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0answers
20 views

Vector fields corresponding to null geodesic congruences in general relativity

I'm working in Minkowski space, and I'm considering some 2D surface, $S$. On each point of the surface, I've computed a null vector, $k^a$, which is orthogonal to it. There will be a unique null ...
1
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0answers
29 views

Potential (which only depends on the length of a position vector) [on hold]

a) A potential $U$ only depends on the length $r=|\vec{r}|$ of the position vector $r$. Show that $$\vec{\nabla}U(r)=U'(r)\frac{\vec{r}}{r}. $$ What properties does this vector field have? ...
-1
votes
0answers
21 views

Killing vectors of AdS space with the metric given in Poincaré coordinate [on hold]

I am trying to solve this problem: Find the Killing vector correspond to the symmetry of the scale invariant for the AdS(n+1) $$ (t,{\bf x}) \rightarrow (at, a{\bf x}) $$ when the metric of the AdS ...
2
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1answer
32 views

Books on Liouville Operator

I am looking for a good book doing classical mechanics and statistical mechanics in terms of the Liouville operator. I have not found a lot on this subject and even books like Mathematical Methods of ...
1
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0answers
30 views

Hamiltonian flow?

I was wondering what the Hamiltonian flow actually is? Here is my idea, I just wanted to know if I am correct about this. So let $(x(t),p(t))' = X_{H}(x(t),p(t))$ are the Hamilton's equations and ...
5
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2answers
101 views

Lie derivative vs. covariant derivative in the context of Killing vectors

Let me start by saying that I understand the definitions of the Lie and covariant derivatives, and their fundamental differences (at least I think I do). However, when learning about Killing vectors I ...
-2
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0answers
22 views

Why is the rotation not zero and the divergence zero in the figures below? [migrated]

Why is the rotation not zero and the divergence zero in figure 1 and figure 2 below? Figure 1 Figure 2
0
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1answer
44 views

Is the flux through A the same as the flux through B?

In the figure below, the amount of field lines through A is the same as the amount of field lines through B, but can you say the flux through A is the same as the flux through B as well?
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2answers
65 views

How is the MHD magnetic field time evolution equation transformed to the vector potential time evolution equation?

Starting from the time evolution equation of the magnetic field for incompressible MHD (magnetohydrodynamics) $$\frac{\partial \vec{B}}{\partial t} = \nabla \times (\vec{v} \times \vec{B}) + ...
2
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1answer
39 views

Parallel Transported Orthonormal Basis

The following argument results in a conclusion that I find strange, and makes me suspect there is something wrong with the reasoning. First, consider a timelike geodesic $\gamma$ with normalized ...
0
votes
1answer
50 views

Show that a solenoidal field is always a curl of a vector field [closed]

Can someone prove that: $$\nabla \cdot \mathbf{B} = 0 \implies \mathbf{B} = \nabla \times \mathbf{A}~?$$ I know that $$\nabla \cdot (\nabla \times \mathbf{A}) = 0$$ identically. But can one prove ...
3
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1answer
67 views

What is the physical cause that circulation on a closed surface is zero?

This is quoted from Feynman's Lectures: We would like to see what happens when the loop shrinks down to a point, so that surface boundary disappears - the surface becomes closed. Now, if the ...
0
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1answer
94 views

What are the mathematical models for force, acceleration and velocity?

In mechanics, the space can be described as a Riemann manifold. Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector ...
1
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0answers
29 views

SR: vector field and change of reference [closed]

If $U$ and $V$ are vector fields, then the derivative of $U$ along $V$ is the vector field $\nabla _V U$ with components $$\nabla _V U^a=V^b \frac{\partial U^a}{\partial x^b}.$$ I would like to verify ...
2
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1answer
53 views

Gradient and curl of a field in polar coordinates

How do we determine the gradient and curl of a scalar/vector field in polar coordinates? For instance, if we have the following potential energy function for a force, $$U = ...
0
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1answer
48 views

If the electric field is not a gradient, can it exist?

We know that the gradient of the electric potential function $V(x,y,z)$ is the electric field. But not all vector fields are gradients, for example $y\hat{i}-x\hat{j}$ is not a gradient. Does this ...
3
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1answer
82 views

Lagrangian vector field expression

The Lagrangian vector field $X_L$ on the tangent bundle is given in page 4 of Marco Mazzucchelli's "critical Point Theory for Lagrangian systems" to be; \begin{equation} ...
2
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1answer
51 views

Electric field in a hollow object

I am currently visiting a course about electrodynamics. In my last lecture it was said that if a hollow sphere is inside of a bigger sphere, but only in the bigger sphere there are charges, the ...
0
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1answer
94 views

What is the Jacobian matrix of Newton's law of gravity? [closed]

I am trying to come up with the gradient of a vector (Jacobian matrix) for Newton's Law of Gravity. In other words, the differential gravitational field in 3-dimension. Here's the initial Law. ...
2
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1answer
51 views

Can we depict the electric field lines between three or more charges?

Last year we studied in school electric field that was created around charges and we were showed how to depict the electric field lines for one charge or between two charges. Then i wondered how would ...
4
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6answers
700 views

Is electromagnetic vector field a sum of E and B?

I have a hard time to fully understand (classical) electromagnetic field theory with respect to Helmholtz's decomposition. Let me start from Helmholtz's theorem: Any vector field of class ...
0
votes
1answer
31 views

How to Model Differential Gravitational Field from Vectors?

The tides are caused by differences in the gravitational field of the moon on the near side and far side of the earth. If I set this gravitational field as a vector field using Newton's Law of ...
0
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1answer
26 views

Force on a dipole

The force exerted by an electric field on a dipole is : $$(\vec{p}.\vec{\nabla})\vec{E}$$ but how exactly do I develop this ? Is it : $$p_x\frac{\partial E}{\partial x}\vec{e_x} + ...
2
votes
1answer
56 views

Estimating divergence of set of vectors

I have a set of points where directions and intensities of a flow are given (in 3D). Is it possible to estimate the divergence of the flow defined by those vectors? I only need a rough estimate and I ...
11
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7answers
255 views

How can a set of components fail to make up a vector?

Many books in Physics insist to define vectors are objects with components with the property that the components transform in a proper way under a change of coordinates. Now, in mathematics, on the ...
0
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1answer
59 views

Lie derivative in this paper [closed]

In this paper http://arxiv.org/abs/1210.2332 it says in (3.19) p. 8 that $$L_{V}z^A =0$$ but I don't know much about Lie derivatives except what I saw now through wikipedia ...
0
votes
1answer
79 views

Geodesic equation

I have a technical question about the geodesic equation. Assume we have a frame $(E_{1},E_{2},E_{3},E_{4})$ (not necessarily a coordinate frame). Assume we have a parametrized curve $\gamma(s)\in M$ ...
1
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1answer
81 views

Understanding the covariant derivative and its relation to parallel transport

I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators $\partial_{a}$ ...
0
votes
2answers
77 views

Divergence of vector potential

I was given the vector potential $$\vec A (\vec r) = - \vec a \times \nabla \frac{1}{r}$$ with a constant vector $\vec a$. Now, I found the $\vec B$ field which is I think $- \vec a \frac{2}{r^3}$, ...
1
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2answers
52 views

Identifying a scalar function

We know that a scalar is invariant under rotations. What about a scalar function? Should it also be invariant under rotations? Therefore, under rotation $\phi(x,y,z)$ must be equal to ...
1
vote
2answers
138 views

Is a vector field not a vector quantity?

I'm trying to make sense of Poisson bracket relation $$\{L_i,A_k\}_{PB}~=~\epsilon_{ikl}A_l,\tag1$$ where $L_i$ is $i$th component of angular momentum, $A_k$ is $k$th component of an arbitrary ...
-2
votes
6answers
474 views

Momentum of stationary electron in a curl free vector potential

The essence of this question is simplicity itself: There is an electron in a curl-free $\vec{A}$ field. The electron is stationary so its m$\vec{v}$ momentum is 0. However, it has "momentum" from ...
0
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1answer
35 views

Vector fields of a physical quantity [closed]

I had this small confusion - The components of a vector field representing a physical quantity must have the same physical dimension right? for example- the radius vector has the unit of length along ...
0
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1answer
78 views

Finding Transversal Components from Longitudinal component for Electric and Magnetic Field in a cylindrical coordinates system

Can someone explain why this two equation are equivalent? $\nabla_T$ denotes the transverse two-dimensional nabla operator: $\nabla_T=\hat{x}\frac{\partial}{\partial ...
0
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1answer
54 views

Duality and 1 forms

If a Killing vector is equal to: $$X= -\frac{1}{\sqrt{2}}\partial _t + \frac{\alpha}{\sqrt{2}}\partial_1.$$ But as far as I know is that the dual of a vector is a 1-form, so can I represent that ...
4
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3answers
103 views

If a Killing vector field is timelike, can it be set to $\partial/\partial t$?

If one has a Killing vector that turned out to be a timelike Killing vector field because of negative norm. Can we set this Killing vector field equal to $\partial/\partial t$?
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0answers
30 views

The first term of Stokes Vector of natural light is zero?

Consider the electric field of a beam of natural light: $$ E(r,t) = E_0 \cos(k·r+wt) $$ Since this beam of light is natural, the vector E has all the components possible that satisfies: ...
0
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0answers
41 views

Norm of Killing vector field

Let us suppose we have a Killing vector field with $X^a = 1/2$ and $X^b = 1/3$ and $g_{ab}=1$ where the other $c$ and $d$ components are zero. Now we want to find its norm: The formula for finding ...
0
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2answers
56 views

EM waves and fields

According to wikipedia, electromagnetic waves are "synchronized oscillations of electric and magnetic fields that propagate at the speed of light". I understand what it means in theory. But in ...
1
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2answers
88 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
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1answer
195 views

What is an “Einstein transformation” in general relativity?

When introducing the vielbein formalism in general relativity, I came across the use of an infinitesimal general transformation, or Einstein transformation. The latter term seems not to be covered on ...
2
votes
1answer
71 views

How to get the linear and angular acceleration generated by a force vector field?

I am working on a physics simulation and I have to calculate the angular acceleration in degrees per seconds squared around the point on the object located relatively to the center of a vector field ...
1
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1answer
104 views

Can a spacetime solution in GR have no Killing vector fields?

Sometimes Killing vector fields in a given spacetime are described as giving information about a symmetry of that particular spacetime solution. If I look at the requirement of a Killing vector field ...
4
votes
1answer
193 views

Conformal Killing fields on Schwarzschild

I am trying to understand which are the conformal Killing Fields on the Schwarzschild spacetime. I say that $X$ is a conformal Killing field on $S$ ($S$ is Schwarzschild) if there exists a function ...
4
votes
1answer
120 views

field solutions for covariant derivative of vector field constrained to zero

Question: What do the solutions of $\nabla_\mu A^\nu = 0 $ look like? And is it possible for spacetime curvature to somehow restrict the solution to $A^\nu = 0$? Here is my current ...
-2
votes
1answer
58 views

If you are not given a metric, which one is more fundamental: a vector or a covector? [closed]

If we do not have the metric $g_{\mu\nu}$ for a given spacetime, are vectors $x^\mu$ more fundamental than covectors $x_\mu$ or vice versa? Why? (if the metric were given we could just raise/lower ...
1
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2answers
161 views

Why is $\nabla\cdot(\hat{\bf r}/r^2)$ giving 0 as answer? [closed]

While I was reading I encountered the statement $\nabla\cdot(\hat{\bf r}/r^2)$ (r cap divided by $r$ square) is 0. Can anyone explain proof of the statement why is it giving 0?
0
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1answer
78 views

Want to know about divergence [duplicate]

Can anyone please explain how to know whether a vector field has divergence or not by seeing its diagram? I have read that a vector field must change for having divergence but why is divergence zero ...
1
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0answers
52 views

Magnetic field derived from a scalar function

Question: Show that $\mathbf B = \hat r \times \nabla g(\mathbf r)$, where $g$ is an arbitrary scalar function, is a plausible magnetic field. What current density $\mathbf J $ can produce ...
3
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3answers
161 views

Line integral definition of work clarification

So I am kind of confused about the role of force when calculating work. Specifically, when defining work using a line integral. There is a paragraph in my calculus book that is really throwing me off ...