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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Intuitive analysis of gradient, divergence, curl

I have read the most basic and important parts of vector calculus are gradient, divergence and curl. These three things are too important to analyse a vector field and I have gone through the physical ...
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2answers
74 views

What is the velocity in the Navier-Stokes equation?

I have been looking at the Navier-Stokes equation, and can't seem to find anywhere a clear description of what velocity it represents. From what I have read it could be any of the following: The '...
2
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1answer
64 views

Why does the divergence of the Ponyting vector have energy flux density?

The poynting vector is defined as $\vec{S}=\mu_{0}^{-1}\vec{E}\times \vec{B}$ Taking the divergence of the poynting vector, one arrives at $\vec{\nabla} \cdot \vec{S}=-\frac{\partial u}{\...
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1answer
62 views

Covariant derivative [closed]

Hi, Could you explain to me why the subtraction of vector at some point and parallel transported vector is covariant derivative vector. How is it possible
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1answer
43 views

Symmetry group of FLRW metric

$$ g = dt^2 - a^2(t) (dx^2+dy^2+dz^2) = dt^2-a^2(t)(dr^2+r^2d\Omega^2)$$ So this is my metric. What is the symmetry group of it? I think that my Killing vectors are 3 translation vectors: $$K_i = \...
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43 views

Help with understanding what is Curl [duplicate]

Yeah, I watched several YT videos and read few articles and my head is spinning. I am trying to get the right understanding of what Curl is. There is this excellent video: Divergence and Curl Now ...
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1answer
53 views

Divergence of $\frac{e^{-br}\hat{r}}{r^2}$ in electrostatics [closed]

My question is how to calculate the divergence of a vector field (Electric field) given as: $\vec{E}(r)=\frac{e^{-br}\hat{r}}{r^2}$. Or more generally how to approach finding the divergence of a ...
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2answers
33 views

How do I draw the force field lines of an isotropic oscillator?

In general, how do I draw the force field lines (in the sense of Faraday, i.e. continuous curves whose tangents give the directions and the density of lines give the intensity of the field) of a ...
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0answers
41 views

Can someone explain me magnetic potential? [duplicate]

I don't understand why we can write the magnetic field as a potential vector A, and what exactly this potential is. Is it just a mathematical thing? And how can it help me calculate the magnetic ...
4
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1answer
35 views

Lie Derivative of Kahler 2-form

Suppose there is a Killing vector $k$ on a Kahler manifold $M$. By definition, $k$ generates isometries of the metric. That is, $L_kg=0$, where $L$ is the Lie derivative. At the same time, there is a ...
2
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2answers
81 views

Lie derivative for a covariant derivative of vector

I would like to calculate the $\mathcal{L}_\xi(\nabla_a K^b)$ for the case where $\mathcal{L}_\xi(K^b)=0$ The only Idea that I have is that $$\mathcal{L}_\xi(\nabla_a K^b)=\mathcal{L}_\xi(\...
3
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1answer
83 views

Metric that is Minkowski plus sum of null vectors

In GR exercises I've often seen metrics of the form $g_{ab} = \eta_{ab} + k_ak_b$ where $k_a$ is null with respect to $g$ (or equivalently $\eta$). I'm happy doing calculations with such metrics, but ...
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1answer
48 views

Wald's Equation 3.3.6

I have an issue with Eq. 3.3.6 of Wald's General Relativity. There he would like to prove that for Gaussian normal coordinates, the geodesic tangent field remains orthogonal to all coordinate basis ...
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0answers
27 views

Quotient Rule in Vector Calculus

Wikipedia gives the quotient rule for (1) the gradient of two scalar fields "$f$" and "$g$" and (2) the divergence of a vector/tensor field and a scalar field "$\boldsymbol{A}$" and "$g$" as $$\nabla ...
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1answer
55 views

Integrating by parts [closed]

I am having little trouble with my professor's note. $$F=-\int{(dr)}{(\vec{\nabla} \cdot \vec{P}) \vec{E} }=\int{(dr)}{(\vec{P} \cdot \vec{\nabla} ) \vec{E} }$$ where F is force, P is polarization, ...
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28 views

Longitudinal Polarization and Spin-0 for Massive Vector Fields

I was wondering if anybody would be willing to explain how a plane wave solution of the form $\vec{B^\mu}=\epsilon^\mu{e^{k_0ct+\vec{k}.\vec{x}}}$ for a massive vector field's equations, say for ...
3
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0answers
22 views

Equivariance Relation - Superconformal Hypermultiplets

I'm concerned with equation 2.24 of http://arxiv.org/abs/1601.00482 The superconformal hypermultiplets in this paper have a conic hyperkahler target manifold and the authors want to gauge some ...
2
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1answer
50 views

Energy conservation around a black hole

In the Schwarzschild black hole, the Killing vector "time translation" $k^a$, so that the following quantity is conserved along a geodesic: $$E = -g_{ab}k^au^b = (1 - \frac{2GM}{r})\frac{dt}{d\tau}.$$...
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3answers
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How is the curl of the electric field possible?

Taking the curl of the electric field must be possible, because Faraday's law involves it: $$\nabla \times \mathbf{E} = - \partial \mathbf{B} / \partial t$$ But I've just looked on Wikipedia, where it ...
0
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2answers
64 views

How to convert electric field from spherical coordinates to cartesian?

I have 3 components, $r$, $\theta$ and $\phi$, for an electric field in spherical coordinates (and the $\phi$ component happens to be zero), let's say I just want to convert the $r$ component into ...
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3answers
42 views

Flux - Scalar Multiplication in Integral?

No textbook and website seems to answer this so here is my question: When we have a scalar flux: I understand that you take the scalar product of the vectors. And I understand the need for using an ...
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1answer
39 views

Coulomb gauge and vector identites

consider a coulomb gauge and the following volume integration: $$\int d^3x{\dot{A}.\nabla A}$$ How can we show that this is zero in coulomb gauge? (A is a vector potential) this is my attempt at ...
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0answers
21 views

Which formulas would tell me the gradient of an electromagnetic field at an arbitrary distance from a pole? [duplicate]

I'm a newbie to physics and was wondering where I can read about electromagnetic gradients. From what I understand (and my intuition) electromagnetic fields create force gradients around its poles. ...
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65 views

Relation between second covariant derivative of Killing vector and Riemann tensor [closed]

I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. $\xi$ is a Killing vector. I have proved that $$D_\mu D_\...
3
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1answer
62 views

What is the importance of vector potential not being unique?

For a magnetic field we can have different solutions of its vector potential. What is the physical aspect of this fact? I mean, why the nature allows us not to have an unique vector potential of a ...
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2answers
59 views

Calculate divergence via partial derivative [closed]

I need to calculate the divergence and curl for a vectorfield. I've done that before so that's no problem :) Or I've done it using partial derivative, maybe there are multiple ways to solve for ...
3
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2answers
220 views

Aharonov Bohm Effect Interaction Energy Interpretation: $\vec E_m = -∇Φ - D\vec A/Dt$?

The Wang paper "An experimental proposal to test the physical effect of the vector potential" proposes an experiment to decide between two interpretations of the Aharonov-Bohm effect: “the ...
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37 views

Great Pacific Garbage Patch Equilibrium Points

I originally posted this question on earth science stack, but the question wasn't getting many views. I was watching the science channel yesterday and the program mentioned the Great Pacific Garbage ...
0
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1answer
55 views

Solenoidal forces

As far as I know a solenoidal vector field is such one that $$\vec\nabla\cdot \vec F=0.$$ However I saw a book on mechanics defining a solenoidal force as one for which the infinitesimal work ...
1
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2answers
32 views

How to find the graph of electric field when the potential is given [closed]

Suppose the electric potential due to an electric field is given as $x^2-y^2$, then what will be the graph of the electric field? My attempt: Differentiating the potential partially with $x$ and ...
2
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Meaning of the Vector Wave Equation

So I thought I would try my luck here on physics stack exchange about an intuitive meaning of the Vector Wave Equation. I know there are a lot of resources out there that explain this equation, but ...
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1answer
44 views

Assumptions in physics for Helmholtz decomposition

A version of the Helmholtz theorem says that, under opportune assumptions on the vector field $\boldsymbol{F}:\mathbb{R}^3\to\mathbb{R}^3$ and on $V\subset\mathbb{R}^3$ the following identity holds: $$...
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1answer
57 views

Killing equation manipulation

Why does the killing equation $$K_{\mu;\nu}+ K_{\nu;\mu} = 0$$ equal $$K_{\mu,\nu}+ K_{\nu,\mu} -2\Gamma^{\rho}_{\mu\nu}K_{\rho} = 0 $$ when in general a covariant derivative $V_{\beta;\alpha} = (\...
0
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1answer
55 views

If the integral is zero when is the integrand zero? [closed]

Using Stoke's theorem we prove that the curl of the Electric field vanishes. This would be possible only if the integrand is zero when the integral is zero.
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1answer
517 views

What is the difference between gravitational force and gravitational field?

I see two different formulas describing gravity: $$F=\frac{GMm}{r^{2}}$$ $$g=\frac{F}{m}$$ But I don't understand the difference between gravity as a force and its field as a vector.
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34 views

Divergence theorem for cylindrical coordinates [closed]

I have a Vector field in a cylinder where x^2+y^2=4 and goes from z=0 to z=3 and a vector field A=(4x)i-(2y^2)j+(z^2)k and I'm trying to verify the divergence theorem for the vector field i set set ...
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34 views

Advection Operator shift in scalar product

Can someone help me with advection operator shifts? I can't figure out the rule for the shift inside of a scalar product. The terms $(u,(v\cdot \nabla)\delta v)_\Omega$ and $(u,(\delta v\cdot \nabla)...
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2answers
71 views

What does it mean to be unique in terms of vector potentials?

I was in an electromagnetism lecture, where we were looking at the magnetostatic Maxwell’s equations: $$\begin{align} \nabla\cdot\mathbf{B} &= 0 \\ \nabla\times\mathbf{B} &= \mu_0\mathbf{J} \...
2
votes
1answer
85 views

Components of dual vectors

(This is a close retelling of Wald, problem 2.4b. Not for homework; just curiosity and an increasingly alarming suspicion that I've never actually understood anything.) Let $Y_1 ... Y_n$ be a ...
3
votes
1answer
101 views

Helmholtz decomposition allows incompressible flow with an irrotational component?

A vector field can be written in terms of irrotational and a divergence-free components. Using a 2D velocity field as an example, $ \vec v = -\nabla \phi + \nabla \times \vec \Psi$ Where $\vec \Psi$ ...
3
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1answer
62 views

contravariant and covariant vectors and their orthogonality in Euclidean space

I am reading this paper Sigma Coordinate - Contravariance and covariance and I understand how covariant and contravariant vectors are defined mathematically Covariance and Contravariance and I had ...
0
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1answer
51 views

Metric components transformation under change of coordinates

I have been studying Lie derivatives and some applications. While searching the web I found a refence with the following statement: For a general Riemannian manifold $M$, take a tangent vector field $...
2
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0answers
82 views

Do divergence and curl of Lorentz force have some physical meaning?

Time ago I started thinking about this: if we take the well known Lorentz Force expression, namely $$\mathbf{F} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)$$ and we operate $\nabla\cdot \...
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Field line Direction and exerted force

Magnetic field lines of a magnetic field have different directions. What information about the force exerted on a charge will give us the direction of field lines?
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1answer
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Dot product and divergence [closed]

Divergence is represented by dot product. How is the divergence related to dot product? And curl is represented by cross product. How is the curl related to cross product?
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34 views

Basic Vector field question about notation [closed]

I am taking my first class in electrodynamics and the problem I am working on has a notation I have never seen before Consider a vector field of the form $V= f(x)y + g(y)x$ Is this essentially the ...
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160 views

When is the event horizon a Killing horizon?

I know the definition of both (event horizon is closure of causal past of future null infinity whilst Killing horizon is a null surface where some Killing vector becomes null e.g. the surface where it ...
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What does the density of points (tail point of the vectors) represent in the geometrical representation of a vector field? [closed]

While trying to understand the divergence of a vector through the geometrical representation of the vector field, I found that pictures can be misleading. Even a vectors field which looks to be ...