All Questions
Tagged with differentiation homework-and-exercises
290 questions
10
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7
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What can go wrong with applying chain rule to angular velocity of circular motion?
Lets say I have a circular motion, like this:
I know that:
$$\omega = \frac{\text{d} \phi}{\text{d}t}$$
Mathematically, what I am doing wrong, when I attempt to apply the chain rule in the following ...
- 597
10
votes
2
answers
1k
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Covariant derivative of the spin connection
I wish to compute $$[\nabla_{\mu}, \nabla_{\nu}]e^{\lambda}_{~~a}. $$
To do so, I make use of $\nabla_{\nu}e^{\lambda}_{~~a} = \omega_{a~~~\nu}^{~~b}e^{\lambda}_{~~b}$, so that I may write
$$\nabla_{\...
- 1,415
8
votes
1
answer
10k
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How is the second-order covariant derivative of a scalar computed?
What is second-order covariant derivative $$\nabla_i\nabla_jf(r)$$ in terms of $r,\theta, g(r)$ and partial derivative, given that the metric takes the form $$ds^2=dr^2+g(r)d\theta^2$$ and $f$ is a ...
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8
votes
2
answers
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What does $\exp\left( ax\frac{d}{dx} \right)$ do on $\psi(x)$?
I'm trying to find out
$$\exp\left(ax\frac{d}{dx}\right)\psi(x)= \ \ ? $$
I tried spending the exponential and then operating the derivatives one by one but I found no pattern. Besides, it gets ...
- 12.1k
7
votes
2
answers
3k
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Covariant derivative of a covariant derivative
I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$.
This is something I've taken for granted a lot in calculations, namely I though that by the ...
- 623
6
votes
2
answers
12k
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Variation of square root of determinant of metric, $\delta g$ [closed]
I am trying to calculate
$$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}},$$
where $g = \text{det} g_{\mu \nu}$.
We have
$$ \frac{\partial \sqrt{- g}}{\partial g^{\mu \nu}} = - \frac{1}{2 \...
- 329
6
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5
answers
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Covariant Derivative of Kronecker Delta
I am reading Carroll's book on GR right now, and I ran into a little trouble in his chapter 3 on curvature. He is establishing the properties of the covariant derivative, and claims that the fact that ...
- 147
6
votes
2
answers
4k
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Advection operator
How are exactly $u_j\partial_ju_i$ and $u_i\partial_j u_i$ related?
And what is their relation to ($\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$ and $\boldsymbol{u}\cdot(\nabla\boldsymbol{u})$ ?
I ask ...
- 1,813
5
votes
4
answers
387
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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):
$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\...
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5
votes
3
answers
1k
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Vlasov equation, Maxwell distribution
I have the Maxwellian distribution:
$$f(v)=n\left(\frac{m}{2\pi kT}\right)^{\frac{3}{2}}\exp\left(-\frac{mv^2}{2kT}\right)$$
I have to show that it is a solution to the Vlasov equation:
$$\frac{\...
5
votes
1
answer
5k
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Second derivative of Dirac delta expression
I have come across the expression
$$ \int f(x) \delta(x-a) \delta''(x-a) \mathrm dx$$
where the prime represents the derivative.
Usually with derivatives of the Dirac delta distribution I'd partially ...
- 9,197
4
votes
3
answers
2k
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What does "Just before" and "Just after" really mean in physics problems?
So I'm stuck in a dynamics problem that asks what is the acceleration of a body just after A, where A is the point that separates the motion of the body from a curvilinear path to projectile motion. ...
4
votes
5
answers
1k
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Problem understanding the use of differentiation
I am new to differentiation. Our physics teacher gave us this example problem:
The radius of a sphere is continuously increasing at the rate of 1 m/sec. Find the rate of change of the volume of the ...
4
votes
2
answers
641
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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
4
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3
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1k
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Why is the covariant derivative of the metric tensor with UPPER indices equal to zero? [closed]
I've shown that $\nabla_{\lambda} g_{\mu\nu} = 0 $ rigorously by the following method:
$ \nabla_{\lambda} g_{\mu\nu} = \partial_{\lambda}g_{\mu\nu} - \Gamma^{\rho}_{\lambda\mu} g_{\rho\nu} - \Gamma^{\...
4
votes
3
answers
354
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Ricci Identity with Torsion Proof
In the notes I'm using for General Relativity, the author begins their proof of the Ricci identity with torsion by writing
$$\nabla_{[\mu}\nabla_{\nu]}Z^{\sigma}=\partial_{[\mu}(\nabla_{\nu]}Z^{\sigma}...
4
votes
2
answers
2k
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Derivatives of Dirac delta function and equation of continuity for a single charge
For a single charge $e$ with position vector $\textbf R$, the charge density $\rho$ and and current density $\textbf{j}$ are given by:
\begin{equation} \rho(\textbf{r},t)= e\,\delta^3(r-\textbf{R}(t))...
- 1,412
4
votes
2
answers
1k
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Trouble with Landau & Lifshitz's expansion of the Lagrangian with respect to $\epsilon$ and $v$ [duplicate]
Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on ...
- 43
4
votes
1
answer
230
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Is there a quick way to calculate the derivative of a quantity that uses Einstein's summation convention?
Consider $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_\nu A_\mu$, I am trying to understand how to fast calculate $$\frac{\partial(F_{\mu\nu}F^{\mu\nu})}{\partial (\partial_\alpha A_\beta)}$$
without ...
- 862
4
votes
1
answer
111
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What does $\mathbf{A}\cdot\nabla$ mean here?
What does $\mathbf{A}\cdot\nabla$ mean in an expression like $(\mathbf{A}\cdot\nabla)\mathbf B$?
I found this in Griffiths’ Classical Electrodynamics book and cannot figure it out.
4
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2
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18k
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Why and when do we differentiate or integrate equations in physics? [closed]
I'm an engineering student and none of my professors ever explained why do we use derivations and/or integrations in physics. So I have this task, it goes like:
The object is moving in a positive ...
- 41
4
votes
1
answer
2k
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How do total time derivatives of partial derivatives of functions work?
Say im trying to prove $\frac{\partial \dot{T}}{\partial \dot{q}^i} - 2\frac{\partial {T}}{\partial {q^i}} = - \frac{\partial {V}}{\partial {q^i}}$ from the Lagrangian equation: $L = T - V$, and the ...
- 75
4
votes
1
answer
1k
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Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives
For my course in General Relativity I am given the problem to find the expressions for the gradient, laplacian, divergence and curl in spherical coordinates using covariant derivatives.
I have tried ...
- 53
4
votes
4
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198
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How to prove $\mathrm{Tr}[(\partial_\mu U)U^\dagger]=0$?
I am studying ChPT by referring to "A Primer for Chiral Perturbation Theory" by Stefan Scherer.
I'm having a problem with the consideration of terms that appear in the Lagrangian.
The ...
- 173
3
votes
2
answers
2k
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For the Yang-Mills field strength defined as a commutator, why does the $A_\nu\partial_\mu - A_\mu\partial_\nu$ term vanish?
In basically every QFT book the Yang-Mills strength tensor $F_{\mu\nu}$ is defined as $$F_{\mu\nu}=[D_\mu,D_\nu]$$
where $D_\mu$ is the covariant derivative $$D_\mu=\partial_\mu-A_\mu$$ and $A_\mu$ is ...
- 1,674
3
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3
answers
880
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Math in deriving Pauli Equation
When deriving the Pauli Equation, it has the following step:
$$i\frac{e}{c}\hbar[\vec{A}\times\nabla+\nabla\times\vec{A}]\phi=i\frac{e}{c}\hbar\space curl\vec{A}\cdot \phi$$
$\phi$ is one of the ...
- 912
3
votes
1
answer
271
views
How is $ \frac{dv}{ dt} = a $?
I know how , in the physical sense -
$$\frac {dv}{dt} = a$$
But, even after thinking a lot, I am not able to see the fault in this -
$$\frac {dv}{dt} = \frac {d(st^{-1})}{dt}
= \frac {sd(t^{-1})}...
- 141
3
votes
4
answers
902
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Multiplicity Identity in Kittel's Thermal Physics
On page 25 of Kittel's Thermal Physics, the author derives the multiplicity of $N$ harmonic oscillators with total quanta of energy $n$, $g(N,n)$.
He writes
\begin{align}
g(N,n) &= \lim_{t\...
- 129
3
votes
1
answer
884
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Derivative of the trace of $e^{-\beta \mathbf{A}}$
I'm trying to compute the derivative with respect to an inverse temperature parameters $\beta$ of a density matrix that has the following form:
$$\rho(\beta,\mathbf{A}) = \frac{e^{-\beta \mathbf{A}}}{...
- 1,277
3
votes
2
answers
93
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What is the meaning of the equation of the change in entropy? [duplicate]
In my chemistry book, the formula for change in entropy is given as :
$$\int{dS} = \int{\frac{δq_{rev}}{T}}$$
What is the meaning of $δq_{rev}$? I know that it is the heat exchanged in a reversible ...
- 869
3
votes
1
answer
138
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Is this covariant derivative identity true?
Trying to work through a textbook derivation of the geodesic deviation equation, I've calculated this identity:$$u_{;\beta}^{\alpha}u_{\alpha}=u_{\alpha;\beta}u^{\alpha}.$$
If this is true, I'm ...
- 3,089
3
votes
2
answers
210
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Applying the exterior differential to the first law of thermodynamics [closed]
I'm working on an exercise for an advanced statistical physics course. The question I'm struggling with is this:
$$TdS=dE+PdV-\mu dN\tag{1}$$
Write the first law $(1)$ as $dS=....$ Applying the ...
- 1,449
3
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1
answer
853
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Divergence of $ \frac{1}{s}\hat{s}$ in cylindrical coordinates
In Griffiths' electrodynamics, the divergence of $\frac{1}{r^{2}}\hat{r}$ is evaluated in spherical coordinates to be $4\pi\delta(r)$. I encounter the same problem in case of $\frac{1}{s}\hat{s}$ ...
- 171
3
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1
answer
420
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Exterior and Covariant Derivatives
Is the following guaranteed to be true for any covariant vector $f_\mu$ (1-form $\boldsymbol{f}$) in the absence of torsion?
$$\nabla_{[\alpha}\nabla_{\beta}f_{\mu]}=\partial_{[\alpha}\partial_{\beta}...
- 307
3
votes
1
answer
1k
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Taylor expansion of a scalar function of a four-vector
Consider a scalar function $\phi(x^\mu)$ of a four-vector $x^\mu=(x^0,x^1,x^2,x^3)=(ct,x,y,z)$. What is the Taylor expansion of $\phi(x^\mu+\delta x^\mu)$ for infinitesinal $\delta x^\mu$?
In ...
- 27.2k
3
votes
1
answer
543
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Time derivative of time-translation Killing vector
I'm working with the spherically symmetric, static black hole metric. In the problem I'm working on, I'm told that $K$ is the time-translation Killing vector, $\frac{\partial}{\partial t}$ or $K = (1, ...
- 569
3
votes
3
answers
116
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Finding the vector potential
$$\nabla\times\mathbf{B}=\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla^2\mathbf{A}=\mu_0\mathbf{J}\tag{5.62}$$
Whenever I try to work this out and ...
- 220
3
votes
1
answer
238
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Is the time derivative of the adjoint equal to the adjoint of the time derivative?
This is hopefully straightforward. Starting from the Schrödinger equation as an axiom, one obtains the operator differential equation for the $U$ such that $| \psi(t) \rangle = U(t,t_0) | \psi(t_0) \...
- 1,271
3
votes
1
answer
454
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Heaviside-Feynman formula derivation
I want to discuss derivation of Feynman-Heaviside formula.
The topic has already been discussed here but I can not put there any question that's why I'm making new post.
Deriving Heaviside-Feynman ...
3
votes
2
answers
722
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Differentiation of the determinant $g$
Let $g$ be the determinant of the metric tensor.
I want to derive the following equation $g_{,\nu}=gg^{\lambda \mu}g_{\lambda \mu,\nu}$. It is said that $gg^{\lambda \mu}$ is a cofactor, but I can't ...
- 551
3
votes
0
answers
396
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Laplacian and Dirac Delta function
Although I find it mathematically dubious, we said that $$\Delta \frac{1}{r} ~=~ -4\pi \delta^3({\bf r}).$$ Now, I was wondering is there a similar relation to the delta function if we look at $$\...
- 1,928
2
votes
4
answers
261
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Show that $d\mathbf{v}^2/dt = 2\mathbf{v}\cdot d\mathbf{v}/dt$ using geometry only
I have just begun reading Modern Classical Physics by Thorne and Blandford and I am trying to wrap my head around their "geometric viewpoint" on classical mechanics. The first exercise in ...
- 510
2
votes
1
answer
2k
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Divergence of inverse of metric tensor
I know that the Levi-civita connection preserves the metric tensor. Is the divergence of the inverse of metric tensor zero, too?!
I'm not so familiar with the divergence of the second ranked tensor. ...
- 119
2
votes
3
answers
198
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Derivation of entropy, I don't understand the relation $ \frac{\partial S_2}{\partial E_1} = -\frac{\partial S_2}{\partial E_2} $
My course guide gives the following derivation for change in entropy w.r.t. energy, where I don't understand a step:
\begin{align}
E & = E_1 + E_2 \\
S & = S_1 + S_2 \\
S(E,E_1 ) & = S_1 (...
- 2,180
2
votes
2
answers
2k
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Commutator of scalar field and its spatial derivative
Consider the usual commutation relations of two scalar fields
$$\left[\phi_{m}\left(t,\boldsymbol{x}\right),\pi_{n}\left(t,\boldsymbol{y}\right)\right]=\boldsymbol{i}\delta_{mn}\delta\left(\...
- 141
2
votes
1
answer
109
views
$x$-derivative of the wave function and its conjugate [closed]
I saw that in order to show that the normalisability of a wave function does not depend on time, there is a necessary step in the calculation that says that:
$$\left(\Psi^*\frac{\partial^2\Psi}{\...
2
votes
2
answers
207
views
Take derivative to a cross product of two vectors with respect to the position vector [closed]
I'm doing classical mechanics about Lagrange formulation and confused about something about vector differentiation.The Lagrangian is given:
$$\mathcal{L}=\frac{m}{2}(\dot{\vec{R}}+\vec{\Omega} \times \...
- 103
2
votes
1
answer
2k
views
Derivatives with upper and lower indices
I'm studying classical and quantum field theory, but evaluating derivatives of fields (scalar and/or vector) described with upper and lower indices is somewhat new to me. I'm trying to evaluate
$$\...
- 187
2
votes
2
answers
4k
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Total time derivative of magnetic vector potential $A$
I am looking at this document, which tries to establish the Lagrangian of the Lorentz force. Everything is fine, but I don't see why:
$$\frac{dA_i}{dt}=\frac{\partial A_i}{\partial t}+\frac{\partial ...
- 604
2
votes
2
answers
318
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Derivation of velocities in the Coriolis force
In Fitzpatrick's Newtonian Dynamics book on the Coriolis force, he states
\begin{align}
v_{x'}&\simeq V_0\cos\theta-2\Omega t V_0\sin\lambda~\sin\theta \tag{433}\\
v_{y'}&\simeq-V_0\sin\...
- 1,629