All Questions
Tagged with covariant-derivatives or differentiation
1,900 questions
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How does the chain rule work in sound wave analysis using fluid mechanics? $\tfrac{d x}{dt}\neq v$?
Context:
I am reading Landau & Lifshitz's book on Fluid mechanics. Specifically its section on Sound waves.
In section 101, the book's authors discuss about nonlinear traveling waves in one ...
- 23
3
votes
1
answer
310
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What's the physical meaning of Curl of Curl of a Vector Field?
The curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$
Now, curl means how much a vector field rotates ...
- 121
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75
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Deriving Klein-Gordon equation from Euler-Lagrangian equation: Taking partial derivative inside [duplicate]
Lagrangian for Klein-Gordon equation is given by
$$L=\frac{1}{2}\partial_\mu \phi \partial^\mu\phi - m^2\phi^2/2.$$
To derive Klein-Gordon Equation I have to Compute derivative in Euler-Lagrange ...
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2
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89
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How to calculate the final position of a particle under variable accelaration and its instantenous velocity?
I'm a first-semester physics student who was recently on a train. On a screen, it said the instantaneous velocity of the train was 176 km / h. We had 4 min left until our destination. I wanted to ...
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1
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2
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325
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Question regarding error analysis of focal length of a lens [duplicate]
The question in whose context i am asking this question is as follows
In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10 \pm ...
- 173
0
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2
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84
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Solving a PDE using $x-vt$ as a variable
So I was reading this Landau and Lifshitz paper:
https://doi.org/10.1016/B978-0-08-036364-6.50008-9
The article can also be found without a paywall by just searching its title, "On the Theory of ...
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0
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56
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Partial derivative operator
It's mentioned in this paper that if $\partial^i \partial^j$ applied on an equation like:
$$
x \delta_{ij} + (\nabla^2 \delta_{ij}- \partial_i \partial_j) y =0
$$
It yields a couple of equations:
$$
...
- 405
0
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1
answer
75
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Lie derivative: moving boat on a flowing river
Lie derivatives signifies how much a vector (Tensor) changes if flown in the direction of some other vector. I am thinking the typical moving boat on a flowing river problem where the river is flowing ...
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1
vote
1
answer
62
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Deriving the Curl of the Magnetic Field, Role of the Nabla Operator
We know that the magnetic field can be written in the following way:
$$\nabla_{\vec r }\times\vec B(\vec r) = \frac 1 c \nabla_{\vec r}\times\int d^3\vec r_q\ \vec j(\vec r_q)\times \frac {\vec r-\vec ...
- 193
1
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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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2
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234
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Why is the differential form of Gauss's Law equivalent to the integral form?
I can understand the Differential form of Gauss's Law ∇⋅𝐄= $\frac{ρ}{ɛ_0}$
as saying that the source of electric field vectors or flow disperse(The divergence of the electric field) is equal to the ...
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0
votes
1
answer
89
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In $a = dv/dt$, is $a$ the net acceleration? [closed]
While going through the calculus approach to accelerate, we have,
$$a = dv/dt, $$
I think, here, v and a should be in the same axis,
is my process correct?
in a planar motion in two dimensions, it ...
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3
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96
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Proof that small change in temperature leads to small change in entropy
I have been trying to find a mathematical proof (or even from a reliable source) which verifies that/proves that:
A small change in temperature leads to a small change in entropy.
However, I was ...
1
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1
answer
61
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Tensor equation
What is a valid tensor equation. In the book by Bernard Schutz, it is often argued that a valid tensor equation will be frame invariant. So the conclusions reached by relatively easy calculation done ...
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0
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43
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Generator normalisation in the covariant derivative
A common convention for the definition of the covariant derivative in the SM is
$$
D_\mu = \partial_\mu - i g_s \frac{\lambda^a}{2}G^a_\mu - \cdots
$$
where $\lambda^a$ are the Gell-Mann matrices. In ...
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1
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1
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142
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First law of thermodynamics: Can we always speak in terms of infinitesimal changes?
While reading lecture notes for the course on thermodynamics I have encountered some tiny details that seem extremely important for the understanding of the topic. However, something seems amiss so, I ...
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1
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105
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Derivation of the state equation of a van der Waals gas. Can I invert the derivative to help me?
The state equation of a van der Waals gas is
$$\left(P+\frac{a}{v^2}\right)(v-b)=RT$$
with $a,b$ and $R$ constant. Find $$\frac{\partial v}{\partial T}\bigg\rvert_P.$$
Finding $\frac{\partial v}{\...
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1
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3
answers
176
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Where to apply $\nabla$ operator when taking curl of a cross product?
In my EM class we went over $$\nabla\times \frac{\vec{d}\times \vec{r}}{r^3}$$ which apparently can be breaken down to $$r(d\cdot \nabla)\frac{1}{r^3}-d(r\cdot\nabla)\frac{1}{r^3}+\frac{\nabla\times(d\...
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2
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142
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Average velocity showing different results
I was solving a question, in which, a particle has travelled a distance $s$, with initial velocity $0$ and constant acceleration.
So the equation of motion becomes,
$$ v = a t \tag{1} $$
and
$$ v = \...
- 56
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3
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363
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How to derive $i=I_0 \sin(wt)$ in alternating current? [closed]
Our teacher taught us today that instantaneous value of current in Alternating Current is
$$i=I_0 \sin(wt)$$
Where $I_0$ is the amplitude and $wt$ is the angular speed times time. Now, she didn't ...
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1
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41
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Is derivative of a primary operator primary?
For primary operators we have $$T(z)O(w)=\dfrac{hO(w)}{(z-w)^2} + \dfrac{\partial_wO(w)}{z-w}+\cdots$$
(the ordering problem can be ignored by setting $|z|>|w|$.)
If we apply $\partial_w$, then a $(...
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3
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125
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Deduction of Kinetic energy operator in quantum mechanics
In Chapters 1 and 2 of Introduction to Quantum Mechanics Third edition, Griffiths and Schroeter state that to get kinetic energy operator one replaces momentum with $p\rightarrow -i\hbar\,\partial/\...
0
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0
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117
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Are eigenvalues of slashed covariant derivative real?
I am trying to demonstrate that the slashed covariant derivative
$$
\gamma^\mu D_\mu = \gamma^\mu(\partial_\mu -iA_\mu)
$$
has real eigenvalues:
$$
\gamma^\mu D_\mu \varphi_m(x)=\lambda_m \varphi_m(x)...
- 161
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1
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98
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Derivation of lagrange equation in classical mechanics
I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
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90
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How to take the second-order gauge covariant derivative in quantum field theory?
I am studying quantum field theory and gauge theory, and I am confused about how to take the second-order gauge covariant derivative of a field.
(1) The first way is to write the second order gauge ...
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3
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107
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Is there no sense of 'absolute' in the universe?
Imagine we are talking about electric potential (e.g. gravitational potential or electric potential or whatever, it doesn't matter), then we have:
\begin{equation}
dV = \textbf{E} \cdot d\textbf{l},
\...
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0
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47
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Finding condition for Adiabaticity
I have a differential equation describing a resonator that looks like this:
$$ \frac{d\alpha(t)}{dt} = [j a - b]\alpha(t) + \sqrt b e^{jct}$$ where I can solve it putting: $$\alpha(t) = \alpha e^{jct}$...
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1
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84
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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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3
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294
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Are we allowed to cancel the units of a derivative?
Since the volume of a sphere $v(r)=\frac{4}{3} \pi r^{3} \left[m^{3}\right]$, its derivative relative to the radius is:
$$
\frac{dv}{dr} =4\pi r^{2} \left[\frac{m^{3}}{m}\right]
$$
Which is also a ...
0
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1
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346
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Isomorphism of the tangent space and the space of directional derivatives [closed]
I have already constructed the tangent space to a manifold, denoted $T_pM$, and I have a good basis for it $\{\hat e_{(\mu)}\}$. (I followed the method of equivalence classes of curves tangent at $p$....
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2
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62
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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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0
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46
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Application of Fermi-Walker derivative to specific problem
I am now reading about the tetrad formalism in GR and I am starting (how not) with the Wikipedia Article:
Frame fields in general relativity.
In this article, as an example, they show how tetrads can ...
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3
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0
answers
82
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Spectrum of Harmonic Oscillator with Ladder Operators [closed]
Is the ladder operator trick specific to the harmonic oscillator or can it be generalized to arbitrary second order operators? If yes, what is the general mathematical theory behind it? Can all ...
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1
answer
48
views
Time taken for a rocket to travel upwards [closed]
My doubt is a rather silly, simple one but i cant seem to understand what's wrong.
Let's assume a rocket is moving up with a constant acceleration of a, is moving strictly vertically(no gravity turns, ...
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2
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259
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Why does $e^{-H}\partial_j e^{H} = \partial_j + \partial_jH$?
I apologize if this is a dumb question but I have really thought about this a while and I can’t understand it. I have tried to prove this using the power series of the exponential function but I did ...
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2
votes
1
answer
156
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Dirac Delta applied to the gradient of a function
The supplementary section of a paper I am reading uses the "substitution" property of the Dirac delta function :
$$\mathbf{v}\left(\mathbf{x}_j\right)\delta\left(\mathbf{x}-\mathbf{x}_j\...
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1
vote
2
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86
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How to use differentials in thermodynamics? [closed]
I wonder how to use and manipulate differential forms in thermodynamics.
I see for
$ U= αPV$, it is written $dU = αPdV + αVdP$
But how this works in terms of differentiation? (Proof)
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107
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When to use multivariable chain rule in thermodynamics?
If I take $U(P,V)$, I can do:
$$ \frac{dU}{dT} = \frac{\partial U }{\partial P } \frac{dP}{dT}+\frac{\partial U }{\partial V} \frac{dV}{dT} \tag1$$
But, I see the following used in books,
$ dU = ...
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0
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Problem in calculation of spherically symmetric Laplacian in electrodynamics
I have come across the following operation in two electrodynamics textbooks, which I find problematic: When evaluating an integral over a Laplacian in a spherically symmetric function, the radial term ...
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1
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190
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Covariant derivative to the metric determinant?
I am reading the paper Alternatives to dark matter and dark energy, but cannot obtain one specific equation no matter how I tried. So I wrote an email to the author, the following is what he replies ...
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4
answers
440
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Variation of a function
I'm studying calculus of variations and Lagrangian mechanics and i don't understand something about the variational operator
Let's say for example that i got a Lagrangian $L [x(t), \dot{x}(t), t] $ ...
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1
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1
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201
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How do I get a derivative of the field inside of the path integral?
I am trying to find the 3-gluon vertex rule in QCD by finding the amplitude of a 1-2 gluon scattering process. I want to find the generating functional of the interaction by taking the functional ...
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1
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82
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Sufficient condition for conservation of conjugate momentum
Is the following statement true?
If $\frac{\partial \dot{q}}{\partial q}=0$, then the conjugate momentum $p_q$ is conserved.
We know that conjugate momentum of $q$ is conserved if $\frac{\partial L}{\...
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D'Alembert Solution to 1+1D wave equation - integration step
I am working through d'Alembert's solution to the 1+1D wave equation using the substitution of canonical coordinates. I have an initial condition of: $$u_{t}(x,0) = g(x) $$ with a general solution ...
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1
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74
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Closed interval in variation of a field
Let's suppose that $\psi$ is a field so that $$\psi\in\Gamma^\infty(\pi):=\{\psi\in C^\infty(M,E)\ |\ \pi\circ\phi=I_M\}$$ where $E$ is a spacetime bundle over spacetime $M$ and $\pi : E\rightarrow M$ ...
user361492
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3
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240
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What is the correct term for $\nabla\phi$? Co-vector or 1-form or both?
In the olden days, $\nabla\phi$ was used to be called a covariant vector (Weinberg used this language in his book Gravitation & Cosmology). But this terminology is considered bad for several ...
- 12.3k
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1
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103
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Time derivative of a "general" vector $\vec A$ in an accelerating frame: what about e.g. velocity $\vec v$?
According to Morin "Classical Mechanics" (Section 10.1, page 459), the derivative of a general vector $\vec A$ in an accelerating frame may be given as
$$\frac{d\vec A}{dt}=\frac{\delta \vec ...
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1
vote
1
answer
75
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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
1
vote
1
answer
109
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Spherical coordinate of a vector when divergence of the vector is zero
$\nabla \cdot \mathbf{\delta u_{perp}} = 0$ where $\mathbf{\delta u_{perp}}$ is a function of both x and y coordinates and perpendicular to z axis. Moreover, $\delta u_{perp}$ along z axis is $0$.
I ...
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1
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115
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Double covariant derivative of a mixed tensor
Let's say, we have a mixed tensor of type (2,1) denoted by $T^{mn}{}_p$ and the goal is to find the expression of $[\nabla_a, \nabla_b] T^{mn}{}_p$ in terms of fundamental tensors.
Firstly, I am ...
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