Tensor Definition and 1000 Threads

How do I calculate the unit normal vector for any metric tensor?

Let ##L/k## be a field extension. Suppose ##F## is a finite separable extension of ##k##. Prove ##L\otimes_k F## is a semisimple algebra over ##k##.

I am reading about symmetries in crystals, and my knowledge in the field of group theory is almost nill. I am reading that, in the worst case, the electrical and thermal conductivity tensors can possess, at maximum, 6 different entries rather than 9, thanks to Neumann's principle which states...

Hi Pfs i have 2 matrix representations of SU(2) . each of them uses a up> and down basis (d> and u> If i take their tensor product i will get 4*4 matrices with this basis: d>d>,d>u>,u>d>,u>u> these representation is the sum equal to the sum of the 0-representation , a singlet represertation with...

I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in following his logic regarding proceeding to derive the components of Angular Momentum and from there the components of the Inertia Tensor ... On page 36 we read the following: In the above text...

My field is physics and I'm very cautious about the "math describing the Nature" attitude, but I can't help admiring the deep richness of mathematics! The other day, I was checking about isotropic tensors. An isotropic tensor keeps its components in all coordinated systems transformed under...

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How to calculate the contraction of second and fourth indices of Riemann tensor?I can only deal with other indices.Thank you！

I came across a statement in《A First Course in General Relativity》:“The only matrix diagonal in all frames is a multiple of the identity：all its diagonal terms are equal.”Why?I don’t remember this conclusion in linear algebra.The preceding part of this sentence is:Viscosity is a force parallel...

I came across a statement in《A First Course in General Relativity》on page 97 which confused me.It read:"if the forces are perpendicular to the interfaces,then##T^i{^j}##will be zero unless ##i=j##". Where the ##T## is stress-energy tensor,##T^i{^j}##is the flux of i momentum across the j surface.

(1,1)or(2,0)or(0,2)?And does a dual vector field have gradient?

ep_{ijkl} M^{ij} N^{kl} + ep_{ijkl}N^{ij} M^{kl} The second term can be rewritten with indices swapped ep_{klij} N^{kl}M^{ij} Shuffle indices around in epsilon ep{klij} = ep{ijkl} Therefore the expression becomes 2ep_{ijkl}M^{ij}N^{kl} Not zero. What is wrong here?

I want to get the stress energy tensor of a scalar field using the Hilbert method (namely, ##T^{\mu v} = \frac{2}{\sqrt{-g}} \frac{\delta S}{\delta g_{\mu v}}##) $$S = \int \frac{1}{2}(\partial_\mu \phi \partial^{\mu} \phi - m^2 \phi ^2)\sqrt{-g}d^4x$$ $$= \int \frac{1}{2}(\partial^{v} \phi...

Let's arrange the rod's axis parallel to the z axis. ##T_{00} = A/\mu## (since it represents the energy density) ##T_{03}=T_{30} = \frac{F\sqrt{\mu / F}}{A}## (It represents the flow of energy across the z direction) ##T_{33} = F/A## (pressure) It seems that ##T_{33}## i have got has the...

We mainly have to prove that this quantity## \bra{\varphi} A^{\otimes n } \ket{\varphi} \pm \bra{\varphi} B^{\otimes n } \ket{\varphi} ## is greater or equal than zero for all ##\ket{\varphi}##. Being ##\ket{\varphi}## a product state it is straightforward to demonstrate such inequality. I am...

Is tensor product the same as dyadic product of two vectors? And dyadic multiplication is just matrix multiplication? You have a column vector on the left and a row vector on the right and you just multiply them and that's it? We just create a matrix out of two vectors so we encode two...

I have the matrix $$ A = \left(\begin{array}{cc} y^2 & -xy\\ -xy & x^2 \end{array} \right) $$ I know that to prove that the matrix is a tensor, it transform their elements in another base. But I still without how begin this problem. Help please! Thanks.

hello, 1. according to Robert Wald, General Relativity, equation (4.2.22) the magnetic field as measured by an observer with 4-velocity ## v^b ## is given by ## B_a = - \frac {1}{2} {ϵ_{ab}}^{cd} F_{cd} v^b ## where ## {ϵ_{ab}}^{cd}##, the author says, is the totally antisymmetric tensor (for...

i am facing problems in the definition of dual oF some objects which has pair of anti symmetric indices e.g. Weyl curvature tensor. Double dual is there in the literature but given that how to find the anti self dual part of that. the problem is written in attached the file.

I am completely stuck on problem 2.45 of Blennow's book Mathematical Models for Physics and Engineering. @Orodruin It says "We just stated that the moment of inertia tensor ##I_{ij}## satisfies the relation$${\dot{I}}_{ij}\omega_j=\varepsilon_{ijk}\omega_jI_{kl}\omega_l$$Show that this relation...

According to Wikipedia, the definition of the Riemann Tensor can be taken as ##R^{\rho}_{\sigma \mu \nu} = dx^{\rho}[\nabla_{\mu},\nabla_{\nu}]\partial_{\sigma}##. Note that I dropped the Lie Bracket term and used the commutator since I'm looking at calculating this w.r.t. the basis. I...

I am a beginner in GR, working my through Gravitation by the above authors. If there is a better place to ask this question, please let me know. I understand (from section 5.7) that the stress-energy tensor is symmetric, and from equation 5.23 (p. 141), it is explicitly symmetric. But...

Galaxies are very large rotating bodies, so it seems that, as with the Kerr model for black holes, there could be an effect of this global rotation on the energy momentum tensor in the more dense regions of the galaxy that could in turn affect the space-time in the vicinity of the object and so...

How do you go about writing down the energy momentum tensor for the 2-body problem. Just looking for the approach.

I am in a course in applied strength of materials and we often use the 3D stress tensor for stress analysis of materials i.e. Mohr's circles, bending, torsion, etc. Is the stress-energy tensor in relativity basically a 4-d extension to the Cauchy stress tensor commonly used in mechanical...

When we compute the stress energy momentum tensor ## T_{\mu\nu} ##, it has units of energy density. If, therefore, we know the total energy ##E## of the system described by ## T_{\mu\nu} ##, can we compute the volume of the system from ## V = E/T_{00}##? If it holds, I would assume this would...

How to write following equation in index notation? $$\nabla \cdot \left( \mathbf{e} : \nabla_{s} \mathbf{u} \right)$$ where ##e## is a third rank tensor, ##u## is a vector, ##\nabla_{s}## is the symmetric part of the gradient operator, : is the double dot product. The way I approached is...

1.) I have the following equation $$\nabla \cdot \left( \mathbf{A} : \nabla_{s}\mathbf{b} \right) - \frac{\partial^2\mathbf{c}}{\partial t^2} = - \nabla \cdot \left( \mathbf{D}^{Transpose} \cdot \nabla \phi \right )$$ Is my index notation correct? $$(A_{ijkl} b_{k,l}),_{j} - c_{i,tt} = -...

If I have an equation, let's say, $$\mathbf{A} = \mathbf{B} + \mathbf{C}^{Transpose} \cdot \left( \mathbf{D}^{-1} \mathbf{C} \right),$$ 1.) How would I write using index notation? Here A is a 4th rank tensor B is a 4th rank tensor C is a 3rd rank tensor D is a 2nd rank tensor I wrote it as...

Hello all, I am hoping to get some feedback on the manner in which I performed computations towards solving the following problem. There are a couple specific points which I am not confident of: 1. Did I properly account for the manifold structure in my computation of the nonzero components...

In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation) ## \nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j} ## I'm trying to prove that this covariant...

I'm still confused about the notation used for operations involving tensors. Consider the following simple example: $$\eta^{\mu \sigma} A_{\mu \nu} = A_{\mu \nu} \eta^{\mu \sigma}$$ Using the rules for raising an index through the (inverse) metric tensor ##\eta^{\mu \sigma}## we get...

In Gravitation by Misner, Thorne and Wheeler (p.139), stress-energy tensor for a single type of particles with uniform mass m and uniform momentum p (and E = p2 +m2) ½ ) can be written as a product of two 4-vectors,T(E,p) = (E,p)×(E,p)/[V(E2 – p2 )½ ] Since Einstein equation is G = 8πGT, I am...

What is the index notation of divergence of product of 4th rank tensor and second rank tensor? What is the index notation of divergence of 3rd rank tensor and vector? div(a:b) = div(c^transpose. d) Where a = 4th rank tensor, b is second rank tensor, c is 3rd rank tensor and d is a vector.

It's given as ##T_{\mu \nu} = - \mathrm{tr}(F_{\mu \lambda} {F_{\nu}}^{\lambda} - \frac{1}{4} g_{\mu \nu} F_{\alpha \beta} F^{\alpha \beta})##. Can somebody explain the notation, i.e. what is the meaning here of the trace? (usually I would interpret the trace of a matrix as the number...

Hi all, I am currently trying to prove formula 21 from the attached paper. My work is as follows: If anyone can point out where I went wrong I would greatly appreciate it! Thanks.

We know that the cofactor of determinant ##A##, is $$\frac{\partial A}{\partial a^{r}_{i}} = A^{i}_{r} = \frac{1}{2 !}\delta^{ijk}_{rst} a^{s}_{j} a^{t}_{k} = \frac{1}{2 !}e^{ijk} e_{rst} a^{s}_{j} a^{t}_{k}$$ By analogy, $$\frac{\partial Z}{\partial Z_{ij}} = \frac{1}{2 !}e^{ikl} e^{jmn}...

I've started reading up on tensors. Since this lies well outside my usual area, I need some clarifications on some tensor calculus issues. Let ##A## be a tensor of order ##j > 1##. Suppose that the tensor is cubical, i.e., every mode is of the same size. So for example, if ##A## is of order 3...

Goldstein pg 192, 2 edIn a Cartesian three-dimensional space, a tensor ##\mathrm{T}## of the ##N## th rank may be defined for our purposes as a quantity having ##3^{N}## components ##T_{i j k}##.. (with ##N## indices) that transform under an orthogonal transformation of coordinates...

Hi, I would like to ask for a clarification about the difference between a differential k-form and a generic (0,k) tensor field. Take for instance a (non simple) differential 2-form defined on a 2D differential manifold with coordinates ##\{x^{\mu}\}##. It can be assigned as linear combination...

What's the derivative of deformation gradient F w.r.t cauchy green tensor C, where C=F'F and ' denotes the transpose?

Hi everyone! I'm having some difficulty showing that the variation of the four-velocity, Uμ=dxμ/dτ with respect the metric tensor gαβ is δUμ=1/2 UμδgαβUαUβ Does anyone have any suggestion? Cheers, Rafael. PD: Thanks in advances for your answers; this is my first post! I think ill be...

Let ##\varphi## be some scalar field. In "The Classical Theory of Fields" by Landau it is claimed that $$ \frac{\partial\varphi}{\partial x_i} = g^{ik} \frac{\partial \varphi}{\partial x^k} $$ I wanted to prove this identity. Using the chain rule $$ \frac{\partial}{\partial x_{i}}=\frac{\partial...

I need to vary w.r.t ##a_{\alpha \beta} ## ##\frac{\partial L}{\partial_{\mu}(\partial_{\mu}{a_{\alpha\beta}})}-\frac{\partial L}{\partial {a_{\alpha \beta}}}## (1) I am looking at varying the term in the Lagrangian of ##\frac{1}{3}A^{\mu} \partial_{\mu}\Phi ## where ##A^{\beta}=\partial_k...

Hello I am reviewing the proof of Cauchy's formula for the stress tensor and surface traction. Without exception, every book I look at gets to the critical point of USING the projection of a triangle onto one of the three orthogonal planes. However, I have never seen this proven. I have...

According to my book, the equation that should meet a vector ##\mathbf{v}=v^i\mathbf{e}_i## in order to be parallel-transported in a manifold is: ##v_{, j}^{i}+v^{k} \Gamma_{k j}^{i}=0## Where ##v_{, j}^i## stands for ##\partial{v^i}{\partial y^j}##, that is, the partial derivative of the...

Homework Statement:: Book suggestion Relevant Equations:: Calculus Book suggestion for tensor calculus.

The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...

I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...