The vector-fields tag has no wiki summary.
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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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137 views
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:
$$j=\lim\limits_{A\to ...
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542 views
Why the heat flux vector at a point must be perpendicular to the temperature isothermal surface? Is it a definition or a deduction?
Before the question: I am working on numerical calculation of three dimension parabolic equation that based on Fourier's Law of which I am a little confused.
Here comes the law in modern mathematics ...
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Null vector fields given Bondi metric
I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric
$g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$
with $d\Omega$-standard metric ...
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1answer
52 views
Regarding Electromagnetic Plane and Maxwell equations
I asked this on the math.stackechange but I was told that it might be a good idea to ask here too since my problem is physics/math! Here is the question:
Hello everybody I am kind of struggling with ...
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$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf u}$
Please see the next link: http://www3.kis.uni-freiburg.de/~peter/teach/hydro/hydro02.pdf
In (2.13), he used:
$$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf ...
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Restriction on vector fields
The 2D vector field (x,-y) does not transform like a vector under rotation(Arfken Vol. 1)! Does this mean we cannot have such a vector field physically?
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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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131 views
Metric coefficients in rotating coordinates
Let $(t,x,y,z)$ be the standard coordinates on $\mathbb{R}^4$ and consider the Minkowski metric
$$ds^2 = -dt^2+dx^2+dy^2+dz^2.$$
I am trying to compute the metric coefficients under the change of ...
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240 views
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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Physical significance of Killing vector field along geodesic
Let us denote by $X^i=(1,\vec 0)$ the Killing vector field and by $u^i(s)$ a tangent vector field of a geodesic, where $s$ is some affine parameter.
What physical significance do the scalar quantity ...
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Killing Vectors of BTZ black hole and their calculation in general
I was wondering what are the Killing vectors of BTZ black hole and how to guess them easily? Will it be the same as of AdS? What then will be Killing vectors for AdS-Schwarzschild e.g.?
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Example of two linearly independent, nowhere vanishing vector fields in $\mathbb{R}^{2}$
I knew that two linearly independent and nowhere-vanishing vector fields provide a basis for the tangent space at each point in $\mathbb{R}^{2}$.
Is it necessary that these two vector fields commute? ...
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722 views
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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The equation of a ferrofluid under a magnetic field?
What is the parametric equation guiding the geometry of a ferrofluid under a magnetic field? See also this Wikipedia page.
From previous research, Maxwell's Equations and Navier-Stokes Equations were ...
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1answer
234 views
Is it possible to prove that the curl of a gradient equals zero in this way?
If $(\nabla\times\nabla\Phi)_i = \epsilon_{ijk}\partial_j\partial_k\Phi$, where Einstein summation is being used to find the $i$th component...
Using Clairaut's theorem $\partial_{i}\partial_{j}\Phi ...
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678 views
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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Longitudinal and transverse part of vector field components
I was reviewing a paper of coupling to vector field and tensor field. I have got stuck with the term $$A_k \varepsilon^{kmn}\partial_mV_n=V^{T}.(\nabla\times A^{T})-\nabla.(A^{T}\times ...
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281 views
Divergence of cross product of transverse component
If I define the vector as $V_i=V^T_i+V^L_i$ and the transverse part is defined by $$V^T_i=\Big(\delta_{ij}-\frac{\partial_i\partial_j}{\partial^2}\Big)V_j$$ then is is obvious that $\nabla.V^T=0$ as ...
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Pseudo scalar mass and Pure scalar mass
Since the only difference between pseudo scalar and a scalar term is just a change of sign under a parity inversion, is it possible that both of them be present in the same field and interact?
For ...
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466 views
Are the field lines the same as the trajectories of a particle with initial velocity zero
Is it true that the field lines of an electric field are identical to the trajectories of a charged particle with initial velocity zero? If so, how can one prove it?
The claim is from a german ...
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How to get an integral formula for the flux time derivative
$$\frac{d}{dt}\int \limits_{A} \mathbf B d \mathbf A = \int \limits_{A} \left( \frac{\partial \mathbf B}{\partial t} + \mathbf v (\nabla \cdot \mathbf B ) + [\nabla \times [\mathbf v \times \mathbf B ...
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265 views
How can a non-conservative field be a scalar multiple of a conservative field?
Okay so I was reading this from University Physics by Freeman and Young and on the topic of inductors as circuit element, they wrote that $\mathbf{E_c} + \mathbf{E_n} = 0$ which makes no sense to me
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A problem on fluid flow
I am extremely weak in visualizing physical problems in mathematical context. Please help me in solving the following problem and please give as much details as possible.
A fluid flows radially ...
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Applying $\nabla\times\mathbf{B} = \mu_0\mathbf{J}$ in the presence of magnetic shielding
2012-06-13 - Revised question in experimental format
(This is a thought experiment for which RF experts may have an immediate answer.)
I'll assume (I could be wrong) the possibility of creating a ...
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1answer
132 views
How to decompose a divergence operator
I am reading a paper, and see someone decompose a divergence operator as follows, could someone judge and see if it is correct?
$$\nabla \cdot {\bf{v}} = \left( {{\bf{n}} \cdot \nabla } \right){v_n} ...
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1answer
196 views
Existence Of Electric Field Lines
Can an Electric Field with field lines Like So Exist:
One Of my friends said it couldn't as the field lines here are not conservative ; so it cannot exist ; Is he right?
Or can it be made to ...
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Wind's Sources and Drains (see live map!!)
I was pointed out by a friend to this website that shows live map of wind in US. It sometimes show interesting places where all the wind seems to converge and vanish.
What's the origin of such "wind ...
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1answer
191 views
Equivalent system in Centre manifold theory
I was studying the centre manifold theory. It says (see Kuznetsov page 155, theorem 5.2) that the system on the left side of the picture is topologically equivalent to the one on the right.
$
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2answers
451 views
Image charges, laplace equation and uniqueness theorem
Consider a well-known problem of the electric field generated by a system composed of a point charge in proximity of a large earthed conductor. It is said that the potential due to an image charge ...
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Divergence of $\frac{\vec{r}}{r^2}$
In David J. Griffiths 'Introduction to ELECTRODYNAMICS' , as one the excercize he gave the following problem.
Sketch the vector function: $$
\vec{v} = \frac{\vec{r}}{r^2}
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Can the field generated by a magnet domain extend to infinity?
As a thought experiment let us assume that we have isolated a magnetic domain. This domain is of finite size and we know its dimensions. Assuming that we can measure an infinitesimal field, will there ...
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A wonky gravitational potential and its critical points
I have tough problem I am not sure how to solve:
For this question, we are confined to a plane. Consider a gravitational field that is proportional to $\frac{1}{r^3}$ instead of $\frac{1}{r^2}$, and ...
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2answers
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In a gas of particles, how is the displacement vector related to the number density?
Suppose I have a gas of particles that is initially uniformly distributed so that the number density is $n_0$ (number of particles per unit volume), and then I displace the particles by the vector ...
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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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Concrete example of divergence of a vector field
I'm studying vector analysis and it is hard for me to understand what divergence of a vector field really is. I know that $divF=\nabla\cdot F$ but I don't understand what kind of quantity it gives and ...
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370 views
Simulation of physics of chains/ropes in force fields resources?
I'm thinking about a project to tackle, and I'd like to make a simulation that allows the user to define a rope or chain of length L, pin it at arbitrary points r1, r2.... etc. and draw the resulting ...
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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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653 views
Electric field at a point from a square surface
I'm trying to determine the electric field at a point P (located on the +Z axis) due to a square of side length [L] and centered at the x-y plane origin. The square has a constant surface density [s]. ...
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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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How can we describe the polarization (of light) coming from an arbitrary angle?
In an optics lab, where all optical beams pretty much reside in a plane, it is fairly simple to describe (linear) polarizations as vertical or horizontal (or s and p).
When we start talking about ...
