Tensor Definition and 1000 Threads

Hello, I have derived two Maxwell's equations from the electromagnetic field tensor but I have a problem understanding the second formula, which is: \partial_{\lambda} F_{\mu\nu} + \partial_{\mu} F_{\nu\lambda}+\partial_{\nu} F_{\lambda\mu} =0 I have a few questions to help me start: 1) Is...

In special relativity, the electromagnetic field is represented by the tensor $$F^{\mu\nu} = \begin{pmatrix}0 & -E_{x} & -E_{y} & -E_{z}\\ E_{x} & 0 & -B_{z} & B_{y}\\ E_{y} & B_{z} & 0 & -B_{x}\\ E_{z} & -B_{y} & B_{x} & 0 \end{pmatrix}$$ which is an anti-symmetric matrix. Recalling the...

hi, I looked up torsion tensor derivation on 2 different books, and encountered 2 different situations, so my mind has been confused. For the first image, I could totally understand how torsion tensor was derived, but for the second image although there are similar things, I can not make a...

hi, I am just curious about, and really wonder if there is a proof which demonstrates that curvature tensor is 0 in all flat space coordinates. Nevertheless, I have seen the proofs related to curvature tensor in Cartesian coordinates and polar coordinates, but have not been able to see that zero...

Without regard to a coordinate system (I only wish to consider special relativity) the stress-energy-momentum tensor defines a linear transformation from a 4-vector to a 4-vector. Let T be the linear transformation then b = T(a), a and b are 4-vectors. What is the physical meaning of a and b...

Tensor Products and the Free Z-module - Bland Proposition 2.2.3 ... ... I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.3 Tensor Products of Modules ... ... I need some help in order to fully understand the nature of the free Z-module...

I am reading Donald S. Passmore's book "A Course in Ring Theory" ... I am currently focussed on Chapter 9 Tensor Products ... ... I need help in order to get a full understanding of the free abelian group involved in the construction of the tensor product ... ... The text by Passmore...

http://hitoshi.berkeley.edu/221a/tensorproduct.pdf I was following the above pdf and got through most of it but am not quite understanding how (41), (42), and (43) are derived. It appears that (31) and (41) are representing the same states and are still orthogonal, but how exactly is (41)...

In Wald's "General Relativity", in his section on abstract tensor notation, he let's g_{ab} denote the metric tensor. When applied to a vector v^a, we get a dual vector, because g_{ab}(v^a, \cdot) is just a dual vector. Okay cool. But then he says that this dual vector is actually g_{ab}v^b...

They are being 2 by 2 matrices and I being the identity. Physically they are Pauli matrices. 1. Is $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$$ = $$(A\otimes I\otimes I)\otimes B + (I\otimes A\otimes I)\otimes B + (I\otimes I\otimes A)\otimes B$$? I...

What is meant by taking the tensor product of vectors? Taking the tensor product of two tensors is straightforward, but I am currently reading a book where the author is talking about tensor product on tensors then in the next paragraph declares that tensors can then be constructed by taking...

Hi all! I'm really interested in the physical interpretation of the Riemann and metric tensors. Is there any program that let's you type in a Riemann or metric tensor (or even better, an Einstein tensor) and then gives you an image of how the space would look (i.e. the curved grid lines)? Thanks!

Homework Statement Verify that ##\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}+\partial_{\lambda}F_{\mu\nu}## is equivalent to ##\partial_{[\mu}F_{\nu\lambda]}=0##, and that they are both equivalent to ##\tilde{\epsilon}^{ijk}\partial_{j}E_{k}+\partial_{0}B^{i}=0## and...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get an understanding of an aspect of Example 10.11 and Definition 10.7 in Section 10.3 ... The relevant text in...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get an understanding of an aspect of Example 10.11 and Definition 10.7 in Section 10.3 ... The relevant text in...

Homework Statement Imagine we have a tensor ##X^{\mu\nu}## and a vector ##V^{\mu}##, with components ## X^{\mu\nu}=\left( \begin{array}{cccc} 2 & 0 & 1 & -1 \\ -1 & 0 & 3 & 2 \\ -1 & 1 & 0 & 0 \\ -2 & 1 & 1 & -2 \end{array} \right), \qquad V^{\mu} = (-1,2,0,-2). ## Find the components of...

I am reading Dummit and Foote: Abstract Algebra (Third Edition) ... and am focused on Section 11.5 Tensor Algebras. Symmetric and Exterior Algebras ... In particular I am trying to understand Theorem 31 but at present I am very unsure about how to interpret the theorem and need some help in...

I am reading Dummit and Foote: Abstract Algebra (Third Edition) ... and am focused on Section 11.5 Tensor Algebras. Symmetric and Exterior Algebras ... In particular I am trying to understand Theorem 31 but at present I am very unsure about how to interpret the theorem and need some help in...

Homework Statement Maxwell's Lagrangian for the electromagnetic field is ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}## where ##F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}## and ##A_{\mu}## is the ##4##-vector potential. Show that ##\mathcal{L}## is invariant under gauge...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Example 10.1 in Section 10.3 ...Example 10.1 plus some preliminary definitions reads as...

If one has two single-particle Hilbert spaces ##\mathcal{H}_{1}## and ##\mathcal{H}_{2}##, such that their tensor product ##\mathcal{H}_{1}\otimes\mathcal{H}_{2}## yields a two-particle Hilbert space in which the state vectors are defined as $$\lvert\psi ,\phi\rangle...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Example 10.1 in Section 10.3 ...Example 10.1 plus some preliminary definitions reads as...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...Definition 10.5 plus some preliminary definitions reads as...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...Definition 10.5 plus some preliminary definitions reads as...

When finding the Von Mises of given a stress tensor who's only element is a single shear component (τ): \begin{bmatrix} 0 & τ & 0\\ τ & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} the result is simply √3×τ. Is the Von Mises criterion not valid when considering a single component as in this example? I...

Homework Statement The energy-momentum tensor ##T^{\mu\nu}## of the Klein-Gordon Lagrangian ##\mathcal{L}_{KG} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}## is given by $$T^{\mu\nu}~=~\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}.$$ Show...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Theorem 10.8 which is a Theorem concerning the direct sum of a family of subspaces as a...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.3 The Tensor Algebra ... ... I need help in order to get a basic understanding of Theorem 10.8 which is a Theorem concerning the direct sum of a family of subspaces as a...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help in order to get a basic understanding of Theorem 10.2 regarding the basis of a tensor product ... ... My apologies if my...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help in order to get a basic understanding of Theorem 10.2 regarding the basis of a tensor product ... ... My apologies if my...

A particle in a 1-D Hilbert space would have position basis states ## |x \rangle ## where ## \langle x' | x \rangle = \delta(x'-x) ## A 3-D Hilbert space for one particle might have a basis ## | x,y,z \rangle ## where ##\langle x', y', z' | x,y,z \rangle = \delta(x'-x) \delta (y-y') \delta(z-z')...

Hi, I understand stress, strain but when it moves on to 3 dimension anisotropic materials using tensors and stiffness matrices I get confused with einstein's notation. can someone please help me out in this regard to undrstand how stiffness and compliance matrices get reduced for monoclinic...

In the middle of the below paragraph: "only if the shift vector ##R## is along one of the principal axes relative to the center of mass will the difference tensor be diagonal in that system." I suppose the difference tensor means new inertial tensor ##-## old inertial tensor. That means the new...

Hello, Is the covariant conservation of the matter energy momentum tensor Tμν ; μ = 0 also valid in a theory of gravity having an action for the gravitational field different from the Einstein Hilbert action ? I'm asking because in GR the einstein field equations require Tμν= Gμν where Gμν;μ=0...

I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ... I have a basic question regarding the extension of the scalars ... Dummit and Foote's exposition regarding extension of the scalars reads as...

I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ... I have a basic question regarding the extension of the scalars ... Dummit and Foote's (D&Fs) exposition regarding extension of the scalars reads as...

Is there a simple geometric interpretation of the Einstein tensor ? I know about its algebraic definitions ( i.e. via contraction of Riemann's double dual, as a combination of Ricci tensor and Ricci scalar etc etc ), but I am finding it hard to actually understand it geometrically...

Homework Statement Suppose A and B are vectors. Show that the object Q with nine components Qij=AiBj is a tensor of rank 2. Homework Equations A tensor transforms under rotations (R) as a vector: Tij'=RinRjmTnm The Attempt at a Solution I wanted to just create the matrix, but I don't know how...

I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector. I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$ I can't figure out a way to get the required...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.2 Properties of Tensor Products ... ... I need help with an aspect of the proof of Theorem 10.3 ... ... basically I do not know what is going on in the second part of the proof...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.2 Properties of Tensor Products ... ... I need help with an aspect of the proof of Theorem 10.3 ... ... basically I do not know what is going on in the second part of the...

In the transformation of tensor components when changing the co-ordinate system, can someone explain the following: Firstly, what is the point in re-writing the indicial form (on the left) as aikTklajl? Since we're representing the components in a matrix, and the transformation matrix is also...

I'm having trouble with the mathematics of tensor products as applied to Bell states. Say I have the state \begin{align*} \left|\psi\right> &= \frac{1}{\sqrt{2}} \left(\left|0\right>_A \otimes \left|0\right>_B + \left|1\right>_A \otimes \left|1\right>_B\right) \end{align*} How would the...

Can someone explain mathematically why do we say Riemann Curvature Tensor has all the information about curvature of Space Thank You

I read Weyl tensor helps on propagating gravitational effects. Ricci is local depending on mass energy at that point and would vanish at other points. Weyl propogates the gravity effects (for example gravity at any point between Earth Moon is due to Weyl Tensor). I didn't quite get it...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.2 Properties of Tensor Products ... ... I need help with an aspect of the proof of Theorem 10.3 regarding a property of tensor products ... ...The relevant part of Theorem 10.3...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.2 Properties of Tensor Products ... ... I need help with an aspect of the proof of Theorem 10.3 regarding a property of tensor products ... ... The relevant part of Theorem...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as follows: I do not...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as...