What is Tensor notation: Definition and 56 Discussions
This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:
Tensor
Tensor (intrinsic definition)
Application of tensor theory in engineering scienceFor some history of the abstract theory see also Multilinear algebra.
Let's say I have the following situation
$$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$
Would I be able to commute the integral and the partial derivative? If so, why is that? In the same line of thought, in the situation I'm able to commute, would the result of...
Hi,
I'm in trouble with the different terminologies used for tensor product of two vectors.
Namely a dyadic tensor product of vectors ##u, v \in V## is written as ##u \otimes v##. It is basically a bi-linear map defined on the cartesian product ##V^* \times V^* \rightarrow \mathbb R##.
From a...
Actually, this is not homework, but I think I need help like homework. It was raised from the notice that there is no tensor form of linear Hooke's law in terms of Young's modulus E, and Poission's ratio, v. For example, if we use lame parameters, we have G, \lambda, like
The linear Hooke's...
So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I, in the position of a complete beginner, am taking notes on it, and I just wanted to make sure I wasn't misinterpreting anything.
At about 5:50, he states that "The array for Q is...
So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I am a complete beginner and just want some clarification on if I'm truly understanding the material.
Basically, is everything below this correct?
In summary of the derivation of the...
So, I have recently been trying to learn how to work with tensors. In doing this, I have come across Einstein Notation. Below is my question.
$$(a_i x_i)_{e}= (\sum_{i=1}^3 a_i x_i)_r=(a_1 x_1+a_2 x_2+a_3 x_3)_r$$; note that the following expression is in three dimensions, and I use the...
Hi, I have some soft body equations that require first order elasticity constants. Just trying to figure out the proper indexing.
From Finite Elements of Nonlinear Continua by J.T. Oden, the elastic constants I am trying to obtain are the first order, circled below:
My particular constitutive...
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...
I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus. They are things that I think I understand how they work, but whose purpose I do not see clearly, so I would appreciate if someone could give me some clue about it.
Tensors. As...
I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways:
##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...
Property (a) simply states that a second rank tensor that vanishes in one frame vanishes in all frames related by rotations.
I am supposed to prove: ##T_{i_1 i_2} - T_{i_2 i_1} = 0 \implies T_{i_1 i_2}' - T_{i_2 i_1}' = 0##
Here's my solution. Consider,
$$T_{i_1 i_2}' - T_{i_2 i_1}' = r_{i_1...
I am struggling with tensor notation. For instance sometimes teacher uses
\Lambda^{\nu}_{\hspace{0.2cm}\mu}
and sometimes
\Lambda^{\hspace{0.2cm}\nu}_{\mu}.
Can you explain to me the difference? These spacings I can not understand. What is the difference between...
So if ##P_{0}## is an event, and I have ##\mathcal {g_{\mu\nu}(P_{0})}=0## and ##\mathcal {g_{\mu\nu,\alpha\beta}(P_{0})}\neq0##, does this notation mean ##\partial\alpha\partial\beta## or simply ##\partial(\alpha\beta)##? And what is the significance of it? Why can't it be zero in curved spacetime?
hi, I'm currently taking a classical field theory class (electromagnetism in the language of tensors and actions and etc) and we have just encountered the gauge symmetry, that is for the 4 vector potential we can add a gradient of some smooth function and get the same physics (if we take Aμ →...
Hello
I have been going through the cosmology chapter in Choquet Bruhats GR and Einstein equations and in definition 3.1 of chapter 5 she defines the sectional curvature with a Riemann( X, Y;X, Y) (X and Y two vectors)
I don't understand this notation, regarding the use of the semi colon, is it...
So the first term of the Lagrangian is proportional to ##{F_{\mu \nu}}{F^{\mu \nu}}##. I wrote out the matrices for ##{F_{\mu \nu}}## and ##{F^{\mu \nu}}## and multiplied at the terms together and added them up, but some of the terms didn't cancel like they should have. Should I have used minus...
Homework Statement
Hi, I can't seem to understand the following formula in my professor's lecture notes:
F_αβ = g_αγ*g_βδ*F^(γδ)
Homework Equations
Where g_αβ is the diagonal matrix in 4 dimensions with g_00 = 1 and g_11 = g_22 = g_33 = -1 and F^(γδ) is the electromagnetic tensor with c=1...
Homework Statement
Show that the path line of a particle at point x currently, and point ξ at time τ is given by
ξ(τ) = x + (τ-t)Lx
Homework Equations
Pathline is solution to
dx/dt = u
x(t)|t=τ = X
L is the velocity gradient and is a 2nd order tensor Lij = dui/dxj
The Attempt at a...
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...
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 doing my M.sc in physics.. In my course I have classical field theory and electrodynamics... I need to learn tensor notations to understand the above subjects...
Please tell me about some good introductory books to learn tensor notation to handle things in electrodynamics and classical...
Suppose we have a system of particles that interact via conservative forces. I wish to prove that ##K+U## is a constant of the system with tensor analysis. Here is my procedure:
The Lagrangian is ##L=\frac{1}{2}m_i\dot{ r_i}^2-\Phi##
Lagrange's equations ##\frac{d}{dt}(\frac{\partial...
Homework Statement
A charged particle of charge q with arbitrary velocity ##\vec v_0## enters a region with a constant ##\vec B_0## field.
1)Write down the covariant equations of motion for the particle, without considering the radiation of the particle.
2)Find ##x^\mu (\tau)##
3)Find the...
Homework Statement
Hi all, I'm trying to learn how to manipulate tensors and in particular to differentiate expressions. I was looking at a Lagrangian density and trying to apply the Euler-Lagrange equations to it.
Homework Equations
Lagrangian density:
\mathcal{L} = -\frac{1}{2}...
When writing
##A_{a}\text{ }^{b}## one means ''The element on the a-th row and b-th column of the TRANSPOSE of A" right?
Such that ##A_{a}\text{ } ^{b}= A^{b}\text{ } _{a}## ?
I would just like a confirmation so I'm not learning the convention in a wrong manner.
My notes read that, the quantity ##K^{2}=K_{uv}V^{u}V^{v}## is constant along geodesics, where ##K## is a killing vector. I know my definition that the quantity on the RHS is conserved, I'm just wondering why do we call it ##K^{2}##, rather than anything else?
In analogy to a killing vector, if...
Hi,
Can someone explain the difference between, say, \Lambda_\nu^\mu, {\Lambda_\nu}^\mu and {\Lambda^\mu}_\nu (i.e. the positioning of the contravariant and covariant indices)?
I have found...
Hi,
I'm teaching myself tensor analysis and am worried about a notational device I can't find any explanation of (I'm primarily using the Jeevanjee and Renteln texts).
Given that the contravariant/covariant indices of a (1,1) tensor correspond to the row/column indices of its matrix...
Hi folks.
Hope that you can help me.
I have an equation, that has been rewritten, and i don't see how:
has been rewritten to:
Can someone explain me how?
Or can someone just tell me if this is correct in tensor notation:
σij,jζui = (σijζui),j
really hope, that...
I have an vector calculus identity to prove and I need to use vector notation to do it. The identity is $$\vec{\nabla}(fg)=f\vec{\nabla}{g}+g\vec{\nabla}{f}$$ I tried starting with the left side by writing $\vec{\nabla}(fg)=\nabla_j(fg)$. Now I look and that and it really looks like there is...
Homework Statement
Hi all,
Here's the problem:
Prove, in tensor notation, that the triple scalar product of (A x B), (B x C), and (C x A), is equal to the square of the triple scalar product of A, B, and C.
Homework Equations
The Attempt at a Solution
I started by looking at the triple...
Homework Statement
We will see (in Chap. 5) that the magnetic field can be derived from a vector potential function as
follows:
B = ∇×A
Show that, in the special case of a uniform magnetic field B_{0} , one possible
vector potential function is A = \frac{1}{2}B_{0}×r
MUST USE TENSOR NOTATIONm...
I am new to tensor notation, but have known how to work with vector calculus for a while now. I understand for the most part how the Levi-Civita and Kronecker Delta symbol work with Einstein summation convention. However there are a few things I'm iffy about.
For example, I have a problem where...
Homework Statement
I'm confused about writing down the equation: \Lambda \eta \Lambda^{-1} = \eta in the Einstein convention.
Homework Equations
The answer is: \eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho\sigma}
However it's strange because there seems...
Greetings,
I would like to find the commutator \left[Lx^2,Ly^2\right] and prove that
\left[Lx^2,Ly^2\right]=\left[Ly^2,Lz^2\right]=\left[Lz^2,Lx^2\right] I infer from the cyclic appearance of the indices that using the index notation would be much more compact and insightful to solve the...
Homework Statement
Prove that (AxB) is perpendicular to A
*We know that it is in the definition but this requires an actual proof. This is what I did on the exam because it was quicker than writing out the vectors and crossing and dotting them.
Homework Equations
X dot Y = 0 when...
2A\mu=-\muoJ\mu
Griffith's Introduction to Electrodynamics refers to this 4-vector equation as "the most elegant (and the simplest) formulation of Maxwell's equations." But does this encapsulate the homogeneous Maxwell Equations? I see how the temporal components lead to Gauss' Law, and I'm...
So I know that the Hodge dual of a p-form A_{\mu_1 \mu_2\cdot \cdot \cdot \mu_p} in d dimensions is given by
(*A)^{\nu_1 \nu_2 \cdot \cdot \cdot \nu_{d-p}} = C\epsilon^{\nu_1 \nu_2 \cdot \cdot \cdot \nu_{d-p}\mu_1 \mu_2 \cdot \cdot \cdot \mu_p}A_{\mu_1 \mu_2\cdot \cdot \cdot \mu_p}
where C...
Is the following the definition of wedge product in tensor notation?
Let A \equiv A_i be a matrix one form. Then
A \wedge A \wedge A \wedge A \wedge A = \epsilon^{abcde}A_a A_b A_c A_d A_e
?
in 5 dimensions. This question is in reference to the winding number of maps.
in the appendix on Group Theory in Zee's book there is a discussion of commutations for SO(3)
two questions
- does [J^{ij},J^{lk}] = J^{ij}*J^{lk}-J^{lk}*J^{ij}?
and there is an expression in the appendix that the commutator equals i(\delta^{ik}J^{jl} ...
i don't understand the why...
Hi,
I am very new to general relativity and have only just started to learn how to do some very basic manipulation of tensors. I can understand the methods I am using and have some idea of what a tensor is but am not sure what the difference between upper and lower indices signifies. I can...
Homework Statement
Show that \nabla \times (a \cdot \nabla a) = a\cdot\nabla(\nabla\times a) + (\nabla \cdot a(\nabla \times a) - (\nabla \times a)\cdot \nabla aHomework Equations
\nabla \times (\nabla \phi) = 0
\nabla \cdot (\nabla \times a) = 0
The Attempt at a Solution
I started with...
Homework Statement
I'm looking at the wikipedia article about four-momentum and I can't seem to get things right. It says
Calculating the Minkowski norm of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the ''c'') to the square of the particle's proper mass...
1. While reading notes on group theory there is a step I could not reproduce although it seems to me it should be straightforward. Probably there is something I am missing on tensor indices notation. Since R is an orthogonal matrix you can...
2 ...go from \epsilon...
Hi,
I have the following term in tensor notation
\frac{\partial{c}}{\partial{x_i}}\frac{\partial{u_i}}{\partial{x_j}}\frac{\partial{c}}{\partial{x_j}}
I'm not sure how to write this in vector notation.
Would it be?
\nabla{c}\cdot\nabla\boldsymbol{u}\cdot{c}
The problem I have...
Homework Statement
1. Establish the vector identity
\nabla . (B x A) = (\nabla x A).B - A.(\nabla x B)
2. Calculate the partial derivative with respect to x_{k} of the quadratic form A_{rs}x_{r}x_{s} with the A_{rs} all constant, i.e. calculate A_{rs}x_{r}x_{s,k} Homework Equations
The...
I've posted this in the Geometry & Topology section, but I believe it will get many more views here, so I'm posting a link
Pictures here:
https://www.physicsforums.com/showthread.php?t=407776
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I really liked Penrose's diagrammatic way of writing tensor algebra, so I spent a while...
I really liked Penrose's diagrammatic way of writing Tensor algebra, so I spent a while learning the basic notation. Unfortunately, it took a very long time for me to learn this because there is so little info on it to begin with. I also didn't see much mention of how to use the notation for...