All Questions
Tagged with differentiation homework-and-exercises
23 questions
6
votes
2
answers
4k
views
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
1
vote
4
answers
407
views
Rotation systems. Problem interpreting an equation
In this equation:
$$
\mathbf a_i\overset{\rm def}{=}\left(\frac{d^2\mathbf r}{dt^2}\right)_i=\left(\frac{d\mathbf v}{dt}\right)_i=\left[\left(\frac{d}{dt}\right)_r+\boldsymbol\Omega\times\right]\left[...
- 1,629
7
votes
2
answers
3k
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Covariant derivative of a covariant derivative
I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$.
This is something I've taken for granted a lot in calculations, namely I though that by the ...
- 623
4
votes
2
answers
1k
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Trouble with Landau & Lifshitz's expansion of the Lagrangian with respect to $\epsilon$ and $v$ [duplicate]
Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on ...
- 43
1
vote
4
answers
3k
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What is proper time, proper velocity and proper acceleration?
I am trying to derive the relativistic rocket equations found here [(4),(5),(6),(7),(8)] but I do not understand proper time, proper velocity and proper acceleration.
Define a point $P$ with ...
- 201
0
votes
2
answers
1k
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Prove the Total Mechanical Energy of the System is Conserved via Differential Equations [closed]
Consider the dynamics of a particle P shown: Particle in 3D space with Radius r
Newton's second law states that:
$$\frac{d}{dt}(m\dot r) = \mathbf F$$
where, $\boldsymbol{r}$ is the position vector ...
- 129
8
votes
2
answers
1k
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What does $\exp\left( ax\frac{d}{dx} \right)$ do on $\psi(x)$?
I'm trying to find out
$$\exp\left(ax\frac{d}{dx}\right)\psi(x)= \ \ ? $$
I tried spending the exponential and then operating the derivatives one by one but I found no pattern. Besides, it gets ...
- 12.1k
6
votes
2
answers
12k
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Variation of square root of determinant of metric, $\delta g$ [closed]
I am trying to calculate
$$ \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
votes
5
answers
8k
views
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
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
3
votes
1
answer
454
views
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 ...
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
2k
views
Commutator of scalar field and its spatial derivative
Consider the usual commutation relations of two scalar fields
$$\left[\phi_{m}\left(t,\boldsymbol{x}\right),\pi_{n}\left(t,\boldsymbol{y}\right)\right]=\boldsymbol{i}\delta_{mn}\delta\left(\...
- 141
1
vote
1
answer
1k
views
Covariant derivative ordering
I was working on a problem involving Bianchi identities, in a particular case I have to take the covariant derivative of the following, which indeed is the Ricci tensor in linearised limit
$$r^{\mu}_{\...
- 278
1
vote
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
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
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
0
votes
3
answers
3k
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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
1
answer
353
views
Geodesic equation proof confusing me
I was looking through this proof and have no idea where the $u$ comes from. Any help is appreciated.
This is from here; I want to know how they got from eqn 5 to eqn 6.
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
-1
votes
1
answer
96
views
Finding solution to this differential equation
In this paper http://arxiv.org/abs/hep-th/9506035 equation (3.11) was written as: $$\frac{\partial L}{\partial u}\frac{\partial L}{\partial v} = -1$$
The author then said p.9 that "approximate ...
- 215
-1
votes
3
answers
493
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The use of the commutators in quantum mechanics: explanations [duplicate]
Considering that I've never studied quantum mechanics before I have need to understand the operator commutator. My start is: $[A,B]=AB-BA \tag{a}$
Now, why must be
$$\left[\frac{\partial }{\partial ...
- 2,575
-1
votes
1
answer
4k
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Lennard-Jones potential, distance $r$ for minimum energy
I'm sorry if the question seems stupid. I found (wikipedia) that the Lennard-Jones potential has it's minimum at a distance of
$$r = 2^{\frac{1}{6}}\sigma.$$
If $U(r)_{min} = -\epsilon$
$$U(r) = 4\...
- 137