All Questions
3,004 questions
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From material derivatives to partial derivatives in the wave equation
Consider the Cauchy momentum equation:
$$\rho \frac{d^2 \mathbf{u}}{d t^2} = \nabla \cdot \boldsymbol{\sigma} + \rho \mathbf{f}$$
where $\rho(\mathbf{x},t)$ is the density, $\mathbf{u}(\mathbf{x},t)$ ...
- 225
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2
answers
36
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Why must the total time derivative only be a linear function of velocity? [duplicate]
I'm hung up on page 7 of Landau & Lifshitz Course on Mechanics. They claim,
$$L(v'^2) = L(v^2)+\frac{\partial L}{\partial v^2}2\textbf v\cdot \epsilon \tag{p.7}$$
The second term on the right of ...
1
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0
answers
32
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How much does classical mechanics depend on the choice of symplectic form?
TlDr; a different choice of symplectic structure on a phase-space $\mathcal{M}$ affects the Hamiltonian mechanics insofar as it could affect what the canonical coordinates are, but is this the only ...
- 230
3
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1
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118
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Covariant derivative acting on Dirac delta function
Pardon my naive computational question. In my calculations, I encounter the following expression:
\begin{equation} \label{eq1}
\frac{\delta}{\delta g^{\gamma \epsilon}(z)} \left( g_{\mu \alpha}(x) \...
- 85
1
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0
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56
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Determine the maximum and minimum acceleration from the velocity field [closed]
If the speed distribution, in m/s, of a flow is given by $v = 2x^3 + 2y^2 - 3z$, then the acceleration of the fluid at the point with coordinates $[2, 1, 5]$, in meters, will be greater than $20, \...
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0
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40
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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,
$$ \frac{\partial n}{\partial t} + \nabla \mathrm{...
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0
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63
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Four gradient relation
I'm doing an exercise in QFT and I have to calculate the energy-momentum tensor for the Klein-Gordon Lagrangian and by doing it I got the following term:
$$ \frac{\partial \ \partial^{\nu}\phi}{\...
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1
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81
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Problem in deriving Killing equation
I am studying derivation of Killing equation by Wald (also reading some other literature) but having some problem in understanding the math.
Let $\chi ^a$ is killing vector on the horizon
$$\chi _{[a \...
- 185
3
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2
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340
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Understanding the definition of the covariant derivative
I'm currently working my way through the book "Mathematical Methods for Physics - An Introduction to Group Theory, Topology and Geometry" and I think I have a very fundamental ...
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1
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59
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Need help in understanding Tangential Acceleration [closed]
I am studying Circular motion and I am confused about tangential acceleration and tangential velocity. I am studying uniform circular motion and it says the tangential acceleration is $0$ in uniform ...
3
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1
answer
67
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"Deriving" the covariant derivative
Suppose we are working in scalar QED with Lagrangian
$$\mathscr{L} = (D_\mu \phi)(D^\mu \phi)^* - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$
I now want to find the form of the covariant derivative $D_\mu$ ...
- 283
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1
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53
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Derivative for the Maxwell field [closed]
I'm struggling with the following expression, which occurs in the derivation of the Maxwell Lagrangian in field theory.
$$\frac{\partial(\partial_{\mu}A^{\sigma})}{\partial(\partial^{\nu}A_{\lambda})}...
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1
answer
80
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The definition of the Lie Derivative
I am aware that an answer to an almost identical question already exist, however, I found the already existing answer not helpful (at least to my current question).
Carroll defines, in his book, the ...
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1
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70
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Why are Weyl's Equations composed of only first-order derivatives?
I'm studying the Weyl's Equations from Section 1.5 of Perkins' Introduction to High Energy Physics.
The author says this:
Dirac set out to formulate a wave equation symmetric in space and time, ...
- 1,637
1
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1
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45
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I can't understand why the divergence of an electric field from a point charge is 0 for points away from the source
I understand that for a positive point charge at the origin, the resulting electric field extends out radially and has magnitude that decreases by $\frac{1}{r^2}$. However, I do not understand why the ...
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4
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4k
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Is it ever possible that the object is moving with a velocity such that its rate of change of speed is not constant but acceleration is constant?
Is it ever possible that the object is moving with a velocity such that its rate of change of speed is not constant, but rate of change of velocity is constant?
Like speed is only the magnitude, so ...
1
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1
answer
50
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Grassmann Numbers, anticommutation and derivative rules
If $\psi(t)$ is a complex Grassmann number and $\psi^*(t)$ is its complex conjugated. The following is true:
$$\frac{\partial (\psi^*\psi)}{\partial \psi}=-\psi^*\frac{\partial \psi}{\partial \psi}=-\...
- 1,628
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0
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70
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Use of $dv/ds$ in defining acceleration [duplicate]
We can write acceleration as either
$dv/dt$ or $v dv/ds$.
And surprisingly the work-energy theorem arrives from the second definition.
I feel it would be fundamentally understanding towards work ...
2
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1
answer
139
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Quantum mechanics in the language of differential geometry [closed]
So I am currently studying differential geometry and start recognizing a lot of concepts familiar from physics in the toolset of manifolds, tangential bundles and vector fields. In particular, we can ...
- 342
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1
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Why the two methods give correct answer for the killing fields?
Consider the embedding of the $\mathbb{S}^2$ in $\mathbb{R}^3$,
\begin{align}
x & = r \sin \theta \cos \phi,\tag1 \\
y & = r \sin \theta \sin \phi,\tag2 \\
z & = r \cos \theta,\tag3
\end{...
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0
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17
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How to prove that a Lie algebra-valued differential form is exact for the covariant derivative [migrated]
Given a differential $p$-form $\omega^A$ over a smooth manifold with values on some Lie algebra, I wanted to know how could one prove that it can be written as an exact form for the exterior covariant ...
- 381
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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}$$
Also if an ...
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1
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66
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Is Stress a Derivative?
On page 289 of the text "Fundamentals of Fluid Mechanics" by Munson et al., the authors give the following definition of the normal stress acting on the surface of a fluid element:
At any ...
1
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2
answers
118
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Why is electric field strength indicated by the density of field lines? [duplicate]
It is well known that the magnitude of the electric field is indicated by the density of field lines. However, is it a physical law or additional rule that helps us to draw informative diagrams?
In ...
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1
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1
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89
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Finding Killing vectors for hyperbolic space [closed]
I want to find the Killing vectors for the hyperbolic space, which is described by the metric
\begin{equation}
ds^2 = \frac{dx^2 + dy^2}{y^2}.
\end{equation}
I have found the Killing equations, which ...
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3
votes
1
answer
480
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Second derivative of unit vector
We know that the second derivative of unit vector (the vector from a point toward the source) is proportional to the Electric field caused by the source in a particular point.
If we imagine that our ...
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3
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1
answer
114
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Relationship between covariant derivative and metric tensor
In general relativity, the covariant derivative of the coordinate vector is a tensor, equal to $$x^{\mu}_{:\rho} = x^{\mu}_{,\rho} + \Gamma^{\mu}_{\rho\nu}x^{\nu},$$ is it meaningful to equate this ...
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1
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67
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Where does $[\delta_{\xi_1},\delta_{\xi_2}]=\delta_{(\xi_2\partial \xi_1-\xi_1\partial\xi_2)}$ come from in CFT?
The authors in the book "Bacis concepts of string theory" say on page 69 that
"Using the commutation properties of infinitesimal conformal transformations $$[\delta_{\xi_1},\delta_{\...
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0
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67
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Is divergence infinite when the source is a single point?
EDIT
Can divergence of a vector field be positive everywhere or almost everywhere?
Can it be positive (but finite) at just one point? Is it the case when the point is the source of gas flow, from ...
- 101
1
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1
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86
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Dot product under parallel transport
I was wondering about simple justification that parallel transport preserves dot product value. I came up with an idea like that:
$u^i$ - 1st vector in covariant basis
$v_i$ - 2nd vector in ...
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4
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2
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243
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Leibniz rule and Nakahara's definition for functional derivatives with respect to Grassmann variables
In Nakahara's book "Geometry, Topology and Physics" in section 1.5.7 (I'm reading the second edition) he defines the functional derivative with respect to Grassmann variables. He does so in ...
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A trick for derivatives of thermodynamic quantities [closed]
Starting from
$$dU=TdS-PdV$$
We can write, for instance $U(T,V)$ and $S(T,V)$ to obtain:
$$\left(\frac{\partial U}{\partial T}\right)_VdT+\left(\frac{\partial U}{\partial V}\right)_T dV=T\left(\frac{\...
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0
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41
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Showing surface integral vanishes at infinity - extinction theorem
Suppose the dyadic Green's function $\overline{G}$ is defined as
\begin{equation}
\overline{G}(r, r') = \left(\overline{I} + \frac{\nabla \nabla}{k^2}\right)\frac{e^{-ik\|r - r'\|}}{\|r - r'\|}
\...
0
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4
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87
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Maxwell's third equation for static fields [duplicate]
Since electrostatic field lines are bending why it's curl is zero
specifically asking about charge distribution with bending field lines (example - Dipole)
Please explain with physical meaning not ...
- 13
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votes
0
answers
30
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Kinematical and dynamical symmetries and their relevance
T.i.l that kinematical symmetries are symmetries generated by a Killing vector field $ \pi_*(X_q) $,
$$ {\cal L}_{\pi_*(X_q)}g=0,$$
which is given by the pushforward $ \pi_*(X_q) $ of $X_q$, where $$ \...
- 319
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0
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36
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Is this mathematically correct that gradient of deformation gradient is equal to deformation gradient?
The deformation matrix is defined as follows, where $x$ is the current location and $X$ is the reference location. It shows the relationship between current $x$s with regard to original $X$s,
$$F = \...
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0
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Conflicting Solutions for Calculating Apparent Speed of Jogger's Image in Convex Mirror
I’m facing a challenge with a physics problem due to conflicting solutions across different sources, and I'd appreciate some clarification.
Problem Statement:
Suppose, while sitting in a parked car, ...
1
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0
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30
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Spacetime symmetry of Noether's theorem and Killing vector field
I guess the term "spacetime symmetry" of field indeed implies the symmetry has a strong connection with the Killing vector field. But I have no idea how to express it formally.
For Arnold's ...
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70
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Total energy and time derivative of it in general relativity
I'm studying general relativity through Sean Caroll's book (An introduction to gr; Spacetime and geometry which must be quite similar to its notes)
It defines total energy as: $$E_T=\int_\sum J^\...
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0
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35
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Momentum density of electromagnetic field
I'm reading a textbook deriving the expression for the momentum densisty of the EM field and it starts with the following concept. Let's define the vector field $\vec{p}_\text{me}(\vec{r},t)$ to be ...
- 371
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1
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What happens to $g^{\alpha\beta}_{,\sigma}=-g^{\alpha\mu}g^{\beta\nu}g_{\mu\nu,\sigma}$ when $g_{\mu\nu}\rightarrow \eta_{\mu\nu}$ (weak field limit)?
The equation
$$g^{\alpha\mu}_{\,\,\,\, ,\sigma}\,g_{\mu\nu} + g^{\alpha\mu}\,g_{\mu\nu,\sigma} = (g^{\alpha\mu}g_{\mu\nu})_{,\sigma} = \delta^\alpha_{\nu,\sigma} = 0 $$
gives the useful relation
$$g^{\...
- 467
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21
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What happens when a car starts moving? The last moment the car is at rest versus the first moment the car moves
Imagine a car that's at rest and then it starts moving. Consider these two moments:
The last moment the car is at rest.
The first moment the car moves.
The question is: what happens between these 2 ...
- 371
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136
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Incorrect proof that four-current $J^\mu$ is a four-vector
This question is inspired by this answer to a question about proving that $J^\mu$ is a four-vector.
The answer uses the continuity equation $\partial_\mu J^\mu = 0$ and the experimental fact that ...
- 949
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0
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38
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Four-divergence of a vector [duplicate]
The covariant derivatives of a four-vector is
$$
\nabla_{\nu}U_{\mu} = \partial_{\nu}U_{\mu} - \Gamma^{\lambda}_{\mu\nu}U_{\lambda}
$$
It has the following identity:
$$
\nabla_{\mu}U^{\mu} = \frac{\...
- 49
1
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1
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481
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Doubt in Verlet's Algorithm
In studying the temporal evolution of a system according to the deterministic model, we begin by considering a Taylor series expansion for the displacement $r$. First, we consider a positive variation ...
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1
answer
96
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Why take the derivative of variables such as area, mass, and radius?
I'm taking a module on stars and the solar system; I've attached notes from our first lecture- hydrostatic equilibrium. I'm confused about the notation $\mathrm{d}$ for $\mathrm{d}A, \, \mathrm{d}r$, ...
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1
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69
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Is 4-velocity a Vector in the Sense of Covariant Derivative along Worldline
The definition of 4-velocity $U^{\mu} \equiv dx^{\mu}(\tau)/d\tau$, however, we've learnt that the covariant derivative for a vector along a curve parametrized by proper time is,
$$\frac{DA^{\mu}}{D\...
- 938
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1
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75
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Derivative wrt retarded time
I am confused by the following statement in footnote of Griffiths 4th edition (page 446):
$$\frac{\partial }{\partial t_r} = \frac{\partial }{\partial t},$$ where $$t_r=t - \frac{\mathscr{r}}{c}$$ ...
5
votes
4
answers
386
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Vector triple product with $\nabla$ operator
I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):
$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
- 61
1
vote
2
answers
44
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Perfect gas relation in differential form [closed]
I have a problem to understand the transformation of the perfect gas relation:
$$ \rho\cdot R\cdot T = P $$
into its differential form:
$$\frac {dp}{p} = \frac {d{\rho}}{\rho} + \frac {d{T}}{T}$$
How ...
- 21