All Questions
Tagged with covariant-derivatives or differentiation
260 questions
2
votes
3
answers
323
views
Lack of rigour in usual derivation of Work-Energy Theorem
The derivation of the Work-Energy theorem usually goes as follows:
You define the work done on a particle under net force $\vec{F}$ as
$$W=\int\limits_C \vec{F}\cdot\mathrm{d}\vec{r}$$
And then you ...
- 3,985
2
votes
2
answers
1k
views
Gauge-covariance of the Yang-Mills field strength $F_{\mu\nu}^a$
Accordingly to Yang-Mills theories, after the introduction of a covariant derivative such that
$$D_\mu = \partial_\mu - igA_\mu, \tag1$$
you can built the kinetic term for the gauge potential $A_\...
- 1,607
2
votes
2
answers
1k
views
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
2
votes
0
answers
86
views
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
2
votes
1
answer
105
views
Work-Kinetic energy theorem derivation
So I came across this derivation in the book Classical Mechanics by Herbert Goldstein. I don't follow from the second step onwards. I understand that there's a dot product, but how do you compute it? ...
- 87
2
votes
3
answers
2k
views
The commutation of partial derivatives in curved spacetime
While following a lecture series on General Relativity, an argument was presented that in order the spacetime to be flat, a vector parallel transported along two different paths should yield the same ...
- 1,222
2
votes
1
answer
346
views
Covariant derivative in field theory
I'm reading Physics from Symmetry by Jakob Schwichtenberg and in Chapter 7 equation 7.18 he introduces the covariant derivative when deriving the interaction Lagrangian density for the spin-$\frac{1}{...
- 432
2
votes
1
answer
535
views
Expectation value of derivative of operator
I was given the following question:
Let $A(\lambda)$ be a Hermitian operator, which is dependent on some real parameter $\lambda$. Let us denote the eigenvalues and corresponding eigenstates of $A$ ...
- 353
2
votes
2
answers
536
views
Exponential of an operator shifted by the derivative operator
Let $p(x)$ and $f(x)$ be sufficiently smooth functions and $D=\frac{d}{dx}$. It is easy to show that $$e^{p(x)D}f(x)=f(e^{p(x)D}x).\tag{1}$$
If $p(x)=a \in \mathbb{R}$ , we have the shift operator as $...
- 213
1
vote
3
answers
437
views
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 ...
- 213
1
vote
1
answer
258
views
Derive interaction lagrangian for KG equation in QED
The free-field KG lagrangian density for complex scalar field is given as $$\hat{\mathcal{L}}_{\text{KG}}=(\partial_\mu\hat{\phi}^\dagger)(\partial^\mu\hat\phi)-m^2\hat{\phi}^\dagger\hat{\phi}$$
By ...
- 862
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
1
vote
3
answers
368
views
Commutation relation of $e^{ikx}$ and $\partial_x$ in Nakahara
I'm reading through Nakahara's Geometry, Topology and Physics and I don't understand the following derivation on pg. 41:
$$
\text{Now we find from the commutation relation of } \partial_x \equiv \frac{...
- 432
1
vote
1
answer
98
views
Proving a Superfunction Identity
I am trying to figure out the proof of the identity given between equations (1.11.7) and (1.11.8) in ref. [1], i.e.
\begin{align}
\Phi'(e^{-K}\,z\,e^K)=e^{-K}\Phi'(z) \tag{1}
\end{align}
where $z=(...
- 49
1
vote
0
answers
93
views
Does car move when instantaneous velocity is zero? [duplicate]
In 3blue1brown: derivative paradox.
supposed car moving with:
$S(t) = t^3$
And velocity is:
$V(t) = 3t^2$
He asked when t = 0 velocity is 0 m/s , does that car move at that time ?
And here his ...
- 311
1
vote
1
answer
6k
views
What is the common difference between partial time derivative and ordinary time derivative? [duplicate]
What is difference between partial and ordinary time derivative?
for example: what is difference between $\frac {\partial v}{\partial t}$ and $\frac {dv}{dt}$?
where the $v$ is velocity.
- 11
1
vote
3
answers
64
views
Showing that intensive parameters obtain by considering molar quantities
In Callen's Thermodynamics textbook, he writes that
$$\left(\frac{\partial u}{\partial s}\right)_v = \left(\frac{\partial U}{\partial S}\right)_{V,N}$$
where $u = U/N$, $s = s/N$, and $v = V/N$ and, ...
- 1,271
1
vote
2
answers
168
views
Questions regarding the derivation of Euler-Lagrange Equation from Taylor's Classical Mechanics
I'm self-studying classical mechanics from Taylor's book. I saw his derivation of the Euler-Lagrange Equation and I'm confused about something, he created a 'wrong' function $$Y(x) = y(x)+\eta(x)\tag{...
- 115
1
vote
0
answers
75
views
Wavefunction from the Hamilton-Jacobi formalism [closed]
I was reading this paper, Wavefunctions and Hamilton-Jacobi Equation by Sabrina Pasterski. The author started with the Hamiltonian formalism and then came up with a connection to the Schrodinger ...
- 11
1
vote
1
answer
269
views
Metric independent affine connections
Normally, the affine connections are objects that define parallel transport. In general Relativity they are the Christoffel symbols of the second kind. Consequently, they depend on the metric tensor. ...
1
vote
2
answers
101
views
Tensors Differentiation
I know that $\frac{\partial x^{\mu}}{\partial x^{\nu}}=\delta^{\mu}_{\nu}$ but a few days back, I read somewhere that $\frac{\partial x_{\mu}}{\partial x^{\nu}}=\eta_{\mu\nu}$. Can someone help me ...
- 161
1
vote
1
answer
197
views
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 ...
1
vote
1
answer
1k
views
Covariant derivative ordering
I was working on a problem involving Bianchi identities, in a particular case I have to take the covariant derivative of the following, which indeed is the Ricci tensor in linearised limit
$$r^{\mu}_{\...
- 278
1
vote
1
answer
606
views
Confused about Navier-Stokes equation
Just look at the L.H.S of the compressible navier-stokes equation from wiki
$$\rho(\partial_t \vec{u}+\vec{u}\cdot\nabla\vec{u})=...$$
How can I add a vector $\partial_t \vec{u}$ and a scalar $\vec{...
- 562
1
vote
2
answers
2k
views
If change in position over time is average velocity, why doesn't change in position over time squared equal average acceleration?
For example, let's say a car is experiencing an acceleration of $1$m/s$^2$, for $6$ seconds so it goes $18$m. Now the average velocity is found through dividing $18$m by $6$s which is in line with the ...
- 1,187
1
vote
0
answers
185
views
Derivative with Respect to Symmetric Tensor [duplicate]
If you have a Lorentz tensor $T$ with components $T_{\mu\nu}$, it seems clear that
$$
\frac{\partial T_{\mu\nu}}{\partial T_{\rho\sigma}} = \delta^\rho_\mu \delta^\sigma_\nu. \tag{1}
$$
However, if $T$...
- 361
1
vote
3
answers
143
views
Passing from curl to vector product
I don't understand how to obtain second equation with first part in the equation
$$
\nabla \times \vec A_0 e^{-j \vec k\cdot \vec r} = -j\vec k\times \vec A_0 e^{-j \vec k\cdot \vec r}.
$$
Can you ...
- 13
1
vote
5
answers
7k
views
Direction of velocity vector in 3D space
According to a well-known textbook (Halliday & Resnick), the direction of a velocity vector, $\vec v$, at any instant is the direction of the tangent to a particle's path at that instant, as is ...
- 113
1
vote
0
answers
95
views
Partial derivatives in Lagrangian formalism [duplicate]
Suppose I have a function $f = xy$. A partial derivative of $f$ with respect to $x$ implies holding $y$ constant:
$$ \frac{\partial f}{\partial x} = y $$
Does this mean that in order to evaluate ...
- 959
1
vote
1
answer
162
views
Swapping expectation value of derived operator with derivative of expectation value
My question is somewhat related to this one. I want to know if
$$ \frac{d}{dk}\left\langle \hat{f}_k \right\rangle_{\psi_k} = \left\langle \frac{d}{dk} \hat{f}_k \right\rangle_{\psi_k} $$
holds for ...
- 953
1
vote
2
answers
228
views
Commutation of position four-vector with spacetime derivatives
I am trying to understand a simple demonstration in Ashok Das' Lectures in QFT. He does the following on p. 134
$$[P_\mu,M_{\nu\lambda}]=[\partial_\mu ,x_\nu\partial_\lambda-x_\lambda\partial_\nu]=\...
- 558
0
votes
3
answers
232
views
Are acceleration and velocity simultaneous? [closed]
I would think yes because, if a rope tied to a swinging rock breaks, the rock flies off in the direction that is perpendicular to the direction of the last instant of the acceleration. The ...
- 71
0
votes
3
answers
1k
views
Christoffel symbol and covariant derivative
I came across the Christoffel symbols via the geodesic equation, and I understand the extrinsic form and the intrinsic form and can prove that they are identical:
extrinsic form:
$$\Gamma^{j}_{~ik}=\...
- 157
0
votes
2
answers
163
views
How to identify $\hat{d}_x$ with $\hat{p}_x/\hbar$?
Isham, in his Lecture on Quantum Theory, Chapter 7, Unitary Operators in Quantum Theory, Section 7.2.2 Displaced Observers and the Canonical Commutation Relations, mentions on page 137 (bottom) the ...
- 1,999
0
votes
1
answer
2k
views
Differentiation of kinematic differential equation for quaternion [closed]
We know the kinematic equation in terms of quaternion is the following:
$$\dot{q}(t) = \frac{1}{2}\Omega(t)q(t)$$ where $q(t)$ is the unit quaternion. Now if I want to differentiate the above ...
- 233
0
votes
3
answers
3k
views
Proof divergence of magnetic field is 0
I work in an R&D role that involves magnetism. I am refreshing my memory of electromagnetic and this stumps me. In polar coordinates, the magnetic field of a current loop for distances $R >>...
- 23
0
votes
2
answers
3k
views
Covariant Derivative of Metric Tensor
I'm an amateur studying General Relativity. I'm reading some notes of lectures by Susskind. In them, it is written that
"we know that [the covariant derivative of the metric tensor] is zero. ...
- 113
0
votes
1
answer
187
views
Density operator as a function of time
Given the density operator $\rho = \sum_iw_i | \alpha^{i} \rangle \langle \alpha^{i}|$, how does the density operator change with time. Apparently I should get $$i \hbar \frac{\partial \rho}{\partial ...
- 1,053
0
votes
3
answers
333
views
Model of road disturbance in term of normal force
According to this article Channe, S.S. and Kshirsagar, S.D., Modeling and Simulation of a Suspension System for Different Road Disturbances, the best model for disturbance of a road is in the form of ...
0
votes
3
answers
2k
views
If the electric field is the gradient of the potential, then can we say that whenever potential is zero, the electric field is zero?
For example, in a dipole, at the center of the two charges making up the dipole, the potential is zero but the electric field is non-zero. But if $E = -\operatorname{grad}V$, then why is $E$ not zero?
...
- 303
0
votes
1
answer
86
views
"$\delta^2 S$" confusion regarding "second variations" in stability conditions
As far as I am aware, for some function of $n$ variables $f$, $\delta^2 f$ represents the third term in its Taylor expansion.
So, I've encoutered the following expression in my thermodynamics book:
...
- 135
0
votes
2
answers
993
views
Question about derivation of four-velocity vector
In order to describe a notion of rate of change of positon, in four-dimensional spacetime, we have to introduce the concept of four-velocity.
So, consider the following:
For a massive particle ...
- 3,159
0
votes
1
answer
397
views
Killing equation in coordinates
In proving that it is possible to write the killing equation in coordinates as $$L_X g=0\iff X_{\alpha;\beta}+X_{\beta;\alpha}=0$$
I have read that the key observation, to write the equation in ...
- 187
0
votes
2
answers
317
views
Generalizing the covariant derivate for gauge theory
Concrete example
gauging the complex scalar field
$\mathcal{L}=(\partial_\mu \phi)(\partial^\mu \phi^*)+m^2 \phi^*\phi$
$\phi(x) \rightarrow e^{-i\Lambda(x)}\phi(x)$
$A_\mu \rightarrow A_\mu + \frac{1}...
- 1,081
0
votes
0
answers
31
views
How to pick a boundary layer coordinate or stretching transformation
I am following Introduction to Perturbation Methods by Holmes and am unsure how I to pick the power in my boundary layer coordinate if my governing equation is the Laplace equation given by
\begin{...
- 647
0
votes
7
answers
4k
views
What happens to velocity when Time equals zero?
I am not formally educated in Science but natural questions have always intrigued me.The way I put it is that I am married to Commerce but Science has been a childhood love. Now I have this very basic ...
0
votes
1
answer
385
views
Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero?
See the bold text for my question.
This question regards Dirac's General Theory of Relativity page 13. In his demonstration that the length of a vector is unchanged by parallel displacement Dirac ...
- 2,052
0
votes
3
answers
2k
views
How to derive kinematics equations using calculus? [closed]
I read derivation of kinematics equations using calculus:
$$a=\frac{\text dv}{\text dt}$$
$$\implies \text dv=a\text dt$$
$$\implies \int_{v_0}^v\text dv=\int_0^t a\text dt$$
$$\implies v-v_0=at$$
$$\...
- 259
0
votes
1
answer
4k
views
What is the physical meaning of divergence? [duplicate]
I want to visualize the concept of divergence of a vector field. I also have searched the web.Some says it is
1.the amount of flux per unit volume in a region around some point
2.Divergence of ...
- 289
0
votes
1
answer
1k
views
Squaring the momentum operator in QM becomes a second derivative. How?
$\frac{p^2}{2m}$ is the Kinetic energy in classical mechanics. However, the same $p^2$ becomes the second derivative $\frac{\partial ^2}{\partial x^2}$ in the Kinetic Energy operator in QM. I mean it ...
- 21