All Questions
397 questions
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Covariant and partial derivative of a vector field (not component)
Is the covariant derivative of a vector field (not the components of a vector) same as the partial derivative?
I am adding a screenshot from page 69 from General Relativity: An introduction for ...
- 1,248
3
votes
2
answers
814
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Partial derivatives vs Covariant derivatives in polar coordinates
Covariant derivatives take into account for both component and basis changes, thereby applicable for curved spaces - where partial derivatives only take component changes into account - is this ...
- 1,248
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0
answers
67
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Confusions about partial and covariant derivatives
Let, we are in 1d cartesian space with metric $g_{xx} = x^2$. Let we have a vector $v = 1/x e_x$. Since the vector is designed to shrink its components as the basis grows - its total length will ...
- 1,248
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0
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63
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Wald: 2-dim Covariant Derivative for Null Hypersurfaces
On pp. 221-222, Wald introduces the 2-dim "hatted" manifold of null geodesics. He moves from 9.2.30 to 9.2.31 and he is allowed to do so because the tensors have the special properties that ...
- 61
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1
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70
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Equating 2 sides of EFE
Can we say if the covariant derivative is zero, the partial derivative is also zero because locally covariant derivative reduces to partial derivatives (since locally spacetime is flat)? Because, in ...
- 1,248
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0
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201
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Commuting material time derivative and material space derivative
Let's note $x$ the coordinates in the current configuration and $\nabla$ the associated gradient; similarly, let's note $X$ and $\nabla_0$ for the reference configuration. I will also note the ...
- 1
2
votes
2
answers
387
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Covariant derivative of gauge theory in curved space
I am reading Witten's article and have a basic question about gauge theory in curved space.
In ordinary flat space (Euclidean space or Minkowski spacetime), covariant derivative of a gauge field $A_{\...
- 111
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82
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Simultaneously raising and lowering indices
Let $U$ be a four-vector and $\nabla$ denote the covariant derivative in the Levi-Civita connection. Is it always true that $$\left(\nabla_{\mu}U^{\nu}\right)U_{\nu}=\left(\nabla_{\mu}U_{\nu}\right)U^{...
3
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153
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d'Alembertian operator in presence of torsion
Consider a Riemann-Cartan 4-dimensional spacetime with torsion. In such a spacetime, I have been asked to compute the d'Alembertian operator acting on a scalar field. Here's what I tried:
$$ g^{\mu\nu}...
- 702
1
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1
answer
192
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Intuition behind parallel transport of vectors as partial derivative operators
Imagining a non-embedded manifold that forces a reformulation of the tangent space at a point as partial derivatives of any arbitrary smooth functions on the manifold along a parameterized curve is ...
- 924
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1
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206
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Are rates a scalar, a vector or both?
Are all rates in physics a scalar, a vector or both?
It seem to me like all rates in science are vectors.
Examples of rate that are vectors are rate of charge flow, rate of heat transfer, rate of mass ...
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5
answers
2k
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Intuition for vector calculus identities
I can follow the proofs for these identities, but I struggle to intuitively understand why they must be true:
$$$$
1. The curl of a gradient of a twice-differentiable scalar field is zero:
$$\nabla\...
- 138
2
votes
1
answer
261
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Infinitesimal coordinate transformation and Lie derivative
I need to prove that under an infinitesimal coordinate transformation $x^{'\mu}=x^\mu-\xi^\mu(x)$, the variation of a vector $U^\mu(x)$ is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$...
- 372
1
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0
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344
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On the definition of the Van Vleck-Morette determinant
Let $M$ be a Riemannian manifold and $\sigma$ the world function. The Van-Vleck-Morette determinant $D$ is defined by
$$D(x,x')=\det(-\sigma_{;\mu\nu{}'})$$
Regarding the semi-colon: In chapter $4.1$ ...
- 1,911
1
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2
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181
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Coordinate basis vectors on tangent bundle (extrinsic view)
Short Version: when we say that $(\pmb{q},\pmb{u}):TQ_{(q)}\to\mathbb{R}^{2n}$ are local coordinates for the tangent bundle of $Q$, which can be viewed as an embedded submanifold of a higher ...
- 477
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1
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116
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Covariant derivative of Weyl spinor
What is the expression for the covariant derivative of a Weyl spinor?
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1
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293
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A covariant derivative construction with torsion
Is there a covariant derivative, $\nabla$, that takes into account torsion, $T^\mu_{\;\;\alpha\beta}$, and covariant derivative of the metric equals zero, $\nabla_\alpha g_{\mu\nu}=0$? If yes, is ...
1
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1
answer
149
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Conformal covariant derivative of a scalar
Suppose that I have a metric $g_{ij}$ with covariant derivative $\nabla_{j}$ and another metric, $\gamma_{ij}$, with covariant derivative $D_{j}$ that is conformally related to $g_{ij}$ as $\gamma_{ij}...
- 137
1
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0
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64
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How to transform a partial derivative to a directional derivative with respect to some affine parameter?
Suppose an affine parameter $\lambda$ is defined along a null geodesic with $dx^\mu/d\lambda=k^\mu$. How could I write the partial derivative $\partial f/\partial x^\mu$ by using $df/d\lambda$? If $k^\...
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601
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Inverse of the covariant derivative
Given the covariant derivative of some tensor, for the sake of this example a covariant vector:
$$\nabla_\mu A_\nu$$
Is there a well-defined inverse operation on the covariant derivative such that it ...
- 613
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1
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285
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Exterior derivatives Leibniz rule
I want to prove Sean Carroll's "spacetime and geometry"'s eq.(2.78):
$$
\mathrm{d}(\omega \wedge \eta)=(\mathrm{d} \omega) \wedge \eta+(-1)^p \omega \wedge(\mathrm{d} \eta) \tag{2.78}
$$
...
- 1,461
5
votes
2
answers
597
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Lie derivative in terms of covariant derivative and the symmetry of Christoffel symbols
I want to verify that if a manifold is torsion-free with a metric compatible derivative operator $\nabla_a$, the Lie derivative of a vector $W^a$ along $V^a$ can be written as
$$L_V W^a = V^\nu\nabla_\...
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1
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2
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322
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Upper index covariant derivative $\nabla^\mu$
In the book Cosmology by Daniel Baumann, the author states that $\nabla^\mu g_{\mu\nu}=0$, where $g_{\mu\nu}$ is the metric tensor considered (usually the one associated to the Minkowski metric or to ...
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1
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1
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207
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Alternate derivation of the covariant derivative of a contravariant vector
In Dirac's “General Theory of Relativity”, he derives the covariant derivative of a contravariant vector (his Eq. (10.7)):
$$ A^\mu_{: \sigma} = A^\mu_{, \sigma} + \Gamma^\mu_{\alpha \sigma}A^\alpha $$...
- 467
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1
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72
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What is the physical meaning of non-commuting tetrads?
I'm reading about the tetrad formalism in GR and one main difference between the coordinate and the tetrad frame is that coordinate derivatives commute $\partial_\mu \partial_\nu = \partial_\nu \...
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3
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1
answer
209
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Connection between covariant derivative operators upon conformal compactification
I'm having trouble determining the connection between two covariant derivative operators. These are: the one associated with the original space-time (and thus with the metric $ \tilde{g}_{ab}$) and ...
2
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2
answers
320
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Is the contracted Christoffel symbol a tensor?
The coordinate transformation law (from coordinates x to coordinates y) for the Christoffel symbol is:
$$\Gamma^i_{kl}(y)=\frac{\partial y^i}{\partial x^m} \frac{\partial x^n}{\partial y^k} \frac{\...
- 613
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2
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295
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Christoffel symbol with conformal time not equal to with cosmic time one when making a change of coordinates for d'Alembertian
I think I am having a misunderstanding that would be nice to clear up.
The covariant d'Alembertian
$$
\Box \phi = g^{\mu\nu}\nabla_\mu\partial_\nu \phi= \left(\partial^2 + \Gamma^\mu_{\mu\lambda}\...
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2
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190
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Covariant Derivative and Cartesian Coordiantes
I have a somewhat weird question regarding some problems I have been encountering lately when dealing with transformations of the general covariant derivative to cartesian coordinates and back.
In the ...
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3
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1
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805
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Covariant derivative vs Ordinary derivative
How correct is the statement:
Covariant derivative, denoted by $\nabla_u v$ = Ordinary derivarive (denoted by $u(v)$) - Normal components, also called second fundamental form.
If the former statement ...
- 1,248
3
votes
2
answers
284
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Covariant Derivative vs Vector Differentiation
I am confused about the difference between a vector operating on another (like $u(v)$, used in Lie Brackets) and covariant derivatives ($\nabla_u v$). Can we not use Christoffel symbols in the vector ...
- 1,248
3
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2
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228
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Associativity of covariant derivatives
I'm having trouble proving that covariant differentiation is an associative operation.
Essentially I'll have to show
$$\nabla_\mu( \nabla_\nu \nabla_\sigma) = (\nabla_\mu\nabla_\nu) \nabla_\sigma. $$...
- 126
2
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2
answers
262
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Confusion about Transforming Christoffel Symbols
I'm trying to understand how transforming Christoffel symbols works. Specifically I'm thinking about the transformation between Schwarzschild and Eddington-Finkelstein coordinates,
$$\Gamma^v_{\;vv}=\...
- 343
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1
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126
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Are differentials of a smooth maps tensors? Connection to Jacobian matrices?
I'm trying to learn about smooth manifolds and differential geometry in order to learn the geometric view of classical mechanics. I'm confused about what kind of "object" the differential of ...
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2
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143
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Gradient of scalar field
On page183 of Rayd'inverno "An introduction to relativity" he says that the right term in parenthesis is a gradient of some scalar field i.e.
When $$\partial_a (\frac{\ X_b}{\ X^2})=\...
4
votes
2
answers
510
views
Converting differential to gradient
Landau & Lifschitz's fluid mechanics book proposes the following statement for an isentropic proccess:
$$dH=vdp \Rightarrow \nabla H=v\nabla p$$
What's the rigorous way to get this result (...
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0
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186
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Derivation of covariant derivative of Spinor
I am following this derivation for the covariant derivative of spinors. I have some questions about this derivation:
On page 3 they use the fact, that
\begin{align*}
V^a(x) = \bar{\Psi}(x)\gamma^a\...
- 527
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votes
3
answers
609
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On varying a tensor with respect to the metric
Upon learning about the Lagrangian formulation of GR, where varying an action with respect to a metric (in order to, for instance, arrive at the Einstein field equations) is common, I can't help but ...
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0
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45
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Covariant Derivative of New Timelike coordinate
In another forum, someone was explaining to me how to define a new time coordinate in a metric tensor to diagonalize it. They said that if I want spacelike hypersurfaces, the covariant derivative of $...
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1
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87
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How do I reconcile these two definitions of acceleration?
How do I reconcile these two definitions of acceleration?
$$a=\frac{d\bar{v}}{dt}=(\frac{dv^k}{dt}+v^i v^j \Gamma^k_{ij})\bar{e}_k \tag{1}$$
and
$$a^k=v^{\small\beta} \nabla_{\small\beta} v^k.\tag{2}$$...
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82
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Scalar curvature from Riemannian metric
I want to compute the scalar curvature for points on an empirical manifold (sampled data).
I have already an algorithm that learns the Riemannian metric and computes geodesics, so from the metric I ...
2
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2
answers
1k
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Action of Lie derivative on 1-forms
In Sean Carroll's spacetime and geometry appendix B he derives the action of the Lie derivative on 1-forms. He finds that $\mathcal{L}_X Y^\mu = [X, Y]^\mu$, which I believe is meant as $\mathcal{L}_X(...
- 2,068
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2
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717
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Is gauge covariant derivative an ordinary covariant derivative?
The formal treatment of the gauge covariant derivative in most reference texts for students is too informal and too ad hoc, so that some general issues remain unclear. For example, the gauge covariant ...
- 1,670
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1
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197
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What is the Lie derivative of the field describing the change of mass?
I'm trying to understand Ch. 3.2 of the paper On Bubble Rings and Ink Chandeliers by Padilla et al.. I'm trying to understand the derivation of equation (15). Right now I'm stuck at the point where ...
2
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0
answers
86
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Doubt on $SU(2)_{L} \times U(1)_{Y}$ covariant derivative and its action on a fermion
I) Introduction
I.1)
The mathematical structure is quite clear: given a spacetime $M$, and a Lie group $G$ (the gauge group), we can construct the Principal bundle $P^{G}_{M}$. The connection $1$-form ...
- 3,159
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1
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124
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Einstein field equations from covariant derivative of a general linear gauge transformation
A general linear transformation is given by
\begin{align}
\psi'(x) \to g \psi(x) g^{-1},
\end{align}
The gauge-covariant derivative associated with this transformation is
\begin{align}
D_\mu \psi=\...
- 1,558
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1
answer
743
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Proof that the Euler-Lagrange equations hold in any set of coordinates if they hold in one
This is a question about a specific proof presented in the book Introduction to Classical Mechanics by David Morin. I have highlighted the relevant portion in the picture below.
In the remark, he ...
anon
0
votes
1
answer
102
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Higher dimension derivatives
In the case of higher dimensions (e.g. 4+1 dimensions) how would the 5 derivative ($\partial_5$) change?
For example if $\\X=(x^{\mu},z)$, would the 5 derivative change as $$\partial_5\partial^5X=\...
1
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0
answers
170
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What is the meaning of $\nabla _{\mu}\nabla _{\nu}\phi(r)$ in general relativity?
I know the covariant derivative of a tensor is
$$\nabla_{\mu} V_{\nu}=\partial_\mu V_\nu-\Gamma_{\mu\nu}^{\lambda}V_{\lambda}$$
Now I want to obtain $\nabla_{\mu}\nabla_{\nu}\Phi(x)$ where $\Phi(x)$...
- 67
1
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3
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437
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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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