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.

learn more… | top users | synonyms

0
votes
0answers
25 views

Vector integration in n dimensions

In n dimensions I want to do an integral of the flux through an n-1 D surface. The usual vector calculus integration theorems say I can integrate around the perimeter of the surface. OK, but that ...
1
vote
2answers
54 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
77 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
64 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
votes
1answer
47 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
votes
1answer
34 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
votes
0answers
50 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 ...
1
vote
0answers
12 views

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?
-2
votes
1answer
60 views

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?
1
vote
0answers
32 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 ...
0
votes
0answers
48 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 ...
1
vote
0answers
28 views

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 ...
1
vote
0answers
29 views

Show that a vector field is a symmetry for a Lagrangian [closed]

Let Lagrange function be $$ L=\frac{1}{2}m(\dot{x_1}^2+\dot{x_2}^2+\dot{x_3}^2)-U((x_1^2+x_2^2,x_3)). $$ Show, that vector field $\vec{Y}(\vec{x})=(-x_2,x_1,0)$ comply $$ ...
1
vote
0answers
50 views

Divergence theorem and discontinuous vector fields in electrostatics

Wikipedia defines Gauss Divergence Theorem for a continuously differentiable vector field; but in many idealized physical situations, we use it for non-differentiable fields. For example, the electric ...
2
votes
1answer
75 views

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 = ...
1
vote
1answer
56 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. ...
0
votes
1answer
64 views

Variation of a tensor

Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity. Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means ...
1
vote
2answers
50 views

Killing field in Minkowski space-time

If we look at the killing equation for a vector field $X$ in $\mathbb{R}^{(p,q)}$ (or on an open subset thereof) in coordinates with constant diagonal pseudo-metric we get: ...
2
votes
1answer
95 views

Hyperbolic flow / vector field - irrotational and divergence-free?

My text book on meteorology claims that a hyperbolic flow pattern is both divergence-free and irrotational: (d) Hyperbolic flow that exhibits both diffluence and stretching, but is ...
1
vote
3answers
119 views

Is there any physical interpretation for $\nabla\cdot(\nabla \times F)=0$?

It is well known that the divergence of the curl is always 0. Mathematically I understand why this happens ($d^2=0$ where $d$ is the exterior derivative) but today I was wondering what is the physical ...
1
vote
1answer
59 views

How can I prove that for a Killing vector $\nabla^a \nabla_a \xi^\mu = -R^b_a \xi^a$? [closed]

I'm taking a course on General Relativity and I'm trying to prove that for a Killing vector field $\xi^\mu$ the following equation holds: $$\nabla^a \nabla_a \xi^\mu = -R^\mu_a \xi^a$$ Where $R_ab$ ...
24
votes
6answers
3k views

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 ...
1
vote
1answer
43 views

Why does a system whose equations of movement are $\lambda^2U^{\alpha} + \partial_{\mu}F^{\mu \alpha} = 0$ have three degrees of freedom?

I'm trying to understand the solution of a problem where I have to study a field ($U^\mu$) which Lagrangian is: $$\mathscr{L} = - \frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{2} \lambda^2 U_{\mu} ...
2
votes
1answer
41 views

Local Coordinate Expressions for Lie Derivatives

I'm currently working through the math chapters of Norbert Straumann's book on General Relativity. I have trouble understanding the coordinate expression of the Lie derivative of a basis vector. The ...
0
votes
0answers
41 views

Electric field of a dipole in cartesian

I have the function that describes the electric field of a dipole in cartesian coordinates: $$E_z= \frac{p}{4\pi\epsilon_0} \cdot \left(\frac{3z^2}{r^5}- \frac{1}{r^3}\right) $$ and ...
1
vote
1answer
75 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 ...
1
vote
2answers
65 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 ...
5
votes
1answer
139 views

Finding diffeomorphism given vector fields [closed]

Given a vector field how do you find the associated diffeomorphisms? Say I am given a vector field in Minkowski space $$\xi = x \frac{\partial}{\partial t} + t \frac{\partial}{\partial x}.$$ How do ...
2
votes
0answers
61 views

Integral curves in null hypersurfaces [closed]

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p S$ ...
0
votes
2answers
54 views

What is the criterion for a vector field (dim > 1) to be conservative?

We know that if the curl vanishes: $$ \nabla \times \vec{v} = 0$$ then the field is irrotational and is conservative, but what about in higher dimensions than 1? The cross product is not defined ...
0
votes
1answer
55 views

Wightman function for massless vector fields in Coulomb gauge

I've been looking for quite some time an expression for the Wightman functions for a massless vector field in the Coulomb gauge $\nabla\cdot\mathbf{A}=0$ (I think it is equivalent to the Feynman gauge ...
1
vote
2answers
109 views

Conservation of energy and Killing-field

In general relativity we have no general conservation of energy and momentum. But if there exists a Killing-field we can show that this leads to a symmetry in spacetime and so to a conserved quantity. ...
1
vote
2answers
58 views

Divergence-free vector field on a non-simply connected domain

We know that divergence-free vector fields are themselves curls of vector fields on simply connected domains. I want to construct a counterexample in the case the domain is not simply connected. So ...
0
votes
1answer
69 views

Finding the electric and magnetic fields from the vector potential $\vec A=\vec{E_0} e^{i(\vec k\cdot\vec r-\omega t)}$

I am trying to find the electric and magnetic fields from the vector potential $$\vec A=\vec{E_0} e^{i(\vec k\cdot\vec r-\omega t)},$$ I know $$\vec B=\vec \nabla\times \vec A$$ and $$\vec ...
0
votes
1answer
81 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 ...
-1
votes
1answer
45 views

A leaf floating on the surface of water (in $x$-$y$ plane) curl along positive $z$ axis [closed]

A leaf floating on the surface of water (in x-y plane) show that for a very small circular leaf ($\nabla \times \overrightarrow v $) is equal to twice the angular velocity of rotation of the leaf, ...
2
votes
1answer
37 views

Details on the magnetic field of a linearly polarized electric wave

Suppose we are in vacuum and we have an electric field $\vec{E}$ which we assume is simple harmonic wave that propagates through $z$ and is linearly polarized in the $x$-$y$ plane along $x$ i.e. ...
1
vote
0answers
63 views

Constructing Killing tensors from Killing vectors

Background: After reading about Carter constant and symmetries in GR, I became interested in Killing tensors. I tried reading this paper by Alan Barnes, Brian Edgar and Raffaele Rani, discussing ...
1
vote
1answer
212 views

Derivation of one-form/vector equation in Carroll confusion

I don't understand the derivation of Equation 2.14$$\mathrm{d}f\left(\frac{d}{d\lambda}\right)=\frac{df}{d\lambda} \tag{2.14}$$ in Carroll's Lecture Notes on General Relativity ...
2
votes
2answers
249 views

Killing Vectors in Schwarzschild Metric

Given the Schwarzschild metric with $(-,+,+,+)$ signature, $$ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$$ the lack of dependence of ...
0
votes
1answer
69 views

Killing vectors in General Relativity?

I'm looking to derive the surface area of the event horizon of a Schwarzschild black hole. I was just wondering if it were possible for someone to explain to me this: $$ ...
1
vote
0answers
57 views

How do I derive Feynman rules for vectors involving derivatives?

Suppose I have a term in the Lagrangian: $$\cal{L} \equiv (\partial_\mu B^+_\nu) B^{-\mu} A^\nu $$, where $B^\pm$ are charged massive vector particles and $A$ is photon. Now, how can we derive the ...
1
vote
1answer
137 views

Gradient, divergence and curl with covariant derivatives

I am trying to do exercise 3.2 of Sean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives. ...
-1
votes
1answer
76 views

To prove, $\nabla.(\nabla\phi \times \nabla\psi)$ =0 [closed]

Please Help me solving the problem using levi-cevita symbol : Prove That, $\nabla.(\nabla\phi \times \nabla\psi)$ =0 where $\phi =\phi(x,y,z)$ & $\psi=\psi(x,y,z)$
0
votes
2answers
37 views

Faraday's law and potential along a closed curve

$$\int(\nabla \times \vec{E}) \cdot d\vec{S} = \oint \vec{E}\cdot d\vec{l} = -\frac{\mathrm{d} \phi_b}{\mathrm{d} t}$$ The second expression is the potential difference along a closed curve. Isn't ...
2
votes
3answers
112 views

Why exactly are fields so crucial to modern physics?

I have read that fields were the mathematical tool that allowed people like Laplace to develop a working model of the Solar System where Newton could not. But my understanding is that fields are ...
0
votes
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 ...
0
votes
3answers
85 views

Work done by a constant vector field is 0?

We know that $$\oint \boldsymbol{F}\cdot d\boldsymbol{r}= \iint (\nabla \times \boldsymbol{F})\cdot d\boldsymbol{s}.$$ Now if $\boldsymbol{F}$ is a constant vector, then $\nabla \times ...
0
votes
1answer
66 views

Clarification about some steps in the derivation of the Lie derivative (mechanics)

First of all, this question may seem to be undefined, because I'm not sure how to connect this (to me) newly introduced concept with the abstract notion of the Lie derivative. I'm not even sure if I ...
1
vote
1answer
36 views

Norm of Dilatation operator [closed]

The dilatation operator is given by $$D=x^{a}\frac{\partial}{\partial x^{a}}+z\frac{\partial}{\partial z}$$ How the norm can be $$D^{2}=\frac{L^{2}}{z^{2}}(\eta_{\mu\nu}x^{\mu}x^{\nu}+z^{2})$$ where ...