Questions tagged [vector-fields]
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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Lie derivative: moving boat on a flowing river
Lie derivatives signifies how much a vector (Tensor) changes if flown in the direction of some other vector. I am thinking the typical moving boat on a flowing river problem where the river is flowing ...
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Directional derivative $(\mathbf{A}\cdot\nabla)\mathbf{B}$ of the vector field $\mathbf{B}$
While reading Introduction to Electrodynamics by David J. Griffiths, I have encountered some issue with the notation of the directional derivative of the vector field and I was wondering if there are ...
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What are streamlines and pathlines
I asked a question earlier about having a vector field and a starting point, and then making a parametric that starts at the starting point and the derivative at any point in the parametric equals the ...
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How to make a parametric that matches a vector field?
So I have a vector field defined as $(X(x,y),Y(x,y))$ and I’m trying to make a parametric $(t,t)$ who’s derivative at a point is equal to the vector field at that point.
for example the vector field $(...
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Inner product defined on a Vector Field on Complex Numbers [migrated]
I have a homework problem that seems wrong. Previous to the problem we showed that the inner product of a vector space $V$ on the field $\mathbb{C}$ defined as
$$ \langle v,w \rangle = a^i(b^j)^*\...
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Is $dJ(V,V)=0$? where $J$ is a 1-form?
So is this always 0?( Where $dJ$ is the exterior derivative and $V$ a vectorial field)
\begin{align}
dJ(V,V)=\partial_jJ_i(dx^j\wedge dx^i)(V,V)=\\
\partial_j J_i (v^kdx^j(\partial_k)v^ldx^i(\...
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Where to apply $\nabla$ operator when taking curl of a cross product?
In my EM class we went over $$\nabla\times \frac{\vec{d}\times \vec{r}}{r^3}$$ which apparently can be breaken down to $$r(d\cdot \nabla)\frac{1}{r^3}-d(r\cdot\nabla)\frac{1}{r^3}+\frac{\nabla\times(d\...
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Vector potential of position field
Consider the position vector field $\vec{r}=(x,y,z)^T$. What would be a vector potential $\vec{A}$ for this field? I was thinking of something like $\vec{A}=(yz,zx,-xy)^T$, which gives
$$\nabla\times ...
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Stokes stream function derivation
I want to know a concrete derivation of 3D Stokes stream function.
The statement is, for example in 3D spherical coordinates (with symmetry in rotation about the $z$-axis), if
$$\nabla \cdot u=0\tag{...
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Taylor expanding onto a deformed sphere [migrated]
Suppose I have a deformed unit sphere whose shape is given by $ r(r, \theta, \phi) = 1 + \epsilon x_iA_{ij}x_j$ for $\epsilon << 1$ and $A_{ij}$ is a symmetric traceless second order tensor.
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Vector Magnetic Potential and pointwise current density source
I am currently studying antennas and I am trying to understand how to solve the following vector equation $$\nabla^2A+k^2A=-\mu J$$ in the case when there is a point current source at the origin.
The ...
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Doubt about the derivation of Liénard-Wiechert Potentials?
When deriving the Lienard-Wiechert Potentials, there is one step that you need to perform:
$$
\nabla_{\mathbf r}|\mathbf r - \mathbf r_s(t_r)|
$$
Where $t_r$ is:
$$
t_r = t - \frac{|\mathbf r - \...
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How can a vector field in $E^3$ be represented by a linear combination of only 2 basis vectors?
In Chapter I.7 of "Einstein Gravity in a Nutshell", Zee introduces the concept of covariant derivatives. I am confused by the first line in this section (see below) as it appears that we can ...
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What is the physical significance of the cross product of curl of a vector field $v$ with another vector field $w$?
I think if curl of a vector field v corresponds to an applied rotation, it's cross product with a velocity vector field w (say) should give something analogous to the resulting torque. Am I close?
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Physical significance of $\vec{w}$ $\times$ $($curl $\vec{v})$
I think if curl of a vector field $\vec{v}$ corresponds to an applied rotation, it's cross product with a velocity vector field $\vec{w}$ (say) should give something analogous to the resulting torque. ...
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Extreme confusion with the Lorentz transformation law for vector fields
Let $\Lambda$ be a Lorentz transformation represented as $4 \times 4$ matrix.
Then, following What does it mean to transform as a scalar or vector? , it seems that a vector field $f : \mathbb{R}^4 \...
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What does the notation $(k \cdot \nabla ) v$ mean? [duplicate]
I am reading a paper and it uses a notation I am not too familiar about. Although I saw it used elsewhere, I don't remember the meaning of it and I don't want to misinterpret it and realize after ...
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Magnetic vector potential in 1+1 spacetime dimensions
In the theory of electromagnetism in 1+1 spacetime dimensions (one temporal and one spatial coordinate), one can define the 2-potential vector (analogous to the 4-potential vector in 3+1 spacetime ...
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What is Dirac's reasoning when saying parallel displacement creates vector field with vanishing covariant derivative?
Section 12 of Dirac's book "General Theory of Relativity" is called "The condition for flat space", and he is proving that a space is flat if and only if the curvature tensor $R_{\...
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What isn't the metric invariant under translation with killing vectors?
I am learning about Killing Vectors in GR class, and I'm testing my knowledge of them as a start with the Minkowski metric.
I used the simple 2d Minkowski metric:
$$ds^2 = -dt^2 + dx^2$$
and got 3 ...
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Does the divergence theorem imply an underlying symmetry?
The divergence theorem connects the flux (through surface) and divergence (in a volume) for any vector field.
This theorem expresses continuity. It isn't clear (to me) whether there is a conserved ...
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Application of Fermi-Walker derivative to specific problem
I am now reading about the tetrad formalism in GR and I am starting (how not) with the Wikipedia Article:
Frame fields in general relativity.
In this article, as an example, they show how tetrads can ...
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Is there an equivalent to the Klein-Gordon and Dirac Equations for Vector and other fields?
The Klein-Gordon equation describes a scalar field, and the Dirac Equation describes a spinor field. Is there an equivalent equation for a vector field? As well as spin 3/2 and spin 2 tensor fields? ...
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What may have caused these bizarre frost patterns on my car windows?
I went to my car a couple days ago in the morning, and I was amazed at the patterns I found on all the side windows, shown in the attached picture. No amount of googling revealed anything similar, so ...
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How can you understand curl of electric field illustratively?
Considering the curl of the electric field of an electric dipole, this will be zero in absence of magnetic effects which is clear to me. I watched a video by 3Blue1Brown (some time ago) who explained ...
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Divergence of $H$ Maxwell equation
In the below screenshot from this paper (link below), why is the 2nd Maxwell equation ($\nabla \cdot H = 0$) not automatically satisfied when the 4th Maxwell equation is satisfied? I don't understand ...
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Explicitly Computing a Lie Transported Vector or Covector
I'm having difficulty understanding how to physically compute how a covector changes under Lie transport.
Suppose on $\mathbb{R}^2$ I have a vector field $V=x\frac{\partial}{\partial x}+y\frac{\...
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How to derive the dimension of conformal Killing vector fields on the Riemann sphere? Is it metric independent?
Context
In all string textbooks and lecture notes, they derive the CKV on the sphere by considering the flat plane first, i.e. $(\mathbb{C},\delta_{\mu\nu})$. Then, write it in complex variables
$$z = ...
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What is the correct term for $\nabla\phi$? Co-vector or 1-form or both?
In the olden days, $\nabla\phi$ was used to be called a covariant vector (Weinberg used this language in his book Gravitation & Cosmology). But this terminology is considered bad for several ...
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Spherical coordinate of a vector when divergence of the vector is zero
$\nabla \cdot \mathbf{\delta u_{perp}} = 0$ where $\mathbf{\delta u_{perp}}$ is a function of both x and y coordinates and perpendicular to z axis. Moreover, $\delta u_{perp}$ along z axis is $0$.
I ...
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Understanding the derivation of Killing horizon surface gravity
In the book "A Relativist's Toolkit" by Eric Poisson, he explains surface gravity in section 5.2.4
The equation 5.40 says
$$ (-t^\mu t_\mu)_{;\alpha} = 2 \kappa t_\alpha \tag{5.40}$$
where $...
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Difference between geodesic vector field and pre-geodesic vector field
A curve $L$ is a geodesic iff the tangent vector field $v$ associated with any parametrization of $L$ obeys
$$\nabla_{v}v=\kappa v$$
where $κ$ is a scalar field along $L $.
A vector field $v$ obeying ...
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Deriving divergence in cylindrical coordinates, using covariant derivatives
Covariant derivatives are normally used to write equations covariantly in curved spaces. But in an exercise, I need to use covariant derivatives to derive Gauss' law: $\nabla \cdot \vec{E} = 4\pi\rho$ ...
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Gradient Operator in Vector Spherical Harmonic Basis
The vector spherical harmonic basis (vector generalization to the scalar valued spherical harmonics) is a convenient spectral basis for problems involving vector fields with spherical symmetry (link ...
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Stationary, static and strictly stationary/static?
Consider spacetimes which are asymptotically flat at null infinity.
How to explicitly show that there exists a hyper-surface orthogonal killing vector field $k^{a}$ that is time-like everywhere in ...
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What do we mean by "generator" in general relativity?
Killing vector field is often referred as "generator" of infinitesimal isometry. However from what I understood this would mean that the exponential map of these generators would be the ...
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The angle between the del operator and electric/magnetic field
If the divergence of the electric field is equal to $\frac{p}{ε}$, it must equal to |$\nabla$||E|cos($\theta$) where $\theta$ is the angle between the electric field and the del operator. Same thing ...
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How to find the double covariant derivative of a general vector?
I have been reading through Carroll's GR textbook and there is a line in the derivation of the Riemann tensor that I do not understand.
$$\nabla_\mu \nabla_\nu V^\rho=\partial_\mu(\nabla_\nu V^\rho) - ...
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Gradient of Schwarzschild horizon
I am studying Schwarzschild horizon and badly stuck with some Math stuff
Consider the scalar field defined on Manifold by
$$
u(t,r,\theta,\phi)=(1-\frac{r}{2m})e^{\frac{r-t}{4m}}
$$
$$
du=\frac{1}{4m} ...
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Resolving an apparent contradiction between Schwarzschild and ingoing Eddington-Finkelstein coordinates
I believe this is basic differential geometry issue. This may be obvious to many, but I was quite confused about it, and it took me quite a while to find the resolution. I'm going to ask and answer ...
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Frobenius identity
I am studying null geodesic generators and came across the following paragraph under the heading Frobenius identity in which ℓ and $\nabla u$ are normal vector fields to null hypersurface:
$ℓ = −e^ρ ∇...
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In an electrostatic field with zero divergence everywhere, where is the charge located?
Purcell in section 2.17 discusses the electric field $E = <Ky, Kx, 0>$, which has field lines in the shape of a hyperbola, $\phi = -Kxy$, zero curl, and zero divergence. Purcell states that ...
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Killing vectors on the unit sphere
I am asked to find the Killing vector fields on $S^2$ where the line element is given by $ds^2=d\theta\otimes d\theta +\sin^2\theta d\phi\otimes d\phi$.
I know how to solve this problem by considering ...
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Intuition about inverse square fields from a mathematical perspective
Context: I'm a topologist, teaching vector calc this term. I was writing some exercises on surface integrals but was intrigued by the following:
The vector field $F(x,y,z)=\dfrac{1}{(x^2+y^2+z^2)^{3/2}...
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Surface Integration [closed]
I've been studying surface integration by myself but I always stuck at the last step.
Consider the above question:
This is my approach:
Calculation of the curl of the given field.
Calculation of unit ...
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Calculation of Killing Spinors on the 2-Sphere
A Killing spinor on a Riemannian spin manifold $\mathcal{M}$ is a spinor field
$\psi$ which satisfies
\begin{equation}
\nabla_{X}\psi = \lambda X\cdot\psi
\end{equation}
for all tangent vectors $...
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What are Killing vectors of (2+1)-dimensional spacetime exibiting spatial spherical symmetry
I quote:
... a $(3+1)$-dimensional spacetime exhibiting spatial spherical symmetry, namely, a manifold with the three-dimensional special orthogonal group $SO(3)$ (representing rotations in three-...
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How do you integrate a physics integral? [closed]
I've only taken Physics I (w/ calc) in uni, so physics is pretty new to me, and I'm only through Calculus 2.
However, I keep seeing integrals like these everywhere in physics, and want to know how to ...
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Do all smooth, differentiable manifolds have an acceleration field? [closed]
I made the claim on a manuscript that all manifolds possess an acceleration field. The referee rejected the idea saying "The nature of this acceleration field has not been seriously discussed and ...
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Static spherically symmetric spacetimes
I would like to better understand a hypothesis that Wald uses to derive the general local formula of a static spherically symmetric spacetime.
A spacetime is said to be spherically symmetric if its ...