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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81 views

Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
7
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98 views

Determinant of a tensor

Is there such a thing as the determinant of a tensor of rank $\gt 2$? I'm tried to think how it might be defined -- potentially like, the determinant of the tensor $A=a_{ijk}$ is ...
6
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152 views

Physical components of a third-order tensor

Aris' book Vectors, Tensors, and the Basic Equations of Fluid Mechanics describes how to convert between covariant, contravariant, and physical components of vectors and tensors. For example, in ...
6
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175 views

What is the difference between tensor calculus and exterior derivative type concepts?

I am trying to clarify terms in order to help me figure out what I'd like to study. I understand that $p$-forms and $p$-vectors are used with things like wedge products, exterior algebras, and a ...
6
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166 views

Mnemonic device for relationships between Hom and Tensor

Probably this is a stupid question, but nevertheless... Let $A$, $B$, $C$ and $D$ be rings, and $M$, $N$ and $K$ be appropriate bimodules between them. There are extremely well-known canonical ...
5
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81 views

Decomposition of order-$n$ tensors

If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$. The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is ...
5
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64 views

The metric and Kronecker's delta

I am reading some lecture notes for GR and it is currently showing how we are going to derive the field equations using a metric for a massive free particle with a metric ...
5
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93 views

Elastic wave equation in curvilinear coordinates: how do you perform a coordinate change?

The essence of this question is that I don't know how to convert an equation from Cartesian coordinates into curvilinear coordinates, and would like to know how, preferably using the language of ...
5
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178 views

What are spinor fields?

For example, a tensor field $T=T_{a,b}^{\ \ \ \ c}$ on a manifold $M$ should be thought as $$ T_{a,b}^{\ \ \ \ c}dx^{a}\otimes dx^{b}\otimes \frac{\partial}{\partial x^{c}} $$ with local coordinate ...
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57 views

Matrix Calculus

My school offers Matrix Calculus class next semester. I have never heard about this subject before and got intrigued. After a short chat with professor I found myself unable to get rid of suspicion ...
4
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77 views

Is there a method of knowing which of the Christoffel symbols of second kind survive and vanish for a given metric?

I'm trying to solve a problem and the given $g_{ij}$ metric is $$\left[\begin{array}{cc}1&0&0\\0&(x^1)^2&0\\0&0&(x^1 \sin x^2)^2\end{array}\right]$$ The non-zero Christoffel ...
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177 views

What is tensor, really?

How can one understands the definition of tensor from the purely formal point of view? To what abstract structure this concept can be generalized?
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398 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
4
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162 views

Use of graph theory to determine tensor contraction ordering

I am considering using a computer program to execute tensor contractions like the following: $\displaystyle \sum_{ij}^{o} \sum_{ab}^{v} \sum_{KL}^{X} B_{ia}^{K} B_{ia}^{L} B_{jb}^{K} B_{jb}^{L} $ ...
4
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0answers
39 views

Is $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in $R^n$?

Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors. Then define $H^{\alpha}_{ij,k}=e_k\langle ...
4
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245 views

Better Tensor Notation

I am in a General Relativity class, and I am finding the usual tensor notation very difficult to think about -- it seems like there are too many names to express something simple. E.g., I think of ...
3
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59 views

Understanding tensors

Locally in a chart, a tensor field looks like $$T= T^{i_1,...i_n}_{j_1,...,j_m} dx^{j_1} \otimes...\otimes dx^{j_m} \otimes \partial_{i_1} \otimes ... \otimes \partial_{i_n},$$ where ...
3
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56 views

Difference of two connections is a tensor

I am currently reading through Jost's Riemannian Geometry and Geometric Analysis and am seeking clarification of the statement in the title. Jost abstractly defines a connection on a vector bundle ...
3
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80 views

Computing volume element in spherical coordinates

Suppose $y = (r, \theta^1, \theta^2)$ are spherical coordinates in $(\mathbb{R}^3,g)$. What is the $d\text{vol}$ in these coordinates? I solved it but I don't know if it's right. My solution: We ...
3
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96 views

Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
3
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134 views

Prove the Curvature Tensor is a Tensor

For an affine connection $\nabla$, prove the curvature R $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$ with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is ...
3
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0answers
125 views

Intuition about the Riemann and Ricci tensors

I have started an attempt to self-study Riemannian Geometry. I well understand all the algebraic properties of the Riemann tensor (With a symmetric connection), and how it gradually becomes less and ...
3
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76 views

dot product between vector and matrix

In my book on fluid mechanics there is an expression $$ \boldsymbol{\nabla}\cdot \boldsymbol{\tau}_{ij} $$ where $\boldsymbol{\tau}_{ij}$ is a rank-2 tensor (=matrix). Given that ...
3
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175 views

Multilinear or Tensor Regression?

Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the ...
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48 views

Equivalent definitions of tensors on finite dimensional spaces.

I have recently been studying various texts on differential geometry, and I am quite puzzled that various authors define the notion of a tensor quite differently. I have come across the following ...
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300 views

Gradient and Einstein summation

Suppose I have a vector in the covariant basis $\bar e_i$ and a metric $g$ and I want to obtain a unit vector $\bar u_i$ parallel to $\bar e_i$. I would write: $$\bar e_i = \hat u_i ||\bar e_i|| = ...
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0answers
26 views

Does $\mathfrak T^r(\Bbb R^m)$ count as an vector space?

Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the ...
2
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24 views

Natural bilinear map $B\colon Alt^p(E^*)\times Alt^p(E)\rightarrow\mathbb R$

$Alt^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\ \ , u \text{ is alternating multilinear map}\}$ $Alt^P(E):=\{ \alpha\colon ...
2
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0answers
34 views

Pullback maps and an equallity

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
2
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0answers
14 views

Curvature of $K$-invariant connection (principal bundles)

Here is a proposition from Kobayashi & Nomizu's Foundations of Differential Geometry. I don't understand how they obtain the final line of the proof. They write: \begin{align} ...
2
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0answers
22 views

Hodge star of second-rank antisymmetric tensor

Say we have a tensor $F$ which just for familiarity's sake, we take to be a second rank antisymmetric tensor. I understand that given the Hodge star operator defined as ...
2
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38 views

Tensor transpose notation

I have a rank 3 tensor $\mathbf{Q}$. What notation should I use to denote the transposition of two of the dimensions? For instance, if I want to transpose the first and second dimensions, one way I ...
2
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0answers
29 views

$T(M) \cong R<x_1,\ldots,x_n>$ isomorphism question

I have that $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$ where $T^k(M) = \bigotimes_{i =1}^{k} M$. In a paper i have that to prove such isomorphism, we define: $$\Phi:R<x_1, \ldots, x_n> ...
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0answers
19 views

An extension of change of variables in double (and $n$-?) integrals - second-order Jacobian?

I'm aware that there are many, many questions regarding changing variables in double and triple integrals. The equation that typically pops up in textbooks is \begin{align} ...
2
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25 views

Tensor notation in vectors

I have the following expression $\partial_{x_a}(\partial_{x_b} \rho \partial_{x_b}\rho) - \partial_{x_b}(\partial_{x_a}\rho\partial_{x_b}\rho)$ How do I write this in vector notation? At least the ...
2
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57 views

Connections and Ricci identity

Given $\nabla$ a torsionless connection, the Ricci identity for co-vectors is $$\nabla_a \nabla_b \lambda_c - \nabla_b \nabla_a \lambda_c = -R^d_{\,\,cab}\lambda_d.$$ Prove $R^a_{[bcd]} = 0$ by ...
2
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48 views

Natural Isomorphism between $V^*\otimes W^*$ and $\mathcal L^2(V,W; F)$.

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and ...
2
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117 views

Geometric meaning of $\nabla_{[i}(x^i \nabla_{j]}x^j)$ and $(\nabla_{[i}x^i )\nabla_{j]}x^j$

While teaching myself tensor calculus I have come up with [this] http://mathematica.stackexchange.com/a/71613/12306 {The proof of the 2-D hairy ball theorem). When trying to generalize this proof ...
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123 views

Tensor Calculus Second Order Derivatives

I'm learning tensor calculus by myself through lectures and texts, and I'm presented with the problem of finding the first and second order derivatives of a scalar function of three variables that ...
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31 views

Objects corresponding to Higher forms

If $Q$ is a quadratic form, then we know there exists matrix $A$ such that $Q=xAx'$ and $Q$ can be expressed as weighted sum of eigenvalues of $A$. If $H$ is a higher order form, then is there an ...
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52 views

Bivector into orthogonal components

Suppose I have a metric $g$ and a bivector $ F $ on a four-dimensional vector space. It seems I can always decompose $ F $ into four mutually orthogonal vectors $a,b,c,d$ $$ F = a\wedge b + c\wedge d ...
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94 views

Calculating Hydrodynamic Interaction Tensor

I'm a bit of a newbie when it comes to Tensor calculus. Please excuse me as I learn... Given the Oseen tensor, $\mathbf{T}(\mathbf{R}) = (8\pi \eta R)^{-1} \left[ \mathbf{I} + ...
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51 views

Objectivity or frame invariant

I am not sure it is maths or physics question. A stress tensor of a nematic liquid crystal is given as follows $$ T_{ij}=-P\delta_{ij}-n_{k,i}n_{k,j}+\widetilde{T_{ij}} $$ where ...
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114 views

Converse of Euler Homogeneous Thm. How to show that $\lambda \mathbf{x}\cdot \frac{d}{d\lambda}(\nabla{f(\mathbf{\lambda x})})=\mathbf{0}$?

So basically I read someone else's answer to a question regarding Euler Homogeneous function theorem source: http://quant.stackexchange.com/questions/8911/what-is-exactly-eulers-decomposition Also ...
2
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77 views

Unnecessary Elements in the Tensor Product?

For vector spaces $U, V$ there exits a unique (up to isomorphism) vector space, denoted by $U \otimes V$, and a bilinear map $\eta : U \times V \to U \otimes V$ such that for every bilinear map $\xi : ...
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55 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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179 views

Tensor derivatives of a second order tensor function with respect to itself

In a continuum mechanical context, let $\mathbf{A}$ be a second order tensor function. We consider our operations in an orthonormal context $$\mathbf{A} = A_{ij}\, ...
2
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0answers
36 views

Index notation interpretation

I'm having some confusion with index notation and how it works with contravariance/covariance. $(v_{new})^i=\frac{\partial (x_{new})^i}{\partial (x_{old})^j}(v_{old})^j$ $(v_{new})^i=J^i_{\ ...
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53 views

Hessian matrix of $g\circ f$

Say, $f:\mathbb R^n\to\mathbb R^k$ and $g:\mathbb R^k\to\mathbb R$ are both $C^2$. I'd like to express the Hessian matrix of $g\circ f$ $$\left( \frac{\partial^2(g\circ f)}{\partial x_i \partial ...
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82 views

Covariant vectors and Dual spaces

There are contravariant vectors and their duals called covariant vectors. But the duals are defined only once the contravectors are set up because they are the maps from the space of contravariant ...