Questions tagged [tensor-calculus]

Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

Filter by
Sorted by
Tagged with
-1
votes
0answers
34 views

Question on covariant derivative calculations

I am trying to show by explicit calculation that $$\nabla_{\mu}T^{\nu}_{\phantom{\nu}\lambda}=g_{\lambda\xi}\nabla_{\mu}T^{\nu\xi}\tag{1}$$ where the covariant derivative commutes with the metric ...
3
votes
1answer
76 views

Analogy of Levi-Civita contraction exists?

In three dimensions we know $$ \epsilon_{ijk} \epsilon_{mnk} =\delta_{im} \delta_{jn} - \delta_{in}\delta_{jm}. \tag{1} $$ Is there any known three index object $\alpha_{ijk}$ such that $$ \alpha_{ijk}...
0
votes
0answers
48 views

Covariant Derivative and Del Operator

I have been writing the divergence of a vector field in spherical coordinates, and I know the transformation rules for the del operator and a vector. It's pretty easy to transform both of them into ...
0
votes
0answers
23 views

Computing the longitudinal and traceless part of the left hand side of Einstein's equation

I am reading a textbook on cosmology. Consider $G^i_j$, the left hand side of Einstein's equation. If $\Psi$ and $\Phi$ are first order perturbations to the time and spatial components respectively of ...
0
votes
1answer
31 views

Basis vector as “Array” choice in tensor calculus / GR, and 3+1 decomposition

In differential geometry and general relativity, once we have chosen a basis on our spacetime, say $ \{t,r,\theta, \phi \} $, we can represent every tensor as an "array" of numbers, so a &...
0
votes
0answers
27 views

What is a good book regarding geometrical basics for GR and SR? [duplicate]

What is a good book regarding geometrical basics for GR and SR? Am about to read "Introduction to the Geometrical Foundations of General Relativity" by B.Wichmann (http://www.tensor-calculus....
0
votes
1answer
35 views

A question on the electromagnetic field tensor

Consider \begin{equation} \delta \left(F^{\mu\nu}F_{\mu\nu}\right)=2F^{\mu\nu}\delta F_{\mu\nu} \end{equation} I am trying to convince myself that the above holds for any arbitrary explicit form of $...
0
votes
0answers
23 views

Relation between $\delta$-system

I'm reading Pavel Grinfeld's book "Introduction to tensor analysis and the calculus of moving surfaces". I've reached the section where the author talks about $\delta$-systems and the ...
-1
votes
0answers
27 views

Third order delta system as determinant of a matrix [closed]

I'm reading Pavel Grinfeld's book about tensors ("Introduction to tensor analysis and the calculus of moving surfaces"). I've reached the point where the author starts talking about $\delta$-...
0
votes
2answers
91 views

Question on tensor contraction

I have seen that the contraction of the electromagnetic field (emf) tensor between covariant and contravariant versions of the electromagnetic field tensor is $-2E^2/c^2 +2B^2$ but my confusion is ...
2
votes
2answers
153 views

What does it mean to say “tensors transform sensibly”?

I have been reading about tensors and they are described as "objects which transform in a physically meaningful/sensible manner" and obey the equation (for rank 2, but generalises) $T'_{ij}=...
0
votes
0answers
35 views

Components of a vector after translation [migrated]

If you have a vector expressed in curvilinear coordinates and you translate it to a different point in space, the components of the vector will change to ensure that it maintains its geometric ...
0
votes
1answer
31 views

A query in a step when deriving Maxwell's equations from stationary action

When varying the Maxwell action, one gets to the following part $$ \begin{align} \delta \left(F^{\mu \nu} F_{\mu \nu}\right) &= 2 F^{\mu \nu} \delta F_{\mu \nu} \\ &= 2 F^{\mu \nu} \left(\...
2
votes
2answers
84 views

Integration of tensor to find potential

I have question given as: $$\partial_k \varphi = -(C_k+ D_{jk}r_j)$$ where $C_k \,\&\, D_{jk}$ are constants and $D_{jk}$ is symmetric and traceless. I have to find $\varphi$. I am getting : $\...
1
vote
2answers
48 views

Taylor's series + Stress Equilibrium Equation (Cauchy's equilibrium equation)

I am now studying theory of plasticity and elasticity and I stumbled upon the derivation of the equation of stress equilibrium (or Cauchy's equilibrium equation) that I would like to have more ...
1
vote
0answers
36 views

Differential forms in projective space

I am currently reading some paper about the Amplituhedron, and it is using projective geometric way to present amplitudes. How can we define forms in projective space to measure volume for a polytope?
0
votes
1answer
73 views

Problem in Tensor calculation

I'm a theoretical physics M.SC. student, I have some problems in extending a tensor. Can anyone open up this tensor? it will be a great help for me. And also I want to know what is the difference ...
1
vote
1answer
92 views

Addition of two angular momenta: general theory

Suppose I have two quantum systems $(1)$ and $(2)$, for each of them the angular momenta $J^{(1)}$ and $J^{(2)}$ are defined. Our purpose is to create a total angular momentum $ J$ which can describe ...
0
votes
1answer
50 views

On inner product for operators on $\scr H \otimes H$

Given two linear operators $A \in {\scr H}_A$ and $B \in {\scr H}_B$, their inner product $A \cdot B \in {\scr H}_A \otimes {\scr H}_B$ is defined to be: $$A \cdot B := \sum_{j} A_j \otimes B_j$$ ...
3
votes
1answer
100 views

On addition of angular momenta and inner product

Suppose I have two quantum systems associated with angular momenta $J_1$ and $J_2$ respectively. I can define the angular momentum of the whole system with the operator $J$ acting on $\scr H:={\scr ...
1
vote
2answers
62 views

Deriving the transformation law for the Christoffel symbols

I am a first year undergraduate teaching myself General Relativity from the book by Bernard Schutz. In one of the problems he asks to derive the transformation law for the Christoffel symbols from the ...
0
votes
0answers
38 views

Functional derivatives of tensors in phase space in the context of General Relativity

In classical mechanics the functional derivative of a scalar function $f(x,\dot{x})$ respect to the trajectory $x^i(t)$ is \begin{equation} \frac{\delta f\big[x(t),\dot{x}(t)\big]}{\delta x^i(t')}=\...
0
votes
1answer
67 views

How to verify the gauge invariance of the term $\mathcal{L}= -\frac{1}{2}tr[(F^{i}_{\mu \nu}\sigma^i)^2]$?

This is equation (15.38) from Peskin and Schroeder. I am unable to compute this term to verify if it is invariant (I know that it is but I'd like to verify that). I would appreciate it if someone can ...
0
votes
0answers
34 views

How to write the wedge product in terms of Levi-Civita symbol [migrated]

Suppose we are in 3 dimensional space and that we have a one form $E_i$ and a two form $\omega_{jk}$.The wedge product between these forms is $$(E\wedge \omega ) _{ijk}=E_i \omega_{jk}-E_j \omega_{...
1
vote
0answers
63 views

Why is the curl of a vector field a pseudovector? [migrated]

There is an invariant definition of curl operation: $rotA = (*d(A^b))^{\text#}$ , where: A - vector field, * - hodge star, and $^\text#$ / $^b$ - lowering/raising the index. Acording this, $rotA$ is ...
0
votes
1answer
41 views

Lorentz transformation of covariant tensors

I want to derive the rule of the Lorentz Transformation of covariant vectors (tensors) from the transformation rule of the metric and contravariant vectors. Starting out one can write the equation: $$...
0
votes
1answer
49 views

Gradient in cylindrical coordinate using covariant derivative

I'm reading a little pdf book as an introduction to tensor analysis ("Quick introduction to tensor analysis", by R. A. Sharipov). I've reached the last section where it is explained how it ...
0
votes
1answer
46 views

Problem 2.1(b) in Peskin and Schroeder's Introduction to QFT

In this exercise the author claims that adding $\partial_\sigma K^{\sigma \mu \nu}$ does not affect the divergence of $T^{\mu\nu}$. In other words the author claims that $\partial_\mu \partial_\sigma ...
0
votes
1answer
54 views

Lorentz Transforming the electric field and the change of its directions

This is a two part question about the Lorentz transformation of the electromagnetic field, the electric field to specific. The Lorentz transformation will be a simple boost in the x direction. first ...
0
votes
1answer
42 views

Deducing Lorentz representation out of symmetry type

Cross-posted from here Lorentz algebra can be proven to be isomorphic to $\mathfrak{su}(2) \oplus \mathfrak{su}(2)$, so every representation can be denoted by two indices or spins, $(j_1, j_2)$. Let's ...
1
vote
1answer
57 views

Four vector for dual field-strength tensor

To generate the electromagnetic field strength tensor, one can use the electromagnetic four-vector using by $F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$. Is there a similar ...
0
votes
1answer
66 views

Relationship between derivatives of tensors in different Cartesian coordinate systems

I'm new to tensor calculus: I'm reading a little introductory book whose title is "Quick Introduction to Tensor Analysis", written by R.A Sharipov. I've reached the section called ...
0
votes
1answer
37 views

Weyl's developement of Euler's Equations for a Spinning Top using tensor calculus

Equation 27 referenced in the quotation is torque equals the time derivative of angular momentum. Equation $\eqref{29}$ is the contraction of a position vector with the angular velocity tensor, giving ...
0
votes
1answer
73 views

Lie Derivative - Obtaining Equations for Tensors and Vectors

I am writing a code to calculate the Lie Derivatives, and so far, I have defined the Covariant derivative for scalar function; $$\nabla_a\phi \equiv \partial_a\phi~~(1)$$ for vectors; $$\nabla_bV^...
0
votes
1answer
30 views

Multiple skew brackets in tensor summation, e.g. $U_{[ab]}V^{[ab]}$

I'm wondering what does $U_{[ab]}V^{[ab]}$ or $U_{[ab]}V^{(ab)}$ usually mean? I thought the double brackets meant to apply the $\text{sgn}(\pi)$ twice so $$U_{[ab]}V^{[ab]}=\frac{1}{2}\sum_{a,b}\left(...
13
votes
6answers
2k views

Vectors as functions?

In my study of general relativity, I came across tensors. First, I realized that vectors (and covectors and tensors) are objects whose components transform in a certain way (such that the underlying ...
0
votes
1answer
69 views

Tensor notation of covariant derivative

I'm trying to apply Wald's General Relativity equation $3.1.14$: $$\nabla_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}=\overline{\nabla}_a{T^{b_1\dots b_k}}_{c_1\dots c_{\ell}}+\sum_i{C^{b_i}}_{ad}{T^{b_1\...
1
vote
1answer
98 views

Riemann tensor in linearized theory of gravity [closed]

Here I present some approximations to Christoffel, Riemann, Ricci tensors when the following perturbed metric is taken into account $g_{\mu \nu} \approx \eta_{\mu \nu} + h_{\mu \nu} + \mathcal{O}(h_{\...
1
vote
2answers
86 views

Change of coordinate vs change of reference axes

Does basis vectors change opposite to coordinate scaling ? For example, suppose I have some oblique coordinate system, and I decide to scale up both 'axes' by a factor of $a$ and $b$ respectively. The ...
0
votes
2answers
44 views

Covariant or contravariant nature of Gradient

I've been having this confusion regarding the gradient being a covariant vector. Intuitively I seem to have understood the concept. However, mathematically, I'm unable to show this, in a single ...
1
vote
1answer
71 views

Index convention according to Schwartz

On page 15 of his QFT book, Schwartz writes that all the following contractions are equivalent as long as the flat metric is used: $$v^\mu w_\mu=v_\mu w^\mu=v_\mu w_\mu=v^\mu w^\mu.$$ Isn't this false?...
1
vote
2answers
225 views

Geometrical representation of Contravariant and covariant vectors

After cruising through a lot of material online, and answers over here, my understanding of contravariant and covariant vectors are, in a finite-dimensional vector space, suppose we have a vector, ...
2
votes
3answers
113 views

Vector representation in dual space

I'm new to tensor analysis, and came across the topic of vectors and duals, and faced a massive confusion. Are vectors and duals different representations of the same object ? I had another doubt ...
1
vote
0answers
31 views

Interpretation of Integral of Differential 3-Form over 3-Manifold in Engineering (or Physics)?

[This is a repost of https://math.stackexchange.com/questions/4164258/interpretation-of-integral-of-differential-3-form-over-3-manifold-in-engineering and https://engineering.stackexchange.com/...
0
votes
1answer
34 views

Compute components of elasticity tensor for isotropic material

In linear elasticity we have, for an isotropic material, $$C[E]=2 \mu E + \lambda \operatorname{tr}(E)I$$ where $\mu,\lambda$ are called Lamè moduli and $E=\frac{\nabla u + \nabla{u}^T}{2}$ I've seen ...
1
vote
2answers
42 views

Compute deformation gradient in an elasticity problem

In the problem "finite bending of an incompressible elastic block, discussed here at pg. 181 deformation map is, given $r = f(x)$ and $\theta = g(y)$ two functions which will be determined later: ...
0
votes
2answers
74 views

Why the components of elasticity tensor are 21?

It's known that the elasticity tensor is such that $$C_{ijkl}=C_{jikl}=C_{ijlk}=C_{klij}$$ The first two equalities imply that we have a $6 \times 6$ symmetric matrix. So far so good. I can't ...
2
votes
2answers
339 views

Notation for covariant derivative in the book Gravitation

I am not familiar with semicolon notation for covariant derivatives that is why I am asking this question. In the book Gravitation on page 566 they have the following exercise Show that $\nabla u$ ...
1
vote
1answer
64 views

Multiplication and contraction of multiple metric tensors

Do metric tensors commute? Can I reduce this equation $$ (g_{hj}g_{ik}-g_{hi}g_{jk})g^{hl}, $$ as $$ (g_{hj}g_{ik}g^{hl}-g_{hi}g_{jk}g^{hl}) $$ $$ =(g_{hj}g^{hl}g_{ik}-g_{hi}g^{hl}g_{jk}) $$ $$ =(A_{j}...
1
vote
0answers
13 views

Correct expression for in-plane right Cauchy-Green tensor for surface embedded in 3D

I have a surface embedded in a 3D Cartesian frame undergoing a deformation and want to know the correct expression for right Cauchy Green tensor in the tangent plane of the surface. A point $\mathbf{X}...

1
2 3 4 5
40