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2
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0answers
53 views

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 ...
3
votes
2answers
222 views

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 ...
3
votes
2answers
370 views

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 ...
11
votes
4answers
570 views

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 ...
1
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0answers
81 views

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 ...
2
votes
2answers
137 views

Physical interpretation of order of tensor indices

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} = ...
1
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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): ...
3
votes
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 ...
4
votes
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 ...
16
votes
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 ...
3
votes
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 ...
2
votes
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 ...
2
votes
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 ...
2
votes
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?
0
votes
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 ...
2
votes
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 ...
2
votes
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 ...
2
votes
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 ...
1
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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 ...
2
votes
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? ...
1
vote
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 ...
3
votes
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 ...
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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 ...
0
votes
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) ...
2
votes
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 ...
0
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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, ...
12
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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 ...
2
votes
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 ...
0
votes
2answers
452 views

What is pseudo tensor?

What is the pseudo tensor in relativity? How do we transform tensor and pseudo tensor under parity?
-1
votes
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 ...
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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 ...
1
vote
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) ...
3
votes
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. ...
5
votes
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 ...
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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 ...
1
vote
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: $$R_{μν}-\frac{1}{2}g_{μν}R=\frac{8πG}{c^4}Τ_{μν}$$ ...
3
votes
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 ...
2
votes
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 ...
1
vote
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?
1
vote
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 ...
1
vote
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 ...
2
votes
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 ...
5
votes
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 ...
1
vote
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 ...
3
votes
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 ...
3
votes
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, $V$, ...
1
vote
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} ...
2
votes
2answers
198 views

Why is the anti-symmetric tensor more important than symmetric tensors?

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 ...
1
vote
3answers
299 views

Why do we need a metric to define gradient?

For me, the gradient of a scalar field (say, in three dimensions) is simply (formally) $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\frac{\partial f}{\partial z} ...
2
votes
1answer
340 views

How to transform material permittivity tensor from Cartesian coordinates to another orthogonal coordinate system?

I have a material specified by a permittivity tensor written in Cartesian coordiantes: $$\begin{pmatrix} \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz}\\ \epsilon_{yx} &\epsilon_{yy} ...