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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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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Why does iron fillings form crevices when placed in a magnetic field? [duplicate]
If we get magnetic field lines on a piece of cardboard, shouldn't it form a gradient of metal pieces across the board (thick layer of metal pieces near the magnet and thin layer of metal away from ...
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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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Killing Vectors and Horizons
Let $\chi^{a}$ be a Killing vector, we want to prove that the surface gravity of a stationary black hole is constant over the horizon. The proof can be found in In page 333 of Robert M. Wald's General ...
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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 ...
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Finding the vector potential
$$\nabla\times\mathbf{B}=\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla^2\mathbf{A}=\mu_0\mathbf{J}\tag{5.62}$$
Whenever I try to work this out and ...
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How can the vortex be an elemental potential flow if there is a point of curl?
Aero is not my speciality at all so apologies if missed anything. But when looking at potential flows, i thought the whole point is for there to be no rotation at any point and its that reason the ...
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Coupling of scattered TE mode to TM mode using vector Helmholtz equation
I solve a scattered field computation problem using the frequency domain Maxwell's/Helmholtz PDEs. Particularly, I'm studying the behavior of light, which is essentially a plane wave propagating in a ...
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Looking for the geometric meaning of the curl of Killing vector fields
From Killing equation
$$\nabla_\nu \xi_\mu + \nabla_\mu \xi_\nu = 0$$
it can be shown that $\nabla_\nu \xi_\mu$ is antisymmetric.
From it we can construct an antisymmetric tensor $\mathcal{A}_{\mu\nu}$...
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Recommendation books on graduate level vector analysis and field analysis with application of electrodynamics?
I am currently a physics and mathematics double major graduate student. Looking for a text book and problem sets on the topic of vector field analysis, would be best if the book is advanced and ...
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Lie derivative of a one-form
I am going through Nakahara's textbook on geometry and topology in physics. Intuitively, I understand the definition of a lie derivative of a vector field
$$\mathcal{L}_xY = \lim_{\epsilon_\to 0}\frac{...
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Phase difference in superconducting ring and gradient theorem
In Feymann's seminar on superconductivity there was this equation (21.28) $\oint_C\nabla \theta \cdot ds=\frac q \hbar\Phi$. Closed line integral $\oint_C\nabla \theta \cdot ds$ was claimed to be non-...
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The BRST variation of the gauge fixing condition
Following Polchinski volume I, p 126 onwards, The BRST variation of fields $\phi^{i}$ is given by
$$\delta_{B} \phi_{i} = - i \epsilon c^{\alpha} {\delta}_{\alpha}\phi_{i} \; .\tag{4.2.6a}$$
My ...
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Intergrating the quantum field on a surface with respect to the full quantum field
The quantum field $\phi(x^\mu)$ or $\phi(x^0,x^1, \dots, x^{d-1})$ exists only for $x^{d-1}\geq 0$. We are in Euclidean signature $g_{\mu\nu}=\delta_{\mu\nu}$. Let $\phi(x^0,x^1, \dots, x^{d-2}, 0)=\...
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Normal derivative of a sheet current
Let's say we have a closed surface $\Gamma$ with a local unit normal vector $\mathbf{n}$, where a surface current (electric surface current) $\mathbf{J}_s$ lies on $\Gamma$.
The normal derivative (...
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What are some ways to derive $\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2$?
For each of the two reference books the constant equations are as follows:
$$
\boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}...
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How to calculate the rotation at a singularity?
An electrodynamics lecture asks me to prove that
$$
\nabla \times \left( \frac{\vec{M} \times \vec{x}}{ |\vec{x}|^3} \right) = \frac{8 \pi}{3} \vec{M} \delta^3(\vec{x})- \frac{\vec{M}}{|\vec{x}|^3}+ \...
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No-slip condition tangential and normal component decomposition
No-slip condition on a corrugated surface (modelled by a sinusoidal function $b(x)$))
$\vec{ u} (x,b(x)) =u \vec{i}+ w \vec{k} = 0 \vec{i} + 0 \vec{k}$
expressing in terms of the stream function :
$$\...
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How to prove This Equation between Riemann Tensor and Killing Vector?
How to prove This Equation between Riemann Tensor and Killing Vector?
$$
[\nabla_\mu, \nabla_\rho]\xi_\sigma = R_{\sigma\nu\mu\rho}\xi^\nu
$$
I know
$$
R(\vec{X},\vec{Y},\vec{Z})=[\nabla_{\vec{X}}, \...
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In general relativity, how to get geometric velocity from 4-vector velocity field?
Currently, I‘m listening to the lecture of Prof. Hughes (https://ocw.mit.edu/courses/8-962-general-relativity-spring-2020/video_galleries/video-lectures/).
In General, I‘m able to follow the lecture, ...
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Why is the Riemann curvature tensor not zero?
The Riemann curvature tensor for a torsion-free connection is given by:
$$R^d_{cab}V^c=(\nabla_a\nabla_b-\nabla_b\nabla_a)V^d$$
Where $\nabla_a$ and $\nabla_b$ are the covariant derivatives in the $a$ ...
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Can a non-zero curl vector force field still do a null amount of work?
I've been given a vector field $\vec{F}=(12xy^2, 12yz, 9z^2)$ in cartesian coordinares and I'm asked to calculate the work it would develop on a particle moving from point $A$ to point $B$ through ...
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Do Killing vectors form a Lie-algebra?
In Arnold's book on "Mathematical Methods of Classical Mechanics" it is said that vector fields on manifolds form a Lie-algebra (see chapter 8 on sympletic geometry/manifolds). I consider ...
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Flux of $B$ through a moving surface
I'm struggling to follow a derivation in Griffiths, where he proves that the emf generated by the motion of a conducting loop in a magnetic field is
$$
\mathcal{E} = -\partial_t \Phi
$$
where $\Phi$ ...
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If fluid flow is in units $m/s$, what units are $div$ in?
Imagine a fluid flow given by the formula $\mathbf{v} = x\mathbf{i} + y\mathbf{j}$, this is a simple radial flow outward and the dimensions would be length/time: At a given point, what is the velocity ...
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Still can't see why $\nabla \times \mathbf J(\mathbf r') = 0$ in Biot-Savart derivation
When deriving the differential form of Biot-Savart's law (which is also Ampere's law under magnetostatic conditions) there is one step which I am still not fully convinced.
We will firstly use this ...
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Understanding definition of flux as a vector field
I'm reading about conservation laws in the book Mathematical Models in Biology by Edelstein-Keshet, and I'm a bit confused by the author's definition of flux. For context this is from section 9.3, ...
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How does a vector field transform in quantum mechanics?
( I understand my question is a bit vague. I will try to make it more precise at the end ).
Consider a vector field $\Phi^i$ in quantum mechanics. The crux of my question is, how does it transform?
In ...
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Proof for why flux is proportional to number of field lines
What is the proof for this (assuming that we draw infinite field lines).
I understand why flux through some area is proportional to the number of field lines through that area only in the case of an ...
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Deriving dipolar coupling in spherical coordinates
I am trying to get to the well-known expression of the tensorial coupling between two dipolar quantities, written as an interaction between two centers $\vec{r}$ and $\vec{r}'$ using their distance $d=...
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Allowed free vector field theory Lagrangians
My understanding is that when constructing Lagrangians in QFT, one generally assumes that one can see ~more or less any term that is Lorentz invariant. As such, my question is: suppose one has a ...
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Can the transformation properties of spinor fields on a manifold be *derived* similar to how vector field transformations can be derived?
I'm asking about transformation properties of vector and spinor fields. I'm trying to better understand statements like "vectors/spinors transform in a certain way under way under [symmetry ...
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How to pick the sign when going from a dot product integral to 1D?
my question is about dot product integrals such as work:
$$ \int_{\vec a}^{\vec b} \vec F(\vec r)\cdot d\vec r $$
I want to write this as
$$ \int_{a}^{b} F(r) dr$$
where F is just dependent on the ...
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Exact test for checking conservative forces
What is the nature of this force on the path $x^2+y^2=1$?
$$\tag1\vec F=\frac{-y\,\hat{i }+x\,\hat{j}}{x^2+y^2}.$$
I tried two methods, but they give different answers:
For this specific path it ...
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Given a divergence free vector Field, how can I find the related vector potential without guesswork?
I am reading through Griffith's Electrodynamics 4e cover to cover, skipping no sections and doing all of the problems (sans the ones for masochists) as a passion project.
I came to problem 1.53, in ...
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Drawing Electric field lines
Suppose there is a medium filled by a charge with the volume density $\rho = \frac{\alpha}{r}$ where $\alpha$ is a positive constant and r is the distance from origin.
Now here if we calculate ...
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Proposition 4.4.5 in The Large Scale Structure of Space-Time
In the text by Hawking and Ellis they set the following proposition (pg. 101):
Proposition 4.4.5
If $R_{ab} K^a K^b \geq 0$ everywhere and if at $p = \gamma(v_1)$, $K^c K^d
K_{[a}R_{b]cd[e}K_{f]}$ ...
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Maxwell's divergence of electric fields
According to Maxwell's equations, if we take a circle close to a positive charge (such that the charge is not inside the circle), the divergence of the circle should be $0$ (because there is no ...
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Time evolution operator in classical mechanics?
Hamilton's equation can be written in terms of Poisson brackets, as follows:
$$\dot{q} = \{q,H\}$$
$$\dot{p} = \{p,H\}$$
where $H$ is the Hamiltonian of the system. Now, wikipedia says that the ...
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Can such a field line exist between two positive charges? [duplicate]
I am aware of the usual diagram of field lines between two positive charges.
My question is that is such a field line wrong, if so why?
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Metric Tensor Grid
Let, we are in a 2d metric where $g_{xx}=1, g_{yy}=x^2$, therefore $|e_x|=1$, $|e_y|=x$. If we try to draw the metric in a grid - it looks something like the image I uploaded. Note that, along the X ...
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Covariant and partial derivative of a vector field (not component)
Is the covariant derivative of a vector field (not the components of a vector) same as the partial derivative?
I am adding a screenshot from page 69 from General Relativity: An introduction for ...
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