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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75 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 '...
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3answers
112 views

Why can the divergence of vector potential be anything?

Purcell in his book was deriving the vector potential $\bf A$ using $\text{curl}\;(\text{curl}\; \mathbf A)= \mu_0 \mathbf J\; .$ After some algebra, he came to this: $$-\frac{\partial^2 A_x}{\...
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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
344 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
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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2answers
644 views

Killing Vectors in Schwarzschild Metric

Given the Schwarzschild metric with $(-,+,+,+)$ signature, $$\text ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$$ the lack of ...
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1answer
72 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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0answers
44 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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4answers
173 views

What is the physical 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 $\nabla\times\...
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1answer
70 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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1answer
110 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
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 ...
2
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1answer
122 views

Are all maximally symmetric spacetimes constant curvature spacetimes?

A $d$ dimensional maximally symmetric spacetime is a spacetime with the maximum allowed number of Killing vectors. This number is $\frac{d(d+1)}{2}$. Constant curvature spacetimes are spacetimes ...
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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
230 views

How can I mathematically describe the parallel transport in the Roman soldiers example?

I've been trying to understand parallel transport. Many of the descriptions present a mathematical version: $\nabla_V X = 0$. And/or they present an example involving soldiers (usually Roman) ...
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(\...
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3answers
641 views

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 ...
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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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 ...
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0answers
23 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 ...
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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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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 ...
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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 ...
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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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1answer
13k views

What is Convective acceleration of flow velocity?

I know that $\frac {dv}{dt}=a$ is acceleration, but: What is convective acceleration of a flow velocity? What is difference between $(v\cdot \nabla) v$ and $v\cdot (\nabla v)$?
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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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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. ...
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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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
64 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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6answers
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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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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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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 ...
2
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0answers
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 ...
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2answers
262 views

Direction of H and B inside and outside a bar magnet

I seem to have encountered a contradiction when thinking about the directions of $\textbf{H}$ and $\textbf{B}$ inside and outside a bar magnet. Suppose that a bar magnet has a roughly constant ...
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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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3answers
98 views

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
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} = (\...
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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
521 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.