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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Physical meaning of divergence

While reading the section on Hamiltonian mechanics in Taylor's Classical mechanics, I realized that I didn't fully understand what he was saying when he was explaining why $$\nabla\cdot\vec{F(\vec{x_0}...
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2answers
69 views

What is the meaning of this definition of potential energy?

The isolated system of particles is being observed. In the coursebook, $\vec F_\mu$ is by definition the vector sum of forces of all other particles acting on $\mu$-th particle. Usually, potential ...
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64 views

Regarding Ampere's Circuital Law

If I am to show that Ampere's Circuital law holds true for any arbitrary closed loop in a plane normal to the straight wire, with its validity already established for the closed loop being a circle of ...
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1answer
194 views

Why are densities not fields?

I have read (in Statistical mechanics of lattice system 2: exact, series and renormalization group methods by D.A. Lavis and G.M. Bell pg 2 ), that intrinsic variables are either fields or densities. ...
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1answer
56 views

Apparent discrepancy between Lagrange field equation and Maxwell equation [closed]

I am deriving Maxwell's equations from a Lagrange field equation and have come across something I can not figure out no matter how hard I try. The problem is in the signs. If we take the Lagrange ...
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1answer
106 views

Differential form of the law of gravitation potential

I have problem understanding transaction (operations and methods applied) for one equation to other equation. It is about gravitational potential. $${\vec F_{grav}=\frac{GMm_{obj}\vec R}{R^3}}$$ If we ...
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1answer
117 views

Difference of connections in the Killing vector equation

For the Killing vector equation, I sometimes see it written in terms of spin connection $\omega$ and other times in terms of the affine connection $\Gamma$. More clearly $$\nabla_{\mu}V_{\nu}+\nabla_{...
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2answers
118 views

How to show flat FRW metric has a time-like conformal Killing vector?

I would like to derive the fact that the flat FRW metric has a time-like conformal Killing vector. Is there an easy way to do this? @ValterMoretti showed how one can do this for metrics with a ...
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0answers
75 views

Conformal time-like Killing vector near null geodesics in all spacetimes?

Is it true that in all spacetimes there is some conformal time-like Killing vector $\tau^a$ in the vicinity of null geodesics? If the above statement is true then can one argue that, for all ...
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1answer
73 views

What exactly is conservative vector field?

I'm studying calculus, but since the example involved a physical concept. I will ask here: This is how it goes: This means that in a conservative force field, the amount of work required to ...
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1answer
57 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 ...
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1answer
71 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 $...
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4answers
932 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 ...
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1answer
50 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
142 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}) + \frac{\...
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1answer
112 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 ...
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1answer
133 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 ...
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1answer
101 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 ...
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1answer
347 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 ...
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1answer
219 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 = \frac{kx}{(x^2+y^2)^{3/2}}...
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1answer
64 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 ...
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1answer
104 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} X_L=\sum^M_{j=1}\bigg(v^j\frac{...
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1answer
241 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. $$\...
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1answer
82 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 ...
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6answers
856 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 $C^{\...
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1answer
57 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 ...
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1answer
37 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} + p_y\frac{\...
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1answer
76 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 ...
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7answers
338 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 ...
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1answer
105 views

Lie derivative in this paper [closed]

Say, $$L_{V}z^A =0$$ but I don't know much about Lie derivatives except what I saw now through wikipedia http://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields#Definitions that it is (if I am ...
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1answer
101 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 $\gamma(s)\in M$ ...
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1answer
160 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 $\partial_{a}$ ...
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2answers
360 views

Divergence of vector potential [closed]

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 $- \vec a \frac{2}{r^3}$, ...
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155 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 $\phi^\prime(x^\...
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2answers
187 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 ...
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6answers
660 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 ...
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1answer
57 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 ...
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1answer
148 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 x}+\hat{y}\frac{\partial}{\...
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1answer
81 views

Duality and 1 forms

How is a dual map defined if we are talking about partial derivatives and 1 forms?
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3answers
257 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
40 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: ...
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0answers
136 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 ...
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2answers
67 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 ...
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2answers
231 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 ...
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1answer
252 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 ...
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3answers
607 views

How to visualize the gradient as a one-form?

I am reading Sean Carrol's book on General Relativity, and I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I ...
2
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1answer
105 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 (...
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1answer
197 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 ...
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1answer
284 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 $f:...
4
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1answer
208 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 ...