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Definition of Irreducible Tensor Parts in an Exercise

I am addressing exercise 23.9 on http://www.pma.caltech.edu/Courses/ph136/yr2011/1023.1.K.pdf. The exercise says that a fluid flowing through spacetime $\vec u(\mathcal P)$ can have its gradient ...
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Stress Force - Understanding Cauchy Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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The definition of transpose of Lorentz transformation (as a mixed tensor)

In the appendix of the textbook of Group Theory in Physics by Wu-Ki Tung, the transpose of a matrix is defined as the following, Eq.(I.3-1) $${{A^T}_i}^j~=~{A^j}_i.$$ This is extremely confusing for ...
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Is partial derivative a vector or dual vector?

The textbook(Introduction to the Classical Theory of Particles and Fields, by Boris Kosyakov) defines a hypersurface by $$F(x)~=~c,$$ where $F\in C^\infty[\mathbb M_4,\mathbb R]$. Differentiating ...
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Learning how to use Levi-Civita symbol

I've recently started my second course in Quantum Theory and am now often required to prove more complex commutation relations. I'm aware that the Levi-Civita symbol often makes this sort of thing a ...
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Using positional index notation with tensors is common. For example, the following simple equation from Carroll's Spacetime and Geometry text (eq. 3.146): $$R = R^\mu_{\,\,\mu} = ... 1answer 153 views Proof that terms in decomposition of a tensor are symmetric and antisymmetric Any tensor of rank 2 can be rewritten as:$$A_{bc} = \frac{1}{2}(A_{bc} + A_{cb}) + \frac{1}{2}(A_{bc}-A_{cb})$$I can understand how that works. My question is: Prove that (independently): ... 1answer 181 views Vector fields and tensors in E&M I'm confused by a very basic property of electric fields. The electric field is a vector field. Vectors are tensors. Wikipedia has the following statement in the article about the electromagnetic ... 2answers 75 views Tensors of rotations about an arbitrary vector in C^2 I'm trying to solve the following equation:$$e^{-i\theta/2 \sigma_{\vec{i}}^A} \otimes e^{-i\theta/2 \sigma_{\vec{i}}^B} |\Psi\rangle_{AB} = e^{i\phi} |\Psi\rangle_{AB} $$where e^{i\phi} should ... 3answers 654 views How to tell that the electromagnetic field tensor transforms as a tensor? Is any matrix a tensor in special relativity? My question is inspired by the definition of the electromagnetic field tensor in Carroll's Spacetime and Geometry book. In equation (1.69), he defines a ... 1answer 92 views Tensor notation I'm trying to understand the Maxwell Stress tensor notation. I'm given that each element in the tensor is given by ... 1answer 101 views Why eigenvector points to principal stress plane? I can represent a tensor by a matrix. Suppose we are talking about a 2nd order tensor, and the matrix is therefore 3x3. If I find one eigenvector of that matrix; that vector represents normal vector ... 1answer 143 views Riemann Curvature Tensor Symmetries Proof I am trying to expand$$\varepsilon^{{abcd}} R_{{abcd}}$$by using four identities of the Riemann curvature tensor: Symmetry$$R_{{abcd}} = R_{{cdab}}$$Antisymmetry first pair of indicies ... 1answer 83 views Why is the full eigenfunction a product of eigenfunctions and not a sum? For example suppose there is a two electron system. Why is the full eigenfunction a product of the spatial eigenfunction and spin-wave-function for the two electron system? 0answers 57 views Relation involving the Lorentz transformation and the inverse of its transpose The relation I was referring to in the title is$${\Lambda_a}^b= \eta_{ac} {L^c}_d \eta^{db}$$where {\Lambda_a}^b is the inverse transpose of L, the Lorentz transformation. I was wondering ... 4answers 235 views Nature of Fields in QFT I'm not exactly an expert in quantum physics, but this seems to be a simple question, and I can't find an answer anywhere! There are specific types of fields used in physics: scalar fields (i.e. as ... 1answer 80 views Moment of inertia as a tensor A professor at my university briefly stated that moment of inertia is a tensor and can be represented by a 3×3 matrix. I don't have a good idea of what a tensor is, so I would be grateful if someone ... 2answers 142 views Basic understanding of stress tensors in a fluid So, after having spent the last 9 hours attempting to understand the basic tenets of stress tensors in fluids, I can honestly say that I think I know less now than when I began. My questions are ... 0answers 13 views Basic understanding of stress tensors in a fluid [duplicate] So, after having spent the last 9 hours attempting to understand the basic tenets of stress tensors in fluids, I can honestly say that I think I know less now than when I began. My questions are ... 1answer 246 views Tensor Product of a Bra and a Ket What does one get if the take the tensor product of a bra and a ket, for instance, \langle\uparrow \rvert \otimes \lvert \downarrow\rangle? What I mean it, what is this object? What does it act on? ... 1answer 122 views Riemann curvature tensor notation in Wald This question is entirely on tensorial notation in Wald's General Relativity. When specifying the properties of the Riemann tensor on pg39, he states: R_{[abc]}^{\quad \ \ \ d} = 0 and For the ... 1answer 88 views How to define tensor contraction without referring to summation? The textbook defines a tensor to be an element in (T^*)^k×T^l→R. It then expresses tensors as arrays of components with respect to a certain basis, and defines tensor contraction using summation ... 2answers 150 views Why does the second Weyl scalar describe electromagnetic radiation? I've been reading about the null tetrad, the Weyl tensor, and the Newman-Penrose identities, and so I found out about the Weyl scalars. While the zeroth, first, third, and fourth scalars describe ... 1answer 167 views Clarification of Tensor, Jacobian Is this correct? Tensors are linear mappings between two coordinate systems on a manifold. The elements of that mapping (which include the different changes of bases at each point of the manifold) ... 0answers 183 views Concepts in Gabriel Kron's later papers Gabriel Kron was an important research electrical engineer known for applying differential geometry and algebraic topology to the study of electrical system. Towards the end of his career he published ... 0answers 73 views Jacobian for Kronecker delta I was revising on a bit of tensor calculus, when I stumbled upon this:$$\delta^i_j = \frac{\partial y^i}{\partial x^\alpha} \frac{\partial x^\alpha}{\partial y^j}$$And the next statement reads, ... 3answers 562 views Is force a contravariant vector or a covariant vector (or either)? I don't understand whether something physical, like velocity for example, has a single correct classification as either a contravariant vector or a covariant vector. I have seen texts indicate that ... 1answer 79 views Considering the theory of special relativity: Is torque still a vector? Considering the theory of special relativity: Is torque still a vector? In classical mechanics it is easy: You have 3 axes and thus 3 planes. Every plane has its own torque so torque has 3 ... 2answers 452 views What is pseudo tensor? What is the pseudo tensor in relativity? How do we transform tensor and pseudo tensor under parity? 1answer 141 views Why can't we do some basic algebra in tensor calculus? I have a very, very stupid question on the basics of tensor calculus. Consider R_{ij} = 0. 1)If I expand the ricci tensor R_{ij}= g^{lm}R_{iljm}=0. Now, my question is that, why can't we divide ... 0answers 116 views Double dot product in Cylindrical polar coordinates - Strain Energy I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows:$$2W=σ_{ij}ε_{ij}$$Where σ and ε are symmetric rank 2 tensors. For cartesian ... 1answer 113 views Interpretations of (r,s) tensors [duplicate] A tensor of type (r,s) on a vector space V is a C-valued function T on V×V×...×V×W×W×...×W (there are r V's and s W's in which W is dual space of V) which is linear in each argument. We take (0, 0) ... 1answer 140 views Killing Equation, trouble with tensor algebra I'm attempting to follow a proof that the commutator of two Killing vectors is itself a Killing vector. The source that I've posted is from my course notes. I've highlighted the part I'm stuck on. ... 3answers 840 views How to understand the definition of vector and tensor? Physics texts like to define vector as something that transform like a vector and tensor as something that transform like a tensor, which is different from the definition in math books. I am having ... 0answers 44 views Just a contraction of indices I came across a contraction which is not giving the desired result. This is a toy problem in how to get a supergravity theory in low energy limit of a superstring theory using the vanishing of beta ... 1answer 93 views Ricci curvature tensor, definition of symbols So I know that$$R_{μν}:=R^λ_{μλν}$$is the Ricci curvature tensor (where R^λ_{μλν} is the Riemann Tensor). This is in Einstein's field equations:$$R_{μν}-\frac{1}{2}g_{μν}R=\frac{8πG}{c^4}Τ_{μν}$$... 3answers 246 views Relation between component and algebraic definition of covariant vectors I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its ... 4answers 602 views Gradient is covariant or contravariant? I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis i, j, k, so by invariance nature of vectors, component of ... 1answer 134 views What kind of physical quantity is angular displacement? Angular Displacement is neither a vector nor a scalar. What type of physical quantity it is? Are there any other examples of that physical quantity? 2answers 123 views How can a gas support tensile stresses? In working through a rigorous derivation of the compressible Navier-Stokes equations, I find that the momentum flux in the X-direction should be driven not only by the normal pressure gradient ... 0answers 140 views Variation of the purely covariant Riemann tensor I need to find the variation of the purely covariant Riemann tensor with respect to the metric g^{\mu \nu}, i.e. \delta R_{\rho \sigma \mu \nu}. I know that, R_{\rho \sigma \mu \nu} = g_{\rho ... 1answer 251 views Hookes law and objective stress rates Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ... 1answer 317 views Levi Civita covariance and contravariance I read some older posts about this question, but I don't know if I'm getting it. I'm working with a Lagrangian involving some Levi Civita symbols, and when I calculate a term containing ... 1answer 117 views Tensors in special relativity [duplicate] I'm trying to understand tensors, but I've come across the following question: Let T^{\mu\nu} by a (2,0) tensor. Give the definitions of T_\mu^{\,\nu}, T_{\mu\nu}, and ... 1answer 196 views Invariance of a tensor under coordinate transformation I know, that a tensor is a mathematically entity that is represented using a basis and tensor products, in the form of a matrix, and changing a representation doesn't change a tensor, is kind of ... 2answers 152 views “Vectors” (i.e. 1-tensors) their definition and motivation for relativity I'm reading Einstein Gravity in a Nutshell (by Zee) and here he defines a vector as an object which is invariant under coordinate representation; concretely, if in one coordinate representation, V, ... 2answers 55 views Terminologies for moment of inertia Perhaps someone can suggest the right terms for the following mathematical objects related to moment of inertia? A inertia tensor I.$$I \equiv \begin{bmatrix} I_{1,1} & I_{1,2} & I_{1,3} ...
In differential geometry, the differential forms are anti-symmetric tensors. So, why is the anti-symmetric tensor like $d x_1 \otimes dx_2 - d x_2 \otimes d x_1$, more important than the ...