All Questions
Tagged with covariant-derivatives or differentiation
1,900 questions
1
vote
2
answers
357
views
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 ...
- 85
2
votes
1
answer
81
views
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
0
votes
1
answer
89
views
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 ...
- 1
0
votes
2
answers
89
views
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 ...
- 127
1
vote
2
answers
142
views
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
-1
votes
3
answers
96
views
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
vote
1
answer
69
views
Material to Study the Definition, Algebra, and Use of Infinitesimals in Physics [closed]
This is going to be a rather general question about suggestions on best supplementary material to properly explain the use of infinitesimals (or differentials?) for the purposes of integration or ...
0
votes
1
answer
86
views
In Lagrangian mechanics, do we need to filter out impossible solutions after solving?
The principle behind Lagrangian mechanics is that the true path is one that makes the action stationary. Of course, there are many absurd paths that are not physically realizable as paths. For ...
- 51.8k
154
votes
9
answers
19k
views
Calculus of variations -- how does it make sense to vary the position and the velocity independently?
In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat ...
- 2,235
0
votes
1
answer
111
views
Solving divergence and curl equations numerically
I've recently come to learn about Jefimenko's general solution for Maxwell's equations as well as the FDTD method in electromagnetic optics, and that has got me thinking whether I myself can solve ...
- 1,880
2
votes
6
answers
274
views
Lagrangian - How can we differentiate with respect to time if $v$ not a function of time?
In the Lagrangian itself, we know that $v$ and $q$ don't depend on $t$ (i.e - they are not functions of $t$ - i.e., $L(q,v,t)$ is a state function.)
Imagine $L = \frac{1}{2}mv^2 - mgq$
Euler-Lagrange ...
- 535
1
vote
1
answer
142
views
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 ...
- 85
0
votes
1
answer
75
views
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 ...
- 85
3
votes
4
answers
638
views
Derivation of covariant derivative
I'm currently doing Introductory QFT and was confused about the origin of the additional terms in the covariant derivate. My understanding is as follows:
If we begin with the Dirac Lagrangian ...
- 100
0
votes
0
answers
58
views
Partial derivatives of Christoffel symbols to Covariant derivatives
I wanted to express this thing: $g^{ab}\partial_c\Gamma^c_{ab} - g^{ab}\partial_a\Gamma^c_{cb}$, in terms of a covariant derivative. I figured out that if you swap $a$ and $c$ in the $\partial \Gamma$ ...
- 23
0
votes
1
answer
76
views
Time derivative of moment of inertia tensor
Suppose that I have some fluid in a fixed volume. Its moment of inertia is given by $I=\int\rho r^2dV$. The derivative of $I$ is given by $\dot{I}=\int\frac{\partial\rho}{\partial t}r^2dV$. Why do we ...
- 105
0
votes
2
answers
414
views
Why does tangential acceleration become 0 when the velocity is max? [closed]
I know that tangential acceleration equal to zero when the circular motion is uniform, but why is it equal to zero, when the velocity is max or min? Because there is no relation between the value of ...
- 11
0
votes
1
answer
105
views
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.$$
Finding $\frac{\partial v}{\...
- 11
0
votes
0
answers
75
views
Deriving Klein-Gordon equation from Euler-Lagrangian equation: Taking partial derivative inside [duplicate]
Lagrangian for Klein-Gordon equation is given by
$$L=\frac{1}{2}\partial_\mu \phi \partial^\mu\phi - m^2\phi^2/2.$$
To derive Klein-Gordon Equation I have to Compute derivative in Euler-Lagrange ...
- 45
1
vote
1
answer
72
views
Covariant derivative for spin-2 field
I have mostly seen the concept of covariant derivative with regard to spin-1 fields. Is it possible to define the covariant derivative for spin-2 fields as well?
- 387
3
votes
3
answers
642
views
Is the rate of change of duration a valid quantity?
I was wondering that, if the duration of a recurring event varies as time goes on, what would the magnitude of this quantity be measured in? For instance, if the time for an oscillation of a weighted ...
- 133
1
vote
1
answer
77
views
Does the divergence theorem require the covariant derivative to be metric compatible?
I know this is more of a mathematical question, but it arises in the context of general relativity and uses its language so I thought it would be best to ask it here. I understand that the divergence ...
- 4,737
1
vote
0
answers
50
views
A covariant derivative computation in General Relativity [duplicate]
I am trying to compute $\nabla^\mu\nabla^\nu R_{\mu\nu}$.
I proceed as follows:
\begin{align}
\nabla^\mu\nabla^\nu R_{\mu\nu}&=g^{\mu\rho}g^{\nu\lambda}\nabla_\rho\nabla_\lambda R_{\mu\nu} \\
&...
- 372
0
votes
3
answers
240
views
What is the correct term for $\nabla\phi$? Co-vector or 1-form or both?
In the olden days, $\nabla\phi$ was used to be called a covariant vector (Weinberg used this language in his book Gravitation & Cosmology). But this terminology is considered bad for several ...
- 12.3k
1
vote
1
answer
61
views
Tensor equation
What is a valid tensor equation. In the book by Bernard Schutz, it is often argued that a valid tensor equation will be frame invariant. So the conclusions reached by relatively easy calculation done ...
- 137
1
vote
1
answer
62
views
Deriving the Curl of the Magnetic Field, Role of the Nabla Operator
We know that the magnetic field can be written in the following way:
$$\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
0
votes
3
answers
107
views
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},
\...
- 421
0
votes
1
answer
32
views
Differentiation of a product of functions
If I have three (vector)functions, all dependent on different (complex)variables:
\begin{equation}
a = X^{\mu_1}(z_1, \bar{z}_1),
b = X^{\mu_2}(z_2, \bar{z}_2),
c= X^{\mu_3}(z_3, \bar{z}_3)
\end{...
- 41
2
votes
2
answers
393
views
How to take derivative of density operator?
I was just trying to confirm to myself that the following density operator
$$\rho(t) = e^{-iHt/\hbar} \rho(0) e^{iHt/\hbar}$$
fulfills the Liouville-von Neumann equation:
$$\frac{d}{dt}\rho(t) = - \...
- 29
0
votes
0
answers
117
views
Are eigenvalues of slashed covariant derivative real?
I am trying to demonstrate that the slashed covariant derivative
$$
\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
1
vote
2
answers
325
views
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
votes
0
answers
56
views
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:
$$
...
- 405
2
votes
1
answer
156
views
Dirac Delta applied to the gradient of a function
The supplementary section of a paper I am reading uses the "substitution" property of the Dirac delta function :
$$\mathbf{v}\left(\mathbf{x}_j\right)\delta\left(\mathbf{x}-\mathbf{x}_j\...
- 49
0
votes
1
answer
36
views
Space-for-time Derivative Substitution in Solving for Elliptical Orbit
I am currently working on a simulation of the Newton's Cannonball thought experiment, in which a stone is launched horizontally from atop a tall mountain at a high speed (in the absence of air) and ...
0
votes
1
answer
99
views
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 ...
- 41
4
votes
2
answers
641
views
Confusion on metric determinant derivative
Maybe it is a stupid confusion. I need to compute the derivative of the metric determinant with respect to the metric itself, i.e., $\partial g/\partial g_{\mu\nu}$, but I have an indices confusion in ...
- 378
0
votes
3
answers
363
views
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 ...
- 1
0
votes
1
answer
219
views
Interchange of Lagrangian/material derivative and volume integral
In hydrodynamics there are two basic approaches. The first is the Eulerian specification where the coordinate system is fixed. In that case, the partial time derivative and volume integral operators ...
1
vote
4
answers
445
views
How to find the double covariant derivative of a general vector?
I have been reading through Carroll's GR textbook and there is a line in the derivation of the Riemann tensor that I do not understand.
$$\nabla_\mu \nabla_\nu V^\rho=\partial_\mu(\nabla_\nu V^\rho) - ...
- 71
9
votes
2
answers
530
views
About the traditional explanation of the continuity of the first derivative of a 1D wavefunction
I would like to receive some clarifications about the traditional explanation of the continuity of the first derivative of a 1D wavefunction (E.g. see the very clear answer by @ZeroTheHero ...
- 78.1k
-2
votes
2
answers
62
views
Can the different differentiation notations be equated and do they have an integral definition? [closed]
Are these all equivalent and is there an extension of this to other notation?
Does anyone have a clear and concise chart equating the different notation dialects?
I am also curious if there are more ...
0
votes
1
answer
83
views
What is the relation between gauge field and Levi-Civita connection?
In field theory, covariant derivative is something like
$$D_{\mu}\phi=(\partial_{\mu}-igA_{\mu})\phi$$
while in differential geometry, covariant derivative is something like
$$D_{\mu}V^{\nu}=\partial_{...
- 101
1
vote
2
answers
86
views
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)
- 73
0
votes
0
answers
90
views
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 ...
- 11
4
votes
2
answers
1k
views
Error analysis via two different methods
We have a quantity $a$ expressed in terms of two quantities $b $ and $c$ as $a = b/c$.
It seems to me that there are two ways of estimating the error on $a$, the "physics" ...
- 109
0
votes
1
answer
84
views
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 ...
- 165
0
votes
1
answer
346
views
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 $p$....
- 713
6
votes
2
answers
1k
views
Inverse Laplacian
I have seen the following operator somewhere in a paper on cosmology
$$
\frac{\partial_i \partial_j}{\nabla^2} - \frac{1}{3} \delta_{ij}.
$$
What is the definition of the inverse Laplacian? What is ...
- 385
5
votes
2
answers
457
views
Meaning of the differential entropy
The definition of differential (or continuous) entropy is problematic. As a matter of fact, differential entropy can be negative, can diverge and is not invariant with respect to linear ...
- 186
0
votes
0
answers
73
views
When can I commute the 4-gradient and the "space-time" integral?
Let's say I have the following situation
$$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$
Would I be able to commute the integral and the partial derivative? If so, why is ...
- 21