All Questions
Tagged with differentiation notation
224 questions
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Mass Conservation in Kinetic Theory
In chapter 9 (The Boltzmann Equation) of Schwabl's 2006 text 'Statistical Mechanics', the author has the following statement of conservation of mass,
$$ \frac{\partial n}{\partial t} + \nabla \mathrm{...
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1
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53
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Derivative for the Maxwell field [closed]
I'm struggling with the following expression, which occurs in the derivation of the Maxwell Lagrangian in field theory.
$$\frac{\partial(\partial_{\mu}A^{\sigma})}{\partial(\partial^{\nu}A_{\lambda})}...
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0
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59
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What's the difference between $dx$ and $\delta x$? [duplicate]
In the process of defining crystal momentum $\hbar k$, I found these formulas below.
By the definition of group velocity,
$$v_g=\frac{d\omega_{nk}}{dk}=\frac{1}{\hbar}\frac{dE_{nk}}{dk}$$
Also if an ...
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1
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1
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98
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Proving a Superfunction Identity
I am trying to figure out the proof of the identity given between equations (1.11.7) and (1.11.8) in ref. [1], i.e.
\begin{align}
\Phi'(e^{-K}\,z\,e^K)=e^{-K}\Phi'(z) \tag{1}
\end{align}
where $z=(...
- 49
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1
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Meaning of colon symbol $:$ in optics
When I was reading some early days nonlinear optics paper/textbooks (particularly between 1960-1985), I often see expressions such as:
$\chi^{(2)}:\textbf{E}\textbf{E}$
or
$\nabla\textbf{E}:\partial \...
- 173
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84
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Odd notation $\stackrel{\leftarrow}{\nabla}$ for a gradient
I've tried working out the Heisenberg EOM for the 4-current operator. Two very beautiful articles (DOI: 10.1103/PhysRevA.84.042107, DOI: 10.1103/PhysRevA.90.012508) present this result, but I have not ...
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159
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What's the difference? $\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$
What's the difference? $$\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$$
In John Dirk Walecka's book 'Introduction to General Relativity',...
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110
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Component notation and matrix notation for gradient of vector
I'm trying to understand vector and tensor notation, but I'm coming across some difficulties. Say I have vector $\vec{u}$ and I compute its gradient $\nabla \vec{u}$. Then I get a tensor $\frac{\...
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2
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357
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Directional derivative $(\mathbf{A}\cdot\nabla)\mathbf{B}$ of the vector field $\mathbf{B}$
While reading Introduction to Electrodynamics by David J. Griffiths, I have encountered some issue with the notation of the directional derivative of the vector field and I was wondering if there are ...
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What does the notation $(k \cdot \nabla ) v$ mean? [duplicate]
I am reading a paper and it uses a notation I am not too familiar about. Although I saw it used elsewhere, I don't remember the meaning of it and I don't want to misinterpret it and realize after ...
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2
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Can the different differentiation notations be equated and do they have an integral definition? [closed]
Are these all equivalent and is there an extension of this to other notation?
Does anyone have a clear and concise chart equating the different notation dialects?
I am also curious if there are more ...
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3
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240
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What is the correct term for $\nabla\phi$? Co-vector or 1-form or both?
In the olden days, $\nabla\phi$ was used to be called a covariant vector (Weinberg used this language in his book Gravitation & Cosmology). But this terminology is considered bad for several ...
- 12.3k
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Definition of the left-right derivative symbol in the Klein-Gordon scalar product [duplicate]
At the start of QFT, studying the Klein-Gordon scalar field, it is often mentioned that the following is the definition of the scalar product in the space of the solutions:
$$\langle f _{\vec{k}}|f_{\...
- 4,635
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1
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289
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What is the difference between $\partial_{\mu}$ and $\partial^{\mu}$? [closed]
I've seen in many books both expressions $\partial_{\mu}$ and $\partial^{\mu}$, which are the covariant and contravariant partial derivatives, respectively, and in one of Susskind's books he defined ...
- 751
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Help with Commutators [closed]
I'm trying to self study quantum mechanics and am having a little trouble manipulating commutators. I get two different answers below, depending on the method I'm using. The second method gives me the ...
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114
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What does an upside down delta mean - covariant vectors? [duplicate]
I was scrolling through a wiki article on terminal velocity when I spotted an upside down delta. What does this symbol mean? How is it applied in other contexts?
EDIT: If possible could someone expand ...
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2
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494
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Difference and meaning of index the derivative operator
I'm a beginner in this type of math, we are just starting to study it, but I need some clarifications about the meaning and the difference of when we write
$$\partial_i \qquad \text{and}\qquad \...
- 383
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1
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113
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How does the $\not{\partial}$ work in the Dirac Lagrangian?
The Dirac Lagrangian (Density) is defined in the text "Quantum Field Theory, An Integrated Approach" by Fradkin as:
$$\mathcal{L}=\bar{\Psi}\left(i\not{\partial}-m\right)\Psi\equiv \frac{1}{...
- 179
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268
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Why the $\Delta$ in the definition of pressure? (fluid mechanics)
I'm an engineering student (first year) studying Physics 1 (now an introduction to fluid mechanics).
Q1
In my physics textbook, the "medium pressure" is defined as:
$$p_m = \frac{\Delta F_{\...
- 141
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1
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153
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Differentiating the index notation
I am always confused with the algebra of differentiating the index notation, and have browsed many other posts but still confused. There must be details I have been missing. It would be really ...
- 41
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231
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Transformation of Yang Mills Field Strength
I am confused about the expression $$F_{\mu \nu} \to F_{\mu \nu}' = U F_{\mu \nu}U^{\dagger}.$$ I found related Phys.SE posts How would one show that a nonabelian field strength tensor transforms in a ...
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Evaluating the commutator of derivative and position [duplicate]
In Zettili's book on quantum, the fully worked problem 2.6 asks to show
$$
\hat{A} = i(\hat{X}^2+1)\frac{d}{dx} + i\hat{X}.
$$
Is Hermitian. Where $\hat{X}$ is the position operator. I took the ...
2
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1
answer
355
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$\nabla$, $\cdot \nabla$, $\nabla \cdot$, $\nabla^2$ - What do they do? [closed]
I'm trying to teach myself Smoothed Particle Hydrodynamics. Unfortunately, my background is in electronics, so the Navier Stokes equations are somewhat alien to me, as is vector calculus. The video I'...
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1
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86
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Bianchi identity contradiction in Abelian case
In non-abelian gauge theory, such as P & S's chapter 15, eq. (15.89), we also have Bianchi identity.
Start with
$$\epsilon^{\mu\nu\lambda\sigma}[D_\nu,[D_\lambda,D_\sigma]]=0$$
and use $[D_\mu,D_\...
- 1,461
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85
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What does $\partial_ν/\partial^2$ mean?
I found such notation in this article link, equations 24-25. I know that $\partial_μ$ is four-gradient, but it does not contain second-order derivatives. Only d'Alembert operator does, $\partial^μ\...
- 357
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2
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122
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Understanding this Lagrangian calculation
I was trying to understand this section of a Wikipedia article:
$$0 = \delta \int \sqrt{2T} d\tau =
\int \frac{\delta T}{\sqrt{2T}} d\tau =
\frac{1}{c} \delta \int T d\tau$$
For the life of me, ...
- 238
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Does this particular notation for derivatives imply anything in particular? [duplicate]
In some physics textbooks (and in those of other sciences that use physics, like soil science), I've seen some derivatives written as:
$$\frac{\delta f}{\delta t} $$
Which is a bit strange. Does this ...
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57
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What does this vertical line notation mean?
Here is the definition of the Noether momentum in my script.
$$I = \left.\frac{\partial L}{\partial \dot{x}} \frac{d x}{d \alpha} \right|_{\alpha=0} = \frac{\partial L}{\partial \dot{x}} = m \dot{x} = ...
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94
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Partial Differential with independent quantities held constant meaning?
$$ \mu_{JT}=\left(\frac{\partial T}{\partial P}\right)_H= \frac{V}{C_p}(\alpha T -1) $$ and
$$\left(\frac{\partial T}{\partial P}\right)_H \left(\frac{\partial H}{\partial T}\right)_P \left(\frac{\...
- 861
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225
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Thermodynamics Chain Rule And Independent Variables
I was reading my textbook and I came up across the entropy $S(T,V,N)$ where temperature $T$, volume $V$, and number of particles $N$ are the independent variables. According to the chain rule the ...
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1
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288
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Question on how to make product rule for differentiation consistent with operators? [duplicate]
By the product rule for differentiation:$$\frac{\partial(\hat A\psi)}{\partial x}=\left(\frac{\partial\hat A}{\partial x}\right)\psi+\hat A\left(\frac{\partial\psi}{\partial x}\right)\tag{1}$$
Where $\...
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Understanding the use of $d$ and $\partial$ in thermodynamics
It seems a hundred variations of this question have been asked, and it's difficult to find which of those questions relates to exactly what I'm asking. My apologies if exactly this question has ...
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2
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119
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I'm having trouble understanding exactly what $δ$ represents in thermodynamics [duplicate]
I know that $δ$ sometimes represents the Dirac delta function but in my book it states "Suppose that equilibrium has been established Then a slight change in the position of the piston should not ...
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1
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152
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Meaning of the notation $(D_\nu F_{\lambda\sigma})^a$ in Bianchi's identity
I'm studying Peskin and Schroeder chapter 15, on page 500, we have the Bianchi's identity in nonabelian gauge theory,
$$\tag{15.89} \epsilon^{\mu\nu\lambda\sigma}(D_\nu F_{\lambda\sigma})^a=0$$
Here $\...
- 859
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1
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164
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Question regarding Energy Interaction of two particles
https://i.sstatic.net/LUsKX.jpg
To give a context as to what I'm asking here ,I am talking about the energy of a two particle system (section 4.9 Taylor's Classical Mechanics) .
My question is what ...
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2
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267
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What does $\dot x$ mean as an operator in quantum mechanics?
I've been looking at a paper titled "Feynman's proof of the Maxwell Equations" by Freeman Dyson (American Journal of Physics 58, 209 (1990); https://doi.org/10.1119/1.16188) and I'm confused ...
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1
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61
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What is $A'$ in the Reissner-Nordstrom metric?
So I was reading this paper on the Reissner-Nordstrom metric and on it they define $A$ as:
But they don't define $A'$. Yet $A'$ still ends up in other equations like defining the Ricci tensors:
So ...
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242
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Commutator between covariant derivative and a field
I have field as an element of a Lie algebra as $\Phi = \phi^at^a$ and I want to calculate the commutator $$[D_{\mu}, \Phi],$$
with $$D_{\mu} = \partial_{\mu} + igA^a_{\mu}t^a,$$ the covariant ...
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161
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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 ...
user89243
2
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3
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198
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Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $
My course guide gives the following derivation for change in entropy w.r.t. energy, where I don't understand a step:
\begin{align}
E & = E_1 + E_2 \\
S & = S_1 + S_2 \\
S(E,E_1 ) & = S_1 (...
- 2,180
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1
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134
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Velocities - Equation 1.46 of Goldstein 3rd edition
In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein
uses the parametrization (equation 1.45')
$$\displaystyle{\vec{r_i}=\vec{r_i}(q_1,q_2, ..., q_n, t)}\tag{1.45'}$...
- 113
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Can we define $\text dW$? [duplicate]
I am currently taking applied thermodynamics at my university, and for the definition of entropy this is the formula used in the book (Thermodynamic for Engineers by Moran, Shapiro, Boettner, Bailey): ...
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1
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64
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Convective derivative N-S
This is probably an easy answer, but I've not been able to find it yet -
Why in some formulations of the N-S equations (for example here https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html), is the $...
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159
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Notation and Terminology Questions from Schwartz' QFT Book
I am finding some of the notation confusing in Chapter 3 on Classical Field Theory in Schwartz' QFT book a bit confusing.
First off, on page 34 he defines a translation of a field to first order as
$$...
- 221
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2
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831
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Maxwell's eq-meaning of del's cross and dot product?
In maxwell's eq there is del whose cross and dot products exist.
So what is del in cross vs dot product.
What's the difference when it's just a partial differential operator.
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89
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Does the expression "$𝑑𝑠^2$..." mean the same thing as "$\Delta 𝑠^2$... "?
I reviewed this question but sometimes I'm unsure about delta ($\Delta$) versus differential ($d$) notation.
Does the expression "$ds^2=-c^2dt^2+a^2(t)[dr^2 + S_k^2(r)d\Omega^2 ]$" mean the ...
- 181
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4
answers
793
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Why do we use different differential notation for heat and work?
Just recently started studying Thermodynamics, and I am confused by something we were told, I understand we use the inexact differential notation because work and heat are not state functions, but we ...
- 283
2
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1
answer
2k
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Meaning of the transpose of a gradient
Sometimes I encounter PDE's with a term like this
$\nabla \cdot c(\nabla \textbf{v} + (\nabla \textbf{v})^T)$
An example are the Navier-Stokes equations. Oftentimes this can be further simplified to $...
- 25
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1
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286
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Covariant derivative with an upper index in terms of Christoffel symbols
I have encountered expression
$$\frac{1}{2}\left(2 \dot{g}_{\mu}{}^{\lambda ; \mu}-\dot{g}_{\mu}{}^{\mu ; \lambda}\right)$$
in a GR paper.
Here we assume to be working with the de Sitter metric $g$ ...
- 1,122
1
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1
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394
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How to compute divergence of a metric tensor?
I am reading a paper where the author defines the divergence to be
$$\left(\delta_{g} \dot{g}\right)_{\mu}:=-\dot{g}_{\mu \kappa;}{}^{\kappa}$$
where $g$ looks like the De Sitter metric,
$$g=(3 / \...
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