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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Text introducing $T^{i,j}$-tensor algebra

I'm reading a lecture note here : http://www.cis.upenn.edu/~cis610/diffgeom7.pdf It introduces $T^{•,•}(M)$ the tensor algebra and says that this is a necessary tool in differential geometry. Well, ...
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0answers
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Self-commuting of the covariant derivative: Menzel's Mathematical Physics

Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set ...
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0answers
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+50

Coordinate-free notation for tensor contraction?

I am not sure if I can prevent this question from being too vague or with too large an overlap with other similar math.SE questions, but I will do my best... A standard linear operation in tensor ...
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1answer
23 views

Tensors and invariance [on hold]

In geometry the study of invariants ( i.e. those entities that do not change their shape by linear transformations ) is of special interest. Examples of invariants concepts are vectors , curves, ...
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28 views

Parallel transport for infinitesimal displacement

I have a question about the following calculation about the parallel transport of an infinitesimal vector. I read the following text but I do not understand where the expression for the components of ...
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1answer
62 views

Strassen's Laser Method Technique AND Tensors in matrix multiplication algorithms

I understand the first algorithm presented by Strassen in 1968, for fast matrix multiplication. This was the first improvement to the naive approach of multiplying matrices. Thereafter, he went on to ...
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0answers
33 views

Levi-Civita symbol (permutation tensor)

I was going over a past exam and the following two questions came up: Show that the Levi-Civita symbol $\varepsilon_{ijk}$ is a tensor. Evauluate the following: ...
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1answer
25 views

Tensor derivative

What is the result of $$ \frac{\partial^2 \left(A^{ij}y^ix^j+B^{ij}x^iy^j\right)}{\partial \bf x\partial \bf y} $$ where $i,j$ obey Einstein summation convention, $A,B$ are constant, ${\bf ...
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29 views

Show a beam is in equilibrium given the stress tensor

Having some trouble with this: https://gyazo.com/0835bdaa8e01cb976765aac94555f6ef I know how to show that at x_2 = -h the surface traction is zero, but I'm not sure how to show it's in equilibrium? ...
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1answer
37 views

An example of tensor product

Let $$ \otimes:R\times R\rightarrow W $$ $$ f:R\times R\rightarrow R~~,~f(X,Y)=XY $$ $\otimes$ is tensor product, $W$ is a vector space, and $f$ is a bilinear may. As I know , we need to find a ...
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0answers
47 views

Why is lowering and raising index not affecting the value of a tensor?

I have two questions to ask about: For example I have a tensor $T(r,s)$ by which it means that $T$ operates on $r$ vectors and $s$ dual vectors. Take for example $T(3,0)$, so in this notation it ...
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0answers
38 views

Symmetry of Rank Two Tensor with Mixed Components

I understand that a rank two tensor (t) is classified as symmetric if $t^{ij} = t^{ji}$ or $t_{ij} = t_{ji}$. Later in my reading, I came across the following quote: It is not useful to speak of ...
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1answer
15 views

Defining covectors when the basis is oblique

Given a $2$-dimensional vector space with an oblique unit length basis, say, $(f_1, f_2)$, what is the dual vector or covector corresponding to $f_1$, call it $\hat f_1$? There appear to me to be ...
2
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0answers
28 views

Tensor Eigenstates

I have the following equation: $$f_i^´(t+1)=\sum_{jk}R_{i|jk}\tilde{f_j}(t)\tilde{f_k}(t)$$ It is about evolution of a population. I use this equation in my python program in the following way: ...
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5answers
87 views

In mathematics, what is an $N \times N \times N$ matrix?

In mathematics, what is an $N \times N \times N$ matrix? I think this is a tensor but definitions of tensors that I have read are so overly complicated and verbose that I have trouble understanding ...
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0answers
23 views

Raising index on covariant derivative

So suppose $X$ is some vector field and $t$ is a tangent vector to some curve on some smooth manifold. Then $t^a\nabla_a X$ gives the directional derivative of the vector field in the direction of ...
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1answer
15 views

What is the intuitive meaning of the partial derivate in coordinate transforms?

We learned that when changing coordinate system from $u^i$ to $u'^i$, a contravariant vector transforms like this (using the Einstein-convencion): $v'^i = \frac{\partial x'^i}{\partial x^j}v^j$, And ...
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1answer
351 views

Difference Between Tensor and Tensor field?

I don't understand the difference between tensor and tensor field. I'm learning from Barret O'neill's Semi-Riemann Geometry and here are the definitions: If $A:(V^*)^r \times V^s\to K$ ...
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2answers
188 views

Tensor Components

In Barrett Oneill's Semi-Riemann Geometry there is a definition of tensor component: Let $\xi=(x^1,\dots ,x^n)$ be a coordinate system on $\upsilon\subset M$. If $A \in \mathscr I^r_s (M)$ the ...
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2answers
30 views

Representing a linear transformation as a tensor

I understand that a linear transformation from a vector space $V$ to a vector space $W$ is a rank-$2$ tensor. What I would like some help with is how exactly to represent specific linear ...
0
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1answer
29 views

Tensor product, coordinates

Find the coordinates of bivector u⊗v with the respect to cannonical basis and basis M = ((1,2),(1,3)), u = (1,1) v=(1,-2). Please help, does it even have the solution? After the tensor multiplication ...
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1answer
41 views

Components of the metric tensor and its inverse

Let $g$ denote a metric tensor. Then Wald writes (in his book on general relativity): "The inverse of $g_{ab}$...is a tensor of type $(2,0)$ and could be denoted as $(g^{-1})^{ab}$. It is convenient, ...
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1answer
73 views

Confusion with abstract tensor notation

I am currently going through the abstract tensor notation in Wald's "General Relativity". I understand the purpose of it, but I need help understanding some of the conventions and definitions. So, ...
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2answers
59 views

Proving the Ricci identity

I'm trying to prove the Ricci identity Let $Z^a$ be a vector field, $R^a_{\,bcd}$ the Riemann curvature tensor and $\nabla$ a torsion-free connection. Then: ...
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0answers
22 views

Definition of the derivative of a 2nd-order tensor with respect to a scalar

The derivative of the (positive definite, symmetric, 2nd-order) tensor $\mathbf{C}(t)$ with respect to the scalar $t$ is defined as: $$ \frac{\partial \mathbf{C} }{\partial t} = \lim_{\Delta ...
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1answer
39 views

Tensor product of dual vectors and vectors

I am reading "General Relativity" by Wald. At first he defines a tensor of type $(k,l)$ to be a multilinear map $T: V^* \times \cdots \times V^* \times V \times \cdots \times V \rightarrow ...
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1answer
27 views

Tensor product with vectors

I just started reading Wald's "General Relativity" and I am on his section regarding tensors. He defines the outer product as an operation on tensors of type of $(k,l)$ and $(k', l')$ which gives a ...
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1answer
26 views

Foiling two tensor products

I have this problem in exterior algebra where I have a function B and B is defined in the following ways. $$ B\left( \left( \begin{array}{c} u \\ v \\ w \end{array} \right), \left( \begin{array}{c} ...
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2answers
2k views

Christoffel Symbols and the change of transformation law.

I have seen it written that the change of co-ordinate form is given by the following: $$ \tilde \Gamma^{i}_{jk} = {\partial \tilde x^i \over \partial x^\alpha} \left [ \Gamma^\alpha_{\beta ...
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1answer
1k views

How can I derive the back propagation formula in a more elegant way?

When you compute the gradient of the cost function of a neural network with respect to its weights, as I currently understand it, you can only do it by computing the partial derivative of the cost ...
2
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1answer
36 views

Basic question about the metric tensor

So I am reading Barrett O'Neil's Elementary Differential Geometry, which defines the metric tensor $g_p$ on a surface $M$ to be a bilinear symmetric positive-definite function on the set of ordered ...
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2answers
58 views

Equivalent definition of metric compatibility for a connection: what does $\nabla g$ mean?

So the definition I know for metric compatibility is: $$Xg(Y,Z)=g(\nabla_XY,Z)+g(Y,\nabla_XZ),$$ which make sense, as $g(Y,Z)$ is a smooth function from the manifold to reals and we think of $X$ as ...
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0answers
27 views

Question about properites of tensor product

For $A$, $B$ and $I$ being 2 by 2 matrices and $I$ being the identity, Is $((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$ = $(A\otimes I\otimes I)\otimes B + ...
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1answer
28 views

Show $\delta_{KL}$ is a Cartesian tensor [closed]

By using the definition of kronecker delta $\delta_{KL}$, show that $\delta_{KL}$ is a Cartesian tensor, that is $$\delta ' _{MN}=L_{MK}L_{NL} \delta_{KL}$$ under the rotation $X_K=L_{MK}X' _M$. I ...
2
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1answer
31 views

Prove the exterior derivative of the following (n-1) form is zero

Let $\omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n}$ be a differential $(n-1)$ form on ...
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4answers
11k views

Intuitive way to understand covariance and contravariance in Tensor Algebra

I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference ...
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2answers
66 views

Sum of low rank tensors

How high can the sum of $k$ low rank $m\times m\times\dots \times m$ tensors of rank $t$ be? Is there a good upper bound?
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1answer
20 views

When do exterior and tensor algebras commute with dual spaces?

Suppose $V$ is a vector space, and $V^*$ is its dual space. Furthermore, let $\Lambda(V)$ be the exterior algebra of $V$, and let $T(V)$ be the tensor algebra. When do the following two statements ...
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1answer
26 views

Finding the Gradient of a Vector Function by its Components

In Multivariable Calculus, we can easily find the gradient of a scalar function (producing a scalar field) $f : \mathbb{R^n} \to \mathbb{R}$, and the gradient function would produce a vector field. ...
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0answers
36 views

symmetric and alternating tensors in differential geometry

The following is an excerpt from Chern's Lectures on Differential Geometry: I don't see how the proof shows the other direction of the set inclusion. Would anybody explain the logic in the ...
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1answer
30 views

Levi civita and kronecker delta properties?

I'm trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I'm reading that have me stumped. 1) ...
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2answers
99 views

Geometrical Interpretation of Tensors (intuition)

How would one describe to a non-mathematician (an undergraduate physicist actually- so please do use mathematics-i am not even sure if they can be described without mathematics!) what do tensors ...
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1answer
12 views

How to rewrite $\frac{\partial \rho u_i u_j}{\partial x_j}$ in vector notation

I want to rewrite this index notation expression to a vector notation /symbolic notation. $$\frac{\partial \rho u_i u_j}{\partial x_j}=\frac{\partial \rho }{\partial x_j}u_i u_j+\rho\frac{\partial ...
1
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1answer
23 views

Lie derivative of two differnt size related tensors

Let $\bar{M}=I\times M$ be a pseudo-Riemannian manifold equipped with metric $\bar g=-dt^2\oplus f^2g$ where $(M,g)$ is a Riemannian manifold, $I$ is an open connected interval and $f$ is a positive ...
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1answer
43 views

Affine manifolds which are *not* locally symmetric

Let $M$ be a manifold equipped with an affine connection $\nabla$. By standard theorems of differential geometry we have convex normal coordinate balls around every point in which geodesics are axis. ...
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0answers
21 views

Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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1answer
28 views

What does this colon in tensor notation mean?

I was reading a paper earlier an found the following: "The tensors satisfy orthogonality $$ <S_{:,j,:,:}|S_{:,j',:,:}> =0 $$ if $j \neq j' $. Here $<S_{:,j,:,:}|S_{:,j',:,:}>$ is the ...
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1answer
28 views

Is this tensor contraction correct?

If I start off with the $\left(p,q\right)$-tensor given by $$T_{i_1,\dots,i_p}^{j_1,\dots,j_q}e_{i_1}\otimes\dots\otimes e_{i_p}\otimes\varepsilon^{j_1}\otimes\dots\otimes\varepsilon^{j_q}$$ and I ...
2
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1answer
46 views

Finding the Gradient of a Tensor Field

Finding the Gradient of a Scalar Field I understand that you can find the gradient of a scalar field, in an arbitrary number of dimensions like so : $$grad(f) = \vec{\nabla}f = ...
3
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1answer
79 views

What is the most general/abstract way to think about Tensors

In their most general and abstract definitions as Mathematical Objects : A Scalar is an element of a field used to define Vector Spaces A Vector is an element of a Vector Space. Since a Scalar ...