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.

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Field of research treating tetrads (vierbeins) as fundamental objects?

After a lot of research on tetrads I think I found the subject I'd like to specialize in for postgrad/phd, as they seem to express many interesting and (perhaps) fundamental physical properties. So i ...
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Proper conceptualization & notation for vectors, $n$-tuples, and matrices in physical space

This is a fairly basic question that I may be making longer than necessary. But it has plagued me for some time. It is essentially this: In what space do abstract physical vectors like a velocity ...
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Advection term for a matrix equation

How can I calculate a quantity like $(\vec{v} \cdot \nabla) M$ where $\vec{v}$ is the velocity vector, and $M$ is some 3x3 matrix? (if one wants, assume $M$ is a tensor) This would be the advective ...
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Covariance of Euler-Lagrange equations under arbitrary change of coordinates

I'm trying to prove that the Euler-Lagrange equation $$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}_i})-\frac{\partial L}{ \partial q_i}=0$$ is invariant under an arbitrary change of coordinates $$...
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Question about Wald's example of a "derivative operator"

I am trying to study Wald's book on General Relativity. I thought the first two chapters were ok, but I'm completely stuck in chapter 3, on page 32 where he gives an example of a "derivative ...
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Index transposing in Einstein notation

When working with Einstein's summation convention, how do I have to transpose the indexes of the tensor? For example, supose I want to take the matrix product with its transpose. Which is the correct ...
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Why does contracting a term with a tensor means a portion of this term is a tensor?

I am looking at a problem in Guidry's Modern General Relativity, and the solution contains the following two sentences: In the scalar product expression $A\cdot B = g_{\mu \nu}A^{\mu} B^{\nu}$, the ...
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Confusion about raising and lowering indices

Is it possible to take the following expression: $$U^\mu U^v\partial_\mu\partial_v$$ Where $U$ is the four-velocity, and simplify it the following way?: $$U^\mu U^v \eta_{\mu v}\partial^v\partial_v =c^...
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Can someone suggest me a book where whole Electrodynamics of master level is explained in tensor form

I am trying to study electrodynamics, can someone please suggest me a book or any other resource where I can get it in tensor form, Topics: waves in medium, resonators, etc.
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Divergence theorem in index notation

From Batchelor's book of fluid dynamics: I guess that's an easy question for anyone having more familiartiy than me in tensor calculus, anyways. First integral argument is the i-component of the ...
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Lorentz transform of Levi-Civita Symbol

I was reading about Lorentz transformations and frequently I hear the notion of Lorentz transforming quantities like $\epsilon^{\mu \nu \rho \sigma}$. But I have never heard an explanation as to why ...
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The displacement gradient tensor transformation rule

The transformation rule of a 2nd rank tensor expresssed in a given basis is often written as follow: $$F' = P^T FP $$ where $F$ is the matrix representation of the tensor in the old basis B, $F'$ its ...
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How do we make sense of $F^{\mu\nu}F_{\mu\nu}$? The book just assumes I understand it

Why are these upper and lower indices and what does that mean. I can't interpret the term with upper indices.
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Confusion on number of component of Cauchy stress tensor

The Cauchy stress tensor is often presented as a tensor having $(2,0)$ tensor having nine components in any given basis. However, I think it should actually be $6 \times 3 =18$ because a cube has six ...
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What does $\delta/\delta t$-derivative represent in tensor calculus?

Some texts, such as Pavel Grinfeld's, talk about a $\delta/\delta t$-derivative whose role (in trajectory analysis of particles using tensor calculus) is pretty obscure to me. For example, the ...
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How to calculate the electric field of a polarization density?

By polarization density here I just assume I have a "blob" of free positive and negative charges, and instead of describing the system with a charge density $\rho(\pmb{r})$ I want to use the ...
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Contravariant Vector Component Transformation from Polar to Cartesian

I am new to tensors and I have just learned that the contravarient components of a vector transforms in the following way (using Einstein summation convention) $$A^{'i}=\frac {\partial x^{'i}}{\...
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Spherical harmonics expansion: from scalars to tensors

It is well known that a scalar field on the unit sphere can be expanded in spherical harmonics, see e.g. this. I am wondering if there is a related concept for vector fields and, in general, for any ...
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How do I expand: $\langle u , \nabla\rangle u$?

I'm studying the Landau (VI) and when he "introduces" the material derivative (he is building up the continuity equation), something like this appears: $(u,\operatorname{grad})u$ (sometimes ...
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Generalizing Fermi-Walker Derivative/Transport to General Vector Bundles

The usual definition of the Fermi derivative (eg as given in Hawking and Ellis) is to consider a Lorentzian manifold $(M,g)$ and a unit timelike curve $\gamma$, a smooth vector field $X$ along $\gamma$...
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Riemann curvature tensor in an inertial frame

My understanding is that the mathematical definition of an inertial frame at $x_0$ is a choice of coordinates s.t: $g_{\mu\nu}(x_0) = \eta_{\mu\nu}(x_0)$ $\partial_\rho g_{\mu\nu}(x_0) = 0$ I've ...
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From the point of view of physics, why is it useful to know the irreps of rotation group?

In 3D, the rank two tensorial physical quantities, for example, the electric susceptibility, the conductivity, the stress tensor etc, are in general, not irreducible representations i.e. neither ...
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Are tensors constructed such that one forms "act" on some complex vector field?

I have some confusion understanding the motivation in constructing tensors (or tensor fields). On a differentiable manifold $\mathcal{M}$ consider a vector field $X$. At any point $p\in \mathcal{M}$, ...
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Expressing Maxwell's equations in tensor form using Electromagnetic field strength tensor [closed]

I have yet another derivation question from Carroll's General Relativity textbook. Given the electromagnetic field strength tensor is of the form: $$ F_{\mu\upsilon} = \left( \begin{matrix} 0 & -...
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Expressing Maxwell's equations in tensor notation

I've been teaching myself relativity by reading Sean Carroll's intro to General Relativity textbook, and in the first chapter he discusses special relativity and introduces the concept of tensors, ...
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Generalization of the Impact Depth Equation

Newtons Impact Depth Equation. $L = l *\dfrac{p1} {p2}$ $L$ is the impact depth $l$ is the length of the projectile $p1$ is the density of the projectile $p2$ is the density of the Target has many ...
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Kerr Solution metric ansatz for EFEs

The Schwarzschild metric ansatz is given by $$ds^2=-A(r)dt^2+B(r)dr^2+r^2d \Omega^2$$ where upon applying the Einstein Field Equations $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ we obtain the normal ...
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Finding the proportionality constant in $\varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho}\propto \varepsilon^{\lambda\rho}$

We can show that the contraction of some arbitrary $2\times2$ matrix $A_{\mu}^{\ \lambda}$ with the Levi-Civita symbol is once again antisymmetric \begin{align*} \varepsilon^{\mu\nu}A_\mu^{\ \lambda} ...
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Show that the contraction of a covector and a vector is Lorentz invariant

I just got Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity a couple of weeks ago, and I have resolved to go through the entire book. In the first chapter, he prompts the ...
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Reference which explains Penrose Diagramatic notation in simple way

In both Penrose's Road to reality and Spinor's and space-time, the following notation is shown: With a lot other examples for doing calculation with Tensors. Could someone give another reference ...
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What is the idea behind 2-spinor calculus?

In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
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A particular contraction of Levi-Civita symbols and tetrads

Consider a four-dimensional spacetime. Consider the following contraction between Levi-Civita symbols and tetrads $$\epsilon_{\alpha \beta i j}\,{\epsilon^{ij}}_k\, e^\alpha\!\wedge e^\beta\!\wedge e^...
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About Volterra's displacement equation on dislocation: cancellation of a surface integral on stress

In the theory of dislocations, the displacement induced by a dislocation in an anisotropic solid media can be expressed by Volterra's displacement equation as follows: $$ u_j(\mathbf{x}) = \int\int\...
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Interpreting stress at the ends of a bar

Consider a bar loaded in tension by distributed loads applied on its ends as shown in the figure. The stress at any cross section of this bar will be $$\sigma = \frac{P}{A}$$ From what I know about ...
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How to calculate the rank of a tensor?

I was studying a little of tensorial calculus and came up with this problem: Given a tensor with a rank of (0,2), $T_{\alpha\beta}$. Calculate the rank of this tensor $T_{\alpha\beta}T_{\gamma}^{\...
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Components of the fully contravariant Kronecker Delta in Schwarzschild Metric

I thought the kronecker delta $\delta^{\mu\nu}$ should always be of value $1$ if both indices are equal and $0$ if they are different. However it seems that the delta has components different from $1$ ...
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Ricci Identity with Torsion Proof

In the notes I'm using for General Relativity, the author begins their proof of the Ricci identity with torsion by writing $$\nabla_{[\mu}\nabla_{\nu]}Z^{\sigma}=\partial_{[\mu}(\nabla_{\nu]}Z^{\sigma}...
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Choosing diffeomorphisms for the pullback metric in the Weak Field approximation

In the weak field approximation of the EFEs $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ we take $g_{\mu\nu}\approx \eta_{\mu\nu}+h_{\mu\nu}$. The $\eta_{\mu\nu}$ term is just the flat space Minkowski metric and $...
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Symmetries of Riemann tensor

Is there a way to show that the symmetries of Riemann tensor are preserved even if the indices are raised or lowered in general. I know how to do it individually for each symmetry but am not sure how ...
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Argument of a scalar function to be invariant under Lorentz transformations

I'm trying to prove that a Lorentz scalar object $\rho(k)$ which is a function of a cuadri-vector $k^{\mu}$ can only have a $k^2$ dependency in the argument. I can imagine that this object has to ...
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Preservation of symmetries of Tensors under lowering and raising indices

How do you go about showing that symmetry properties of tensors are preserved during lowering and raising indices in a metric space? I know how do do it for individual tensors with given symmetries ...
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Correlation function of 4-currents on a general QFT

Given $V^\mu(x)$ a 4-current of a general unitary, Poincaré invariant QFT, I need to show that the correlation function: $$iC_{\mu\nu}(x-y) = \langle {\tilde{0}}|T\left[V_\mu(x)V_{\nu}^\dagger(y)\...
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Why is a Lorentzian metric still Lorentzian after a general coordinate transformation?

In my GR course, we define a lorentzian metric $g_{\mu\nu}(x)$ as a symmetric $(0,2)$ tensor field having 3 positive and 1 negative eigenvalue. Now given a general coordinate transformation described ...
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Software recommendation for a tensor calculation

What is the best software/package to calculate $$2R_{\alpha\mu\beta\nu}R^{\mu\nu}-\nabla_\alpha\nabla_\beta R + \Box R_{\alpha\beta}-\frac12g_{\alpha\beta}\Big(R_{\mu\nu}R^{\mu\nu}-\Box R\Big)$$ for a ...
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Maxwell's stress tensor and pressure

I am studying Electromagnetism from Griffiths and in the book it is stated that diagonal elements of Maxwell's tensor represent pressure. I want to calculate pressure on the wirings of an infinitely ...
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What are the significances of the contravariant and covariant four-momentum and their corresponding four-forces in General Relativity?

In General Relativity, the four-momentum is defined as $$p^r=\frac{dr}{d\tau}$$ where $\tau$ is the proper time. Here, we find that the contravariant index is used. However, I am confused on the ...
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How is it that adding a random field to the partial derivative results in a tensorial operation?

We know that the partial derivative of a tensor is not a tensor. But how is this problem fixed by adding to the partial derivatives, a field of Christoffel symbols? Christoffel symbols are a ...
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Inverse of a metric under variation

Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as: $$g_{\mu\nu}+h_{\mu\nu}$$ or equivalently as: $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$ The given metric has the ...
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General relativity algebraic manipulation help

I'm having difficulty understanding a lot of the fundamentals behind the algebra of general relativity. I have a specific question I'm trying to understand but any pointers about how any of it works ...
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Tensor products in Howard Georgi's "Lie Algebras in Particle Physics"

My question is regarding eq.(3.39) in the second edition of Georgi's book (for those who have the book:)). The section deals with tensor product states where the states comprising the product ...
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