answer
21 views

Show that getting parallel transported does not change angle between them

I must tell you that I have never seen this kind of question in Tensor Analysis. Our professor had set up this question in our exam, but I don't know whether it belongs to Tensors or not. The question ...
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1answer
87 views

Which Subspaces do Antisymmetric Tensors Represent?

So antisymmetric tensors represent volumetric subspaces (I've asked this here instead of on phys.stackexchange because it seems like more of a math question)? How exactly would one know WHICH ...
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50 views

Relation between componenet and algebraic definition of covariant vectors

I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its ...
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3answers
86 views

Can you illustrate the use of coordinate-free notations that serve as an alternative to Einstein summation notation with an example?

"Abstract index" and "coordinate free notations" are often submitted as alternatives to Einstein Summation notation. Could you illustrate their use using an example? Here's a sum written in ...
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85 views

Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis i , j , k so by invariance nature of vectors, component of gradient ...
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1answer
115 views

Space of Alternating $k$-Tensors Notation

I will be taking a Differential Geometry class in the Fall, so I decided to get somewhat of a head start by going through Spivak's "Calculus on Manifolds." Before reading, though, I saw the Addenda at ...
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3answers
418 views

Are vectors and covectors the same thing?

In Euclidean space, we usually don't distinguish between vectors and covectors (or dual vectors or 1-forms or whatever you want to call them) -- because the spaces overlap. However, a physicist ...
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1answer
156 views

Invariant tensors in adjoint representation

Suppose we have a simple Lie group $G$ with algebra $\mathfrak{g}=\{X_a\}$, where the generators $X_a$ are in some matrix representation. Is it true that the only invariant rank $n$ tensor in the ...
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1answer
78 views

8th order isotropic tensor

Does anyone knows what is the general form of an $8$th order isotropic tensor? $2$th order is $\delta_{ij}$, $4$th order it is $\lambda \delta_{ij} \delta_{kl}+\mu(\delta_{ik} \delta_{jl} + ...
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1answer
53 views

formula of square of the covariant derivative

I am stuck with the calculation of $(\nabla ^2 \beta)(X,Y,Z_1,\dots,Z_r)$. In the following, capital letters are arbitrary vector fields. Suppose $\beta$ is an $(r,0)$ tensor. Denote $(\nabla ...
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1answer
74 views

Change of Eigenvalues of Ellipsoid Tensor during Rotation

I have an ellipsoid defined by the semiaxes $a,b,c$ and the orthonormal vectors $v_x, v_y, v_z$ describing the directions in which the axes point. The matrix ...
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1answer
47 views

A seemingly counterintuitive result on active and passive transformations of vectors

Let $\mathbf{v}$ be an element of a vector space with Euclidean $R^3$ as the underlying set. Assume the standard Cartesian basis $\{\mathbf{e^{(1)}, e^{(2)}, e^{(3)}}\}$ on it. Let $\mathbf{v^* = R ...
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2answers
177 views

Classical tensor analysis and Tensors on Manifolds

I learned tensors the bad way (Cartesian first, then curvilinear coordinate systems assuming a Euclidean background) and realize that I am in very bad shape trying to (finally) learn tensors on ...
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0answers
49 views

Let $K$ a field, $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor. Prove that $f \wedge f =0$

Let $K$ a field, with $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor ($f \in {\mathcal T}_n(V):=\Lambda^{n}(V)$), i.e., $f$ is an multilinear ...
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17 views

Decompose matrix multiplication?

Sorry, I can break my imagination... I would like to decompose matrix multiplication into some tensor-like operations. As we know, matrix multiplication is done the following way $a_i^j b_j^k$ ...
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1answer
179 views

Building intuition for tensors in machine learning

I'm trying to understand tensors in the context of machine learning, but all the resources that mention tensors that I've found so far were building the intuitions through physics applications. As ...
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1answer
98 views

A question about vector fields and divergence

I am reading the paper http://www.goshen.edu/physix/mathphys/gco/TensorGuideAJP.pdf in order to gain a basic understanding about tensors. I had some difficulties about understanding some definitions. ...
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1answer
119 views

Introductory questions about tensors

I am trying to understand the concept of tensors. I seem to understand that they are generalization of vectors: They are subject to similar basis transformations with vectors but I am somewhat ...
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49 views

Time Evolution of Deformation Gradient Tensor in Lagrangian Frame

I found the following proof in a paper: $\frac{D\mathbf{F}}{Dt} = \frac{D\frac{\delta\mathbf{x}}{\delta\mathbf{X}}}{Dt} = \frac{\delta\frac{D\mathbf{x}}{Dt}}{\delta\mathbf{X}}=\frac{\delta ...
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47 views

Dimension of the space of tensors obtained by making partial symmetrizations and skew-symmetrizations.

Let $A=(a_{i_1\dots i_k})_{i_1,\dots,i_k=1}^n$ be a higher order cubic tensor or hypermatrix. The following two facts are well-known and are easy to prove: ${(\bf 1) }$ The dimension of the ...
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73 views

Planetary motion integral

I was reading Planetary Motion (page 117) in Barry Spain's Tensor calculus, and stupidly enough, I didn't understand this. The equations are: $$\frac{d^2\psi}{d\sigma^2} + ...
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1answer
25 views

Boundedness of Riemannian curvature gradient

I'm reading a paper by Wan-Xiong Shi "Deforming the metric on complete Riemannian manifolds". And there is a statement without proof. It can be summarized as follows: Let $B(x_{0},\gamma)$ be a ...
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1answer
74 views

Question Symmetric 2-Tensors on Vector Fields

I have a question on a direct computation. How would one compute the following $$ (dx \odot dy ) \left(\frac{1+uv+xy}{1+xy} \frac{\partial}{\partial v}, \frac{\partial}{\partial y} \right) $$ and$$ ...
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78 views

Construct tensors from differential forms?

Let $(M,g)$ be a Riemannian manifold, differential forms are defined using tensors, could we define a tensor using a differential form? For example, if $\omega$ is a two-form on $M$ which is expressed ...
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1answer
22 views

sum representation by/ determinant of elementary tensors

Consider the bijective linear map: $\alpha : K^2 \otimes K^2 \to Mat(2 \times 2,K), \alpha(v \otimes w) = vw^t, v, w \in K^2$ , where $K$ is an arbritrary field. First I want to show that every ...
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1answer
55 views

Quick question about contravariant and covariant tensors

I have seen many different notations to denote contravariant/covariant and mixed tensors. For example, I think the notation $\omega^{v}_{\,\,\,\mu}$ stands for a mixed tensor, where one index ...
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1answer
33 views

Covariant vectors

As far as I'm aware, covariant vectors are defined by how they transform: But I've also heard that the covariant components of a vector are defined as the dot product of the vector and the various ...
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1answer
56 views

Tensor Notation

I'm just starting to learn about tensors, and have a question. I'm looking at the statement $\Lambda_{\mu}\,^{\alpha}= \eta_{\mu\nu}\eta^{\alpha\beta}\Lambda^{\nu}\,_{\beta}$ What is the difference ...
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29 views

Characterization of the derivative as a tensor field

I was thinking about the derivative, and I wanted to make sure I’m thinking about it the right way. Suppose we have a $C^{\infty}$ function $f: {V}\to \mathbb{R}$, where $V$ is a finite-dimensional ...
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1answer
118 views

what's the relationship of tensor and multivector

what's the relationship of multivector in geometric algebra and tensor? Is tensor a subset of multivector?
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5answers
201 views

Kronecker delta versus identity matrix

How should $\delta_k^j$ be regarded? Is it a scalar that takes on variable values? A 3x3 identity matrix (in 3 dimensions)? The wikipedia article on raising and lowering indices with the metric ...
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1answer
60 views

$(V^*)^{\otimes n} \cong (V^{\otimes n})^*$

We assume that $V$ is finite dimensional. Make $\theta: (V^*)^n\to (V^{\otimes n})^*$ by $$ \theta(\alpha_1,\cdots,\alpha_n)(v_1 \otimes \cdots \otimes v_n ) := \prod_{i=1}^n \alpha_i(v_i). $$ Then, ...
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148 views

matrix inverse in tensor notation

Suppose there is a matrix $A$ that transforms vectors, $$ Y = A x $$ Now express this in some other coordinate system, with $x = B z, \,\, y = B w$, so \begin{align*} & Bw = A B z \\ ...
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43 views

Would a set of tensors be an algebraic group closed under some operation?

Could a set of tensors be known as an algebraic group or why would that not have a group properties? The reason I'm asking is to understand different tensors.
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30 views

Why this doesn't transform properly?

We are in $\mathbb R^n $, with a tensor field of components $T_\nu$, and being $e_\mu$ the vectors of the basis: $e_\mu \equiv \partial_\mu$, then I'm asked to show that $\partial_\mu T_\nu$ can't ...
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26 views

Does the definition of the rank of a tensor change in the component-free treatment of tensors?

I was looking for the definition of the rank of a tensor. I found 2 different definitions depending on whether we use the component-free approach of tensors: http://en.wikipedia.org/wiki/Tensor: ...
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1answer
29 views

Linear transformation from endormorphism to real number

For a finite dimensional vector space $V$, is there a linear transformation between its endomorphism and real number, please? I suspect that since the element of the endomorphism can be represented by ...