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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Why the heat flux vector at a point must be perpendicular to the temperature isothermal surface? Is it a definition or a deduction?

Before the question: I am working on numerical calculation of three dimension parabolic equation that based on Fourier's Law of which I am a little confused. Here comes the law in modern mathematics ...
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2answers
677 views

Time-like Killing vector in FRW metric?

The spatially flat FRW metric in cartesian co-ordinates is given by: $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$ As I understand it there are Killing vectors in the $x$, $y$, $z$ directions implying ...
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5answers
800 views

Is there a fourth component to the electric field and magnetic field?

The Question If the three vector electric and magnetic fields come from the four component four-potential, then is there a fourth component to the electric and magnetic field? Related Question I ...
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373 views

Field created by varying Gravitational field

Changing Electric Field causes Magnetic filed and changing Magnetic Field causes Electric Field. Is there anything similar in relation to Gravitational Field? What sort of field is created by varying ...
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Uniqueness of Helmholtz decomposition?

Helmholtz theorem states that given a smooth vector field $\pmb{H}$, there are a scalar field $\phi$ and a vector field $\pmb{G}$ such that $$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$ ...
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Electromagnetic field and continuous and differentiable vector fields

We have notions of derivative for a continuous and differentiable vector fields. The operations like curl,divergence etc. have well defined precise notions for these fields. We know electrostatic and ...
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Decomposition of a vectorial field in free-curl and free-divergence fields

Is it always possible to do that decomposition? I'm asking it because Helmholtz theorem says a field on $\mathbb{R}^3$ that vanishes at infinity ($r\to \infty$) can be decomposed univocally into a ...
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423 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 ...
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Killing vector fields

I am facing some problems in understanding what is the importance of a Killing vector field? I will be grateful if anybody provides an answer, or, refer me to some review or books.
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5answers
933 views

What is divergence?

What is divergence? I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To ...
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1answer
54 views

How do we know if a Killing Spinor is Time-like or Null?

How to know whether a Killing spinor orbit is time-like or null? This is present in a paper like this 29/39 here. I'm not asking for a technical answer, just a logical cliche answer chit-chat answer. ...
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1answer
250 views

Amplitude and phase in vector wave field

Is it possible to make some separation of amplitudes and phase for a general vector-wave field? For example, like a paraxial approximation of a complex scalar field of the form $$\Phi(x,y,z) = ...
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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 ...
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Bondi-Metzner-Sachs (BMS) symmetry of asymptotically flat space-times

I started studying the BMS symmetry in connection with the paper: http://arxiv.org/abs/1312.2229 and there are a few strange things I noticed. First of all, from reading the original papers by Bondi, ...
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350 views

Does Hamilton Mechanics give a general phase-space conserving flux?

Hamiltonian dynamics fulfil the Liouville's theorem, which means that one can imagine the flux of a phase space volume under a Hamiltonian theory like the flux of an ideal fluid, which doesn't change ...
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Why is there no Gravitational Magnetic Field?

We think that the electric field and gravitational field operate similarly with their corresponding charges/masses. With just a difference that the electric field is sometimes attractive and sometimes ...
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320 views

What do the names “E mode” and “B mode” mean? Where do they come from?

This has been bugging me a bit since the BICEP announcement, but if there are any resources that answer my question in a simple way, they've been buried in a slew of over-technical or over-popularized ...
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1answer
106 views

Trajectories piecewise smooth?

In my studies of calculus and real analysis I have found the proofs of several theorems, commonly used in physics, such as those concerning the conservativity of fields, for example like If ...
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486 views

Must every isometry have an associated Killing vector?

I understand that the flows of Killing vector fields are isometries, and that one-parameter groups of isometries have an associated Killing vector which generates them, but are your Killing vectors ...
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What is a Killing vector field?

I recently read a post in physics.stackexchange that used the term "Killing vector". What is a Killing vector/Killing vector field?
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Difference between spinor and vector field [duplicate]

How do we distinguish spinors and vector fields? I want to know it in terms of physics with mathematical argument.
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2answers
215 views

Hilbert action's invariance under general coordinate changes

In an article, when considering invariance of the Hilbert action under a general coordinate change this formula appears for how the metric changes ...
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1answer
582 views

Finding the Basis vectors of a Killing field vector space

I have solved the Killing vector equations for a 2-sphere and got the following answer. $A,B,C$ are three integration constants as expected. $$\xi_{\theta}=A \sin{\phi}+B\cos{\phi}$$ ...
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375 views

Interpreting Vector fields as Derivations on Physics

I have a subtle doubt about the physical interpretation of the mathematical definition of vector field as a derivation. In basic physics we understand a vector quantity as a quantity that needs more ...
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713 views

Feynman's subscript notation

Consider this vector calculus identity: $$ \mathbf{A} \times \left( \nabla \times \mathbf{B} \right) = \nabla_\mathbf{B} \left( \mathbf{A \cdot B} \right) - \left( \mathbf{A} \cdot \nabla \right) ...
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How to show that every Killing vector field is a matter collineation?

Various texts make this claim, but no proof is given. Explicitly, let $L$ denote the Lie derivative. Suppose $L_X g_{ab} = 0$ for some vector field $X$, called a Killing vector field. Suppose that ...
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1answer
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Killing vector and one-form [closed]

p. 21 in this paper (http://arxiv.org/abs/0704.0247) $V$ is Killing vector, where $V^2 = −4b\bar{b}$, which means it is timelike Killing vector. The authors say: From $V^2 = −4|b|^2$ and $V = ...
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How could flux can be a vector and a scalar?

Here is the General mathematical definition of Flux on Wikipedia: The frequent symbol is $j$, and a definition for scalar flux of physical quantity $q$ is the limit: $$j=\lim\limits_{A\to ...
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1answer
633 views

Conservative vector fields

I was always told that to find whether or not a vector field is conservative, see if the curl is zero. I have now been told that just because the curl is zero does not necessarily mean it is ...
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1answer
498 views

Vectors of polarizations from vector boson field solution

Let's have the solution for vector boson Lagrangian in form of 4-vector field: $$ A_{\mu } (x) = \int \sum_{n = 1}^{3} e^{n}_{\mu}(\mathbf p) \left( a_{n}(\mathbf {p})e^{-ipx} + b_{n}^{+} (\mathbf p ...
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1answer
234 views

Equivalent system in Centre manifold theory

I was studying the centre manifold theory. It says (see Kuznetsov page 155, theorem 5.2) that the system on the left side of the picture is topologically equivalent to the one on the right. $ ...
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4answers
128 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 ...
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3answers
187 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 $\left(\nabla \times \vec A \right)$ ...
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1answer
78 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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Obtain the same Maxwell's equation after a change of coordinates

In the usual $(x,y,z)$ system of coordinates, if we expand the Maxwell's curls equations for phasors $$\nabla \times \mathbf{E} = - \mathbf{J}_m - j \omega \mu \mathbf{H}$$ $$\nabla \times \mathbf{H} ...