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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1answer
16 views

Index notation with non-commuting matrix entries

Just a contradiction I came across working with matrix multiplication in index notation: I'm probably using some rule wrong, but I can't figure out which one. Consider the expression $A_{ij} B_{ik}$, ...
3
votes
1answer
44 views

differential geometry : basic query about tensor notation and tensor products

I have a few very basic queries. I've been studying differential geometry as part of a course on General Relativity, so I don't have a very well grounded understanding of the mathematical formalism; ...
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0answers
11 views

Show that the trace of the operator $S \wedge T$ is zero

I have some difficulties with the following problem: Let $V$ be a finite dimensional vector space over $\mathbb{K}$. Let $S,T \in L(V,V)$. Show that the trace of the operator $S \wedge ...
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0answers
16 views

Question about tensor products, decomposable tensors, …

I need some help with the following problem: Let $V_1,\ldots,V_m$ be finite dimensional vector spaces over $\mathbb{K}$. Let $\varphi \in L(V_1,\ldots,V_m;U)$ such that $Im(\varphi)=U$. ...
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1answer
17 views

Field extension of a vector space

If $V$ is a vector space over the field $k$, and $K$ is a field extension of $k$, then $(V)_K$ over $K$ is a vector space. How this new vector space is constructed? and how are the linear ...
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0answers
16 views

How does the $10$ dimensional irrep (tensor) of $SU(3)$ look like?

We know that for $SU(3)$ the following tensors furnish the $\mathbf{d}$ dimensional irreducible representation: $$\phi^i\hspace{1cm} (\mathbf{3})\\ \phi^{ij}\hspace{1cm} (\text{asymmetric in ...
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0answers
16 views

Cauchy Equations and Navier Stokes

I'm attempting to take the Navier Stokes Equation and coming up with an expression that will allow me to numerically determine the velocity of non-Newtonian fluid flow. The text I'm using is Cengel ...
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0answers
29 views

Does there exists known special cases of a zero Riemann tensor for 3D metrics?

In two dimensions, if one has a flat metric $g_{ab}$, then one can transform $g_{ab}$ to another flat metric $h_{ab}=e^{2\varphi}g_{ab}$, when $\nabla^2 \varphi =0$ and the Riemann tensor remains ...
0
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1answer
23 views

Do we write a metric tensor as a matrix?

The metric tensor is an (0,2) tensor that is denoted by $g_{\mu\nu}$ in general relativity. I often see people write the metric field in matrix form like \begin{equation} g_{\mu\nu} = ...
-2
votes
1answer
23 views

Is the finding trace of the Riemann tensor the same thing as contracting two indices?

To form the Ricci curvature tensor, we have to take the trace of the Riemann tensor. But I also know \begin{equation} R_{ij} := R_{kij}^{\phantom{kij}k} \end{equation} Can someone show me why ...
3
votes
3answers
139 views

Proof in Hamilton: Divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} ...
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0answers
30 views

Generic rank of tensors

Let the tensor product of the type $$ \underset{k=1} { \overset{m} \bigotimes } v_k$$ denote a simple tensor. As underlying fields, take $$ \underset{k=1} { \overset{m} \bigotimes } ...
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0answers
25 views

Stiffnes tensor, Hooke's law

Let's have a deformed body of an isotropic homogenous material. How is it possible that we can write the free energy in the form $$F=F_0+\frac12\lambda\left(\sum_i ...
0
votes
1answer
35 views

coordinate transformation and tensor

A 2 dimensional Euclidean space is represented by two different coordinate systems: the Cartesian system $(x_1,x_2)$ and an alternative system $(\xi^1,\xi^2)$ where ...
1
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2answers
44 views

Question concerning tensors

As some of you may have seen from my previous question(s), I am working through Spivak's Calc on Manifolds and this happens to be the first time I've been introduced to tensors formally, though I have ...
2
votes
3answers
106 views

Why is a linear transformation a (1,1) tensor?

Wikipedia says that a linear transformation is a $(1,1)$ tensor. Is this restricting it to transformations from $V$ to $V$ or is a transformation from $V$ to $W$ also a $(1,1)$ tensor? (where $V$ and ...
1
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2answers
25 views

Rank of a Decomposable Tensor

I'm independently studying Stephen Roman's Advanced Linear Algebra, and I came across a line of reasoning that appears obvious but that I don't understand, and was hoping someone might help me ...
2
votes
1answer
47 views

Relationship between Levi-Civita symbol and Grassmann numbers?

The multiplication rule for Grassmann numbers $\theta_i$ is $$ \theta_i\theta_j = - \theta_j \theta_i $$ so that $\theta_i\theta_i = 0$. Multiplying three Grassmann numbers yields $$ ...
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0answers
48 views

Identity involving Riemann tensor

I'm reading about the Ricci tensor, and I've found the following statement that is given without proof: For a point $p$ on a Riemannian manifold, and coordinate vector fields $X_{\alpha}$, ...
0
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0answers
37 views

If a tensor space is a vector space, then why isn't a tensor a vector?

On page 78 of Tensor Analysis on Manifolds, Bishop and Goldberg state: Let $V$ be a vector space. The scalar-valued multilinear functions with variables all in either $V$ or $V^*$ are called tensors ...
7
votes
1answer
75 views

Tensors as Multilinear maps?

Today I learned about Tensors as multilinear maps. I usually think of tensors as a multidimensional array of numbers with fixed transformation laws, and I am having trouble understanding how tensors ...
6
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0answers
116 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 ...
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votes
3answers
21 views

Index Notation Divergence of f grad g

I'm struggling to figure this out. How do you write this in index notation? \begin{equation} \nabla \cdot (f\nabla g) \end{equation} I started with $$f\nabla g = f_k \partial_kg_j$$ but when I take ...
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0answers
43 views

Integral Curve of the vector field

If we have a 2-sphere with coordinates $x=r \cos \theta \cos \phi$, $y= r \sin \theta \sin \phi$ and $z=r \sin \theta$ and the vector field $X= (-r\sin \theta \cos\phi, r \cos \theta \sin \phi, r \cos ...
0
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1answer
37 views

3d tensor multiplication

I'm new to tensor theory, and I have a question. It's easy to know how to multiply a $2d$ tensor with another $2d$ tensor: $$A_{ij} \times B_{jk} = C_{ik}$$ However, can I multiply a $3d$ tensor with ...
2
votes
3answers
89 views

Covariant derivative geometric interpretation

I'm having some trouble understanding what the covariant derivative means geometrically. I know the definition which states that for a tensor T with any number of indices: $ \nabla_j T = ...
1
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1answer
56 views

Vectors and Tensors

I am currently studying Reliability engineering and hence need to deal with material properties like elasticity modulus and poisson's ratio. I am basically an electrical engineer and hence never had ...
0
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0answers
43 views

Local expression of a differential form

During the course, we have defined differential forms as maps $$ \omega : D(M) \times \cdots \times D(M) \rightarrow \mathcal{C}^\infty(M), $$ where $D(M)$ are the global derivates of the manifold ...
3
votes
1answer
64 views

4-Vectors and four-tensors

I want to show that if $\Gamma_{\mu \nu} a^{\mu} a^{\nu}$ is a scalar for any four-vector $a^{\nu}$, then $\Gamma_{\mu \nu}$ is a four-tensor. It is $a^{\nu} = g^{\nu \mu} a_{\mu}$, and so I would ...
2
votes
1answer
43 views

Inverse of covariant tensor of rank two is contravariant.

I'm studying tensors on my own, using "Tensor Calculus" from David C. Kay, and there is this theorem in page $29$: Suppose that $(T_{ij})$ is a covariant tensor of order two. If the matrix ...
2
votes
1answer
51 views

Multilinear algebra some basics.

The wedge product of $p$ vectors in vector space $V$ is called a $p$-vector and the vector space generated by all $p$-vectors is denoted $\bigwedge^p V$ with the basis $e_I:=e_{i1}\wedge\dots\wedge ...
1
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1answer
69 views

Is it possible for a manifold to have a normal vector that is zero everywhere, if so, would this indicate that the manifold is non-orientable?

Basically I've been thinking about defining a non-orientable three-dimensional metric space via defining the normal vector and looking to see if there is two possible vectors for the same point. I'm ...
0
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1answer
54 views

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions?

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions? Needed for the irrep decompositon of $3\otimes 3\otimes 3$ in here. No idea where to start to ...
2
votes
1answer
31 views

Integrals in Index Notation and Orientation

I am wondering what is the correct way to write integrals in index notation. At first I thought $$\int_M f \varepsilon_{ij}$$ would be the index equivalent of $$ \int_M f dx \wedge dy $$ but I started ...
2
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0answers
44 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 ...
4
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0answers
67 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 ...
6
votes
2answers
72 views

Does a $p$-form eat $p$-vectors or $p$ number of vectors?

A bilinear form is another term for a $2$-form. So does it eat $2$ distinct vectors or a single $2$-vector?
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0answers
70 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 ...
2
votes
1answer
41 views

Geometry of the Covariant Derivative

Taking the standard covariant derivative from tensor calculus $ \nabla_{\mu}T_{s} =T_{s|\mu}=T_{s,\mu} $. Could this be geometrically interpreted as the directional derivate of a tensor that has extra ...
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0answers
68 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 ...
8
votes
1answer
487 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 ...
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0answers
46 views

Order of Riemann tensor indexes and the Ricci Identity

I have seen the Ricci identity written variously as $R_{ijk}{}^l x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R_{ij}{}^l{}_k x^k = (\nabla_i\nabla_j- \nabla_j\nabla_i) x^l$ $R^l{}_{kij} x^k = ...
1
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1answer
28 views

Writing in Cartesian tensor form

Write the following in Cartesian tensor form $$(1) \nabla (\operatorname{div} G) \times \nabla\Omega$$ $$(2) (\operatorname{curl}(F)\times G)\cdot \nabla(Φ)$$ I have answers for these two questions, ...
0
votes
1answer
13 views

Definition of rank of an element in tensor algebra

Let $V$ be a vector space over a field $K$ and let $T(V)$ be its tensor algebra; that is, $$T(V)=\displaystyle\bigoplus_{k=0}^{\infty}V^{\otimes k}=K\oplus V\oplus V\oplus(V\otimes V)\oplus(V\otimes ...
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3answers
76 views

Intuitive transition from matrices to tensor-concept

I would like to know how to build intuition for the concept of a tensor using the following reasoning: If I conceive of a vector as an extension of the scalar concept, i.e. an $N \times 1$ "array of ...
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0answers
17 views

Matrix operation repeat matrix members

I am going to use C++ Armadillo library which handles matrices to generate matrix $B$ and $C$ from matrix $A$. $$ A=[M_0,M_1,\ldots,M_{n-1}]^T $$ $$ ...
4
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0answers
44 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 ...
2
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2answers
54 views

Raising and Lowering Through Differentiation

I'm calculating the Christoffel symbols of the second kind which is of course defined as multiplying the symbol of the first kind multiplied by the contravariant metric. I was thinking of how to make ...
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0answers
21 views

Extrinsic curvature tensors

I risk of sounding too vague, but I am interested if there are other tensors reflecting the extrinsic geometry of a submanifold other than the second fundamental form? The first fundamental form ...
2
votes
1answer
51 views

Riemannian curvature tensor of product manifolds

Let $(M_{1},g_{1})$ and $(M_{2},g_{2})$ be two Riemannian manifolds. Let $% R_{1}$ and $R_{2}$ be the (1,3)-type Riemannian curvature tensors of $M_{1}$ and $M_{2}$, respectively. Finally, let $R$ be ...