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Questions tagged [vector-fields]

Vector-fields are vector valued functions which define a vector at each point in space. Examples of the vector field include the electric field and the velocity of a fluid.

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Two contradictory derivations of Killing equation

In David Tongs lecture notes he derives the Killing equation by showing that the charge $Q=\xi_\mu \frac{\mathrm{d}x^\mu}{\mathrm{d}\tau}$ is conserved $$ 0=\frac{\mathrm{d}Q}{\mathrm{d}\tau}=\frac{\...
Silas's user avatar
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Cone vs. small circle parallel transport

I'm having trouble reconcile the following two seemingly contradicting conclusions (in 2d space): A cone is flat, because you can unfold it and it's a flat 2d surface. A cone as shown in the picture ...
Cosmo's user avatar
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2 answers
69 views

Isn't the induced electric field vector always tangent to the loop?

In the law of electromagnetic induction: $$\mathcal E_\text{ind}\equiv \oint_\Gamma \mathbf{E_\text{ }}d\mathbf l = -\dot \Phi_\mathbf{B}(t)$$ Here, $d\mathbf l = \hat{\mathbf t}dl$ is taken along the ...
Awe Kumar Jha's user avatar
2 votes
1 answer
87 views

How to compute the vector field from a potential in the complex plane?

I am watching this Youtube video and I have the following dumb question around 1:18:00: How do you draw the vector field for a given potential in the complex plane? He gives the potential $V(x) = x^4-...
Wyatt Kuehster's user avatar
1 vote
3 answers
106 views

The conservative force [closed]

I read about the definition of the curl. It's the measure of the rotation of the vector field around a specific point I understand this, but I would like to know what does the "curl of the ...
Dirac-04's user avatar
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0 answers
36 views

When is the derivative of Hamilton flow respect to initial conditions independent of time?

Consider a Hamiltonian system with coordinates $\Gamma^A=(q^i,p_i)$ and let $X^A(s,\Gamma_0)$ be the Hamiltonian flow (i.e. a solution to Hamilton's equations) parametrized by time $s$ and initial ...
P. C. Spaniel's user avatar
1 vote
1 answer
50 views

How to get vector field from Poisson brackets?

Steinacker defines the Hamilton vector field as any field s.t.: $$\{f,g\}=V_f[g].$$ I really can't understand this. The Poisson algebra is closed with respect to Poisson brackets (i.e. $\{\cdot ,\cdot\...
polology's user avatar
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How do I find the absolute maximum and minimum values of the Lamb-Oseen Vortex?

For an angular velocity function derived by Navier-Stokes, $$ \omega \left(r,t\right)=\frac{\omega _0R_0^2}{R\left(t\right)^2}exp\left(-\frac{r^2}{R\left(t\right)^2}\right)$$ from which the azimuthal ...
Tayler Montgomery's user avatar
-1 votes
1 answer
57 views

Why does the timelike killing vector become spacelike inside the ergoregion?

Why does the timelike killing vector become spacelike inside the ergoregion? Some textbooks make this claim and move on to explain negative energy, but I could not find any proof for this claim. I can'...
Gene's user avatar
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4 votes
3 answers
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What is the proof that the Schwarzschild metric is not static inside the horizon?

In Lecture Notes on General Relativity, Sean M. Carroll shows that the Schwarzschild metric is not only stationary but also static (Chapter 7, page 169, Eq. 7.20 and following interpretation). On the ...
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How does one prove that the electric field is conservative when it is not defined on simply-connected regions?

I apologize for the ignorance and the rough English in advance, I have an issue understanding how to match both what happens in physics and what I am seeing in calculus. In my calculus class we ...
Some random guy's user avatar
1 vote
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68 views

In the frame field construction in GR, how do you get the vector field dual to a co-frame?

I am trying to understand the frame-field construction in General Relativity. We basically have four point-wise orthonormal vector fields, one of them being timelike and the other three being ...
Moustafa M. Kamel's user avatar
1 vote
0 answers
69 views

A preposterous abuse of notation involving Helmholtz decomposition theorem

Take what I am about to present with a light heart, since the mathmetically inclined may find it too out-of-the-world and devastating. The above diagram (this is drawn by me, but the original is very ...
Jonathan Huang's user avatar
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Killing Vectors AdS$_3$ (Solved)

I have been trying to understand the Killing vectors of AdS$_3$,i.e. Anti de Sitter in three dimension: $$ ds^2 = dX^2 + dY^2 -dU^2 -dV^2 $$ In this case the generators are given by $J_{ab}=x_a \...
Leonardo Sanhueza Mardones's user avatar
3 votes
1 answer
115 views

A theorem on page 72 in The Large Scale Structure of Space-Time [closed]

In chapter 3 of the book, page 72, a static observer is defined as $V^{a}\equiv f^{-1}K^{a}$, where $K^{a}$ is a timelike Killing vector field and $f^{2}=-K^{a}K_{a}$. Then, Hawking & Ellis claim ...
Rui-Xin Yang's user avatar
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Method of image charges for ungrounded conductive sphere seems to have charge of $q$ and not $(r/a) q$?

Using the 2d scenario for simplification Vector field of a point charge $q=1$ at (-4.1,0): $$\vec F_1\left( {x,y} \right) = \left(\frac{x+4.1} {{\sqrt{(x+4.1)^2 + (y-0)^2}}^2}\right) \vec e_x + \left(\...
Lewis Kelsey's user avatar
3 votes
1 answer
131 views

Laplace-Beltrami operator for a vector field

For a scalar field $\varphi$, the "wave" operator is defined as follows: $$\Box \varphi \equiv g^{ab}\nabla_a\nabla_b~\varphi = \frac{1}{\sqrt{|g|}}\partial_a\left\{\sqrt{|g|}~g^{ab}~\...
newtothis's user avatar
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Expanding the barotropic nondivergent potential vorticity equation: Which vector calculus property/identity to apply for dot product and del operator?

I am trying to expand the barotropic nondivergent potential vorticity (PV) equation [link] $$\frac{\partial \zeta}{\partial t} = -\vec{V} \cdot \nabla(\zeta + f)$$ where $\zeta$ is the relative ...
Brian Añano's user avatar
-2 votes
1 answer
153 views

Exact definition of divergence. Is it really the dot product of nabla operator with a vector? [closed]

I was trying to understand the derivation for divergence in cylindrical and spherical coordinate system, and I am a bit confused here. https://www.gradplus.pro/deriving-divergence-in-cylindrical-and-...
DocAi's user avatar
  • 33
1 vote
1 answer
103 views

Challenging Cauchy's Stress Tensor: Objectivity and Generalization of Divergence Theorem

I'm investigating the limitations of the Cauchy stress tensor model in classical continuum mechanics, specifically focusing on its compliance with the principle of material frame indifference (MFI) ...
Foad's user avatar
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2 votes
3 answers
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$\int \vec{E} \cdot \vec{dA} = (E)(A)$?

I've seen this kind of simplification done very frequently in Gauss's law problems, assuming E is only radial and follows some "simple" geometry: $$\oint\vec{E}\cdot\vec{dA}=\frac{Q_{enc}}{\...
JBatswani's user avatar
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0 votes
3 answers
131 views

The Curvature of Electric Field Lines

I have been practicing many questions regarding electrical field lines. However, I can't seem to understand when electrical field lines remain straight and when they start to curve. Does it depend on ...
improvement dude's user avatar
0 votes
0 answers
90 views

Component notation and matrix notation for gradient of vector

I'm trying to understand vector and tensor notation, but I'm coming across some difficulties. Say I have vector $\vec{u}$ and I compute its gradient $\nabla \vec{u}$. Then I get a tensor $\frac{\...
John Vector's user avatar
-1 votes
2 answers
415 views

Accurate drawings of field lines in three situations

I am looking for accurate drawings of the electric and magnetic field lines in three situations: The electric field lines formed between a positive point charge and negative point charge. (i.e. the ...
Euclid Looked On Beauty Bare's user avatar
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0 answers
19 views

How to transform the volume integral of a curl of the product of a potential with a magnetic field into a surface integral?

Problem 6.5 of Jackson electrodynamics says: A localized electric charge distribution produces an electrostatic field, $\vec{E} = - \vec{\nabla}\Phi$. Into this field is placed a small localized time-...
Ron Stean's user avatar
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0 answers
65 views

Hamiltonian flows and Poisson Brackets confusion

I have been using results from this paper in calculations. In sections 2.4 and 3.4 they perform a canonical transformation into new coordinates consisting of constants of motion. My question is ...
Geigercounter's user avatar
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0 answers
16 views

Logistic Growth and the Use of Changing Units

I am having a hard time understanding how this unit change is used to get rid of the need to use specific values for $ N $ and $ \mu $. Could somebody explain? I just can't figure it out. Help would ...
j.primus's user avatar
3 votes
1 answer
179 views

What's the physical meaning of Curl of Curl of a Vector Field?

The curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$ Now, curl means how much a vector field rotates ...
Plague's user avatar
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0 votes
1 answer
45 views

Shapes of waves in the surface of a pond when the breeze blows

When I throw a stone into a pond while a breeze is blowing across its surface and the water is at rest, what would be the shapes of the waves? Before the wind, the shape of the waves is circular. Then ...
Majid's user avatar
  • 159
0 votes
1 answer
58 views

Stokes' theorem and vector continuity equations

I have been working with homogeneous continuity equations of the general form: $$\frac{\partial \rho}{\partial t}+\vec{\nabla}\cdot \vec{J}=0$$ This has me wondering whether we can formulate other ...
Lagrangiano's user avatar
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-1 votes
2 answers
71 views

Extracting the dimension of an operator from algebra

I may misinterpret the question. In the lecture note of conformal field theory, arXiv:2207.09474, it says the following where for $P^\mu=i\partial_\mu$ and $D=ix^\mu \partial_\mu$. I am confused ...
Tanmoy Pati's user avatar
0 votes
1 answer
68 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 ...
spacetime's user avatar
1 vote
2 answers
101 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 ...
Tomasz P's user avatar
0 votes
1 answer
43 views

What are streamlines and pathlines

I asked a question earlier about having a vector field and a starting point, and then making a parametric that starts at the starting point and the derivative at any point in the parametric equals the ...
GIORGI GOGIBERIDZE's user avatar
-1 votes
1 answer
51 views

How to make a parametric that matches a vector field?

So I have a vector field defined as $(X(x,y),Y(x,y))$ and I’m trying to make a parametric $(t,t)$ who’s derivative at a point is equal to the vector field at that point. for example the vector field $(...
GIORGI GOGIBERIDZE's user avatar
0 votes
2 answers
54 views

Is $dJ(V,V)=0$? where $J$ is a 1-form?

So is this always 0?( Where $dJ$ is the exterior derivative and $V$ a vectorial field) \begin{align} dJ(V,V)=\partial_jJ_i(dx^j\wedge dx^i)(V,V)=\\ \partial_j J_i (v^kdx^j(\partial_k)v^ldx^i(\...
Guillermo Fuentes Morales's user avatar
1 vote
3 answers
155 views

Where to apply $\nabla$ operator when taking curl of a cross product?

In my EM class we went over $$\nabla\times \frac{\vec{d}\times \vec{r}}{r^3}$$ which apparently can be breaken down to $$r(d\cdot \nabla)\frac{1}{r^3}-d(r\cdot\nabla)\frac{1}{r^3}+\frac{\nabla\times(d\...
sasssu's user avatar
  • 33
2 votes
2 answers
266 views

Vector potential of position field

Consider the position vector field $\vec{r}=(x,y,z)^T$. What would be a vector potential $\vec{A}$ for this field? I was thinking of something like $\vec{A}=(yz,zx,-xy)^T$, which gives $$\nabla\times ...
Riemann's user avatar
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2 votes
1 answer
135 views

Stokes stream function derivation

I want to know a concrete derivation of 3D Stokes stream function. The statement is, for example in 3D spherical coordinates (with symmetry in rotation about the $z$-axis), if $$\nabla \cdot u=0\tag{...
Zjjorsia's user avatar
  • 311
1 vote
1 answer
64 views

Vector Magnetic Potential and pointwise current density source

I am currently studying antennas and I am trying to understand how to solve the following vector equation $$\nabla^2A+k^2A=-\mu J$$ in the case when there is a point current source at the origin. The ...
edoverg's user avatar
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0 votes
1 answer
114 views

Doubt about the derivation of Liénard-Wiechert Potentials?

When deriving the Lienard-Wiechert Potentials, there is one step that you need to perform: $$ \nabla_{\mathbf r}|\mathbf r - \mathbf r_s(t_r)| $$ Where $t_r$ is: $$ t_r = t - \frac{|\mathbf r - \...
Álvaro Rodrigo's user avatar
0 votes
0 answers
47 views

How can a vector field in $E^3$ be represented by a linear combination of only 2 basis vectors?

In Chapter I.7 of "Einstein Gravity in a Nutshell", Zee introduces the concept of covariant derivatives. I am confused by the first line in this section (see below) as it appears that we can ...
Reuven's user avatar
  • 1
0 votes
0 answers
37 views

What is the physical significance of the cross product of curl of a vector field $v$ with another vector field $w$?

I think if curl of a vector field v corresponds to an applied rotation, it's cross product with a velocity vector field w (say) should give something analogous to the resulting torque. Am I close?
Benjamin Kurian's user avatar
0 votes
1 answer
56 views

Physical significance of $\vec{w}$ $\times$ $($curl $\vec{v})$

I think if curl of a vector field $\vec{v}$ corresponds to an applied rotation, it's cross product with a velocity vector field $\vec{w}$ (say) should give something analogous to the resulting torque. ...
Benjamin Kurian's user avatar
0 votes
2 answers
146 views

Extreme confusion with the Lorentz transformation law for vector fields

Let $\Lambda$ be a Lorentz transformation represented as $4 \times 4$ matrix. Then, following What does it mean to transform as a scalar or vector? , it seems that a vector field $f : \mathbb{R}^4 \...
Keith's user avatar
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0 votes
1 answer
74 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 ...
tommy1996q's user avatar
1 vote
1 answer
77 views

Magnetic vector potential in 1+1 spacetime dimensions

In the theory of electromagnetism in 1+1 spacetime dimensions (one temporal and one spatial coordinate), one can define the 2-potential vector (analogous to the 4-potential vector in 3+1 spacetime ...
Daniel Vainshtein's user avatar
1 vote
2 answers
85 views

What is Dirac's reasoning when saying parallel displacement creates vector field with vanishing covariant derivative?

Section 12 of Dirac's book "General Theory of Relativity" is called "The condition for flat space", and he is proving that a space is flat if and only if the curvature tensor $R_{\...
Lewis Kirby's user avatar
0 votes
0 answers
43 views

What isn't the metric invariant under translation with killing vectors?

I am learning about Killing Vectors in GR class, and I'm testing my knowledge of them as a start with the Minkowski metric. I used the simple 2d Minkowski metric: $$ds^2 = -dt^2 + dx^2$$ and got 3 ...
Habouz's user avatar
  • 1,324
3 votes
2 answers
488 views

Does the divergence theorem imply an underlying symmetry?

The divergence theorem connects the flux (through surface) and divergence (in a volume) for any vector field. This theorem expresses continuity. It isn't clear (to me) whether there is a conserved ...
AppliedAcademic's user avatar

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