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1
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3answers
178 views

What is the physical meaning of the Levi-Civita connection?

I'm taking a course in General Relativity and I have studied the fundamental theorem of Riemannian geometry: Let $M$ be a manifold with metric $g$. Then exists an unique torsion-free connection ...
1
vote
3answers
134 views

Is there any physical interpretation for $\nabla\cdot(\nabla \times F)=0$?

It is well known that the divergence of the curl is always 0. Mathematically I understand why this happens ($d^2=0$ where $d$ is the exterior derivative) but today I was wondering what is the physical ...
1
vote
2answers
95 views

Two different formulas

My problem is simple : given a particle of mass $m$, charge $q$ and velocity $\bf{v}$. If $\bf{A}$ denotes the magnetic potential satisfying $\bf{B}= \nabla \times \bf{A}$. I want to etablish the ...
25
votes
6answers
3k views

Why are Killing fields relevant in physics?

I'm taking a course on General Relativity and the notes that I'm following define a Killing vector field $X$ as those verifying: $$\mathcal{L}_Xg~=~ 0.$$ They seem to be very important in physics ...
0
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0answers
41 views

Force of Leaky Bucket over time

Indie Lab Procedure: I attached a cylindrical can to a force sensor. On the bottom of the can I drilled holes of various diameters. I then added a volume of water while collecting data from the force ...
2
votes
1answer
47 views

Local Coordinate Expressions for Lie Derivatives

I'm currently working through the math chapters of Norbert Straumann's book on General Relativity. I have trouble understanding the coordinate expression of the Lie derivative of a basis vector. The ...
2
votes
1answer
334 views

How is it possible to pull out derivatives of a wavefunction?

In an early derivation, the following equation was stated: $$\frac\partial{\partial t}\lvert\psi\rvert^2 = \frac{i\hbar}{2m}\biggl(\psi^*\frac{\partial^2\psi}{\partial x^2} - ...
0
votes
1answer
45 views

Why trajectories approach to origin tangent to the slower direction?

I am reading non-linear dynamics from Strogartz. Suppose, I have two solutions of a non linear system: $x(t) = x_0e^{at}$ and $y(t) = y_0e^{-t}$, where $a\in \mathbb{R}$. Now it is clear that,for ...
1
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3answers
144 views

Can a particle have no instantaneous velocity at all points of the path taken but a finite average velocity?

I have a question on kinematics. Say the path traced by a particle is given by a Koch curve or Koch snowflake. Now consider the particle starts from some arbitrary point $A$ on the curve and ...
0
votes
2answers
118 views

Is resistance the gradient in a $V/I$ graph?

We have a circuit where there is a variable resistor, and we increase this resistance at a steady rate, while increasing current. Thus we have increasing voltage. The gradient is defined by $dy/dx$. ...
1
vote
0answers
30 views

Commutativity Variation and derivative

When in general is true that $\nabla_{\mu}\delta=\delta\nabla_{\mu}$ where $\nabla_{\mu}$ is covariant derivative? I was thinking this when one has to find the equations of motion for example in ...
1
vote
1answer
108 views

Full time-derivative, Poisson brackets and Hamilton's equations (classical mechanics)

While studying Poisson brackets in classical mechanics and the derivation of $\dot{q_j}=\{q_j,H\}$ and $\dot{p_j}=\{p_j,H\}$ form of Hamilton's equations I encountered a surpsing identity, which led ...
1
vote
1answer
128 views

Difference between $dM/dt = 0$ and $\partial M/\partial t=0$ [duplicate]

$\frac{dM}{dt} = 0$ represents a constant of motion $M.$ Why not $\frac{\partial M}{\partial t}$ represent a constant of motion $M$?
0
votes
0answers
47 views

A Lie derivative $\mathcal{L}_{\alpha^A}$ with respect to a spinor $\alpha^A$?

Suppose we work with Minkowski flat space $M$ (just to make things easy). If $\textbf X$ is a Killing vector field it is possible to define the Lie derivative of an spinor $\alpha^A$ with respect to ...
1
vote
1answer
219 views

Derivation of one-form/vector equation in Carroll confusion

I don't understand the derivation of Equation 2.14$$\mathrm{d}f\left(\frac{d}{d\lambda}\right)=\frac{df}{d\lambda} \tag{2.14}$$ in Carroll's Lecture Notes on General Relativity ...
2
votes
1answer
38 views

Confirmation of Uncertainty in Indices New Formula? [closed]

I am experimenting relations with regards of the value with uncertainty raised to the $n$th power. I came up with this formula: $$(A\pm\alpha)^n=A^n\pm(A^{n-1}n\alpha)$$ Anyone here able to ...
1
vote
1answer
225 views

Gradient, divergence and curl with covariant derivatives

I am trying to do exercise 3.2 of Sean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives. ...
1
vote
1answer
64 views

$\frac{d}{dr}=0$ and $\frac{d}{dz}=0$ (cylindrical coordinates) for a 1D ring

In http://ritchie.chem.ox.ac.uk/Grant%20Teaching/2010/Lecture%204%202010.pdf slide 21 of 26, he says "Radius of ring is fixed and so derivatives in $r$ are 0." Presumably this goes for ...
0
votes
1answer
56 views

In central-force mechanics, how do we substitute $ξ=\frac{1}{r}$?

I have taken a look at central-force mechanics in the past, but I still cannot understand how $ξ=\frac{1}{r}$ is substituted to find $\frac{d^2r}{dt^2}$ in terms of ξ. So I know from $F=ma$ that: ...
2
votes
1answer
54 views

What does the zero in the differential operator $\partial_0$ mean?

I have noticed the differential operator $\partial_0$ in a lot of equations whilst studying quantum field theory. I am used to the notation $\partial_x$ meaning $ \frac{d}{dx} \\\\ $ etc. but just a ...
-1
votes
1answer
90 views

To prove, $\nabla.(\nabla\phi \times \nabla\psi)$ =0 [closed]

Please Help me solving the problem using levi-cevita symbol : Prove That, $\nabla.(\nabla\phi \times \nabla\psi)$ =0 where $\phi =\phi(x,y,z)$ & $\psi=\psi(x,y,z)$
0
votes
4answers
134 views

Significance of curl ($\nabla\times\boldsymbol{V}$)

What is the physical significance of curl $$\nabla\times\boldsymbol{V}~?$$ I mean I read 'curl V represents the rotation of the vector $V$. My question what is it about the term ...
4
votes
4answers
209 views

Do $\vec r$ and $d \vec r$ have the same direction?

One question is bugging me for a long time but I couldn't make out anything nor could my friends. Here it goes: We know, $\vec r$ is regarded as the position vector. So we can say, $$\vec r \cdot\vec ...
0
votes
1answer
71 views

Clarification about some steps in the derivation of the Lie derivative (mechanics)

First of all, this question may seem to be undefined, because I'm not sure how to connect this (to me) newly introduced concept with the abstract notion of the Lie derivative. I'm not even sure if I ...
1
vote
1answer
56 views

Derive an equation related to magnetism [closed]

Solve the equations for $v_x$ and $v_y$ : $$m\frac{d({v_x)}}{dt} = qv_yB \qquad m\frac{d{(v_y)}}{dt} = -qv_xB$$ by differentiating them with respect to time to obtain two equations of the ...
1
vote
2answers
119 views

Gauge covariant derivative of a creation operator

Suppose we define the (gauge) covariant derivative or as $$\tilde{\nabla}=\nabla+ie\textbf{A},$$ where the vector potential $\textbf{A}$ has a matrix structure where only the diagonal has nonzero ...
0
votes
0answers
27 views

Evaluating derivatives with respect to certain vector axis

So, I am trying to work in Spherical coordinates. I have a predefined fixed axis, $\hat{v}_0$, so that $\alpha=\vec{r}.\hat{v}_0$ Now, I am interested in the following: \begin{equation} ...
1
vote
1answer
268 views

What is meant by 'probability of transition per unit time'?

Today I came across a term used by Feynman in his thirteenth lecture: 'probability per unit time' to go from $| 1\rangle$ to $|2\rangle$ while initially being at $|1\rangle$. This is the excerpt fom ...
0
votes
1answer
94 views

Trouble understanding Landau & Lifshitz writing about Lagrangians and Galilean Relativity [duplicate]

We have two inertial coordinate systems, $K'$ and $K$. $K$ is moving with infinitesimal velocity ${\epsilon}$ relative to $K'$. Using Galilean relativity we can transform this into $v'=v+{\epsilon}$. ...
4
votes
1answer
151 views

Gradient one-form [duplicate]

I am trying to understand what gradient one-form means actually. In the book that I'm following (A first course on General Relativity by Schutz) it's told that gradient is a one-form and it's ...
3
votes
1answer
143 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 ...
0
votes
1answer
155 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 ...
-1
votes
1answer
114 views

Differentiating displacement with respect to speed in order to obtain time

I have this problem where I am trying to calculate $d(t)$ and $v(t)$ of a mass m on a spring, dropped from a displacement $A$, without using anything else than Hooke's law and energy calculations. ...
0
votes
1answer
69 views

Connection between heat capacity and the derivative of enthalpy

One can define the heat capacity of isobaric processes as $$ c_P = \left( \frac{\partial H}{\partial T} \right)_P . $$ Now, we know that the unit of heat capacity is Joule per Kelvin, i.e., I need to ...
0
votes
2answers
124 views

What is the derivative of $\dot{\theta^2}$? [closed]

$$\frac{d}{dt}(\dot{\theta^2}) =? 2\dot{\theta}\ddot{\theta}$$ is this correct, or am I missing something?
0
votes
0answers
33 views

Usage of delta operator [duplicate]

So I've always thought that "$\Delta$" when applied to an n-tuple or scalar was the change of that n-tuple or scalar relative to a previous state in time and proportional to the amount of time or ...
0
votes
1answer
85 views

What is the difference between $\frac{DA^\mu}{D\lambda}$ and $\frac{DA^\mu}{d\lambda}$?

I earlier asked this question How can you have $\frac{DA^\mu}{d\tau}$? I am now wondering: What is the difference between $\frac{DA^\mu}{D\lambda}$ and $\frac{DA^\mu}{d\lambda}$? In the linked ...
0
votes
1answer
90 views

How can you have $\frac{DA^\mu}{d\tau}$?

If a covariant derivative is given by: $$D_\nu A^\mu=\partial_\nu A^\mu +\Gamma^\mu_{\nu \lambda} A^{\lambda}$$ Then how does $\frac{DA^\mu}{d\tau}$ make any sense? Since there are no 'differentials' ...
1
vote
2answers
83 views

Units of the derivative of a function

I have a function $\phi(\mu, \sigma)$. $\mu$ and $\sigma$ are voltages (in mV in my case), so $\phi$ is a function of two voltages. $\phi$ itself, however, is in units of time (ms in my case). ...
2
votes
1answer
201 views

Differentiation of a vector with respect to a vector

Does differentiation of a vector with respect to a vector make any sense? Even if it makes sense, how does it make any physical meaning? I mean what is the physical interpretation?
0
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0answers
70 views

Can I Wick-contract terms with derivatives with terms without derivatives?

Consider for example the QCD three point vertex, can I contract a gluon field with the gluon field with a derivative in the vertex?
0
votes
3answers
235 views

Curl of a vector field [closed]

What is the physical interpretation of curl of a vector field? Just as divergence implies flux through a surface. I mean if $\vec A$ is a vector field, what does $\left(\nabla \times \vec A \right)$ ...
-1
votes
1answer
83 views

Forced damped harmonic motion, angular frequency at which amplitude is maximum. differentiation [closed]

$$A_0 = \frac{(F_0/m)}{\sqrt{(\omega_0^2-\omega_d^2)^2+b^2\omega_d^2/m^2}}$$ How would I differentiate this with respect to the driven angular frequency (equating to zero) in order to obtain the max ...
0
votes
2answers
528 views

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 ...
2
votes
2answers
80 views

What does it mean to differentiate a spinor-valued field?

Peskin and Schroeder, equation 3.28, states that the Klein-Gordon equation $$(\partial^2+m^2)\psi=0 \tag{3.28}$$ is a valid choice of equation for a Dirac spinor field. Their explanation makes sense ...
8
votes
3answers
420 views

How does uncertainty/error propagate with differentiation?

I have a noisy temperature (T) vs. time (t) measurement and I want to calculate dT/dt. If I approximate $dT/dt = \Delta T/\Delta t$ then the noise in the derivative gets too high and the derivative ...
1
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1answer
123 views

Geometric meaning of spin connection

A very short question: Does the spin connection that we encounter in General Relativity $$\omega_{\mu,ab}$$ have a geometric meaning? I am supposing it does because it comes from mathematical terms ...
1
vote
1answer
60 views

Issues with the chain rule in derivatives in chiral perturbation theory [duplicate]

I realize that this is a purely math question, which however has arisen in a physics computation. The reason to post it here is that I want a fast dirty answer. Not something cluttered with ...
0
votes
2answers
106 views

Changing coordinates in partial derivatives, re: Hydrogen atom

At around 11:35 in this video https://www.youtube.com/watch?v=q99ygFeWGv4 the instructor is using a "standard way of changing coordinates" in partial derivatives relating to the new variables of ...
13
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5answers
941 views

What does it mean for a physical quantity if its mixed second partial derivatives are not equal?

This goes for every problem (either in electromagnetism or fluid dynamics) that has to do with vector fields. Say we have a fluid flowing in a closed circular pipe (or an electromagnetic field, the ...