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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Intuitive understanding of what coordinate basis elements of $T_p$ mean in relation to the manifold
I am trying to have a "visual" or intuitive understanding of the description that was made in my GR lecture about the object $\partial_\mu$. The following was said:
We have a manifold $M$, a ...
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Covariant Derivative of a Vector Field in Flat Space but Curvillinear Coordinates
Let $\vec{V}$ be a vector in flat Euclidean space. In curvilliner coordinates, using Einstein summantion convention, $$\vec{V}=V^j\vec{e}_j$$ where $\vec{e}_j$ are the basis vectors and $V^j$ are the ...
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Is there an elementary magnetostatic example of a current distribution which does not produce magnetic field lines which are closed loops?
It is often incorrectly asserted in elementary textbooks that magnetic field lines must form closed loops. This is often deduced from the non-existence of magnetic monopoles. Unfortunately, ...
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Problem with vector calculus - electric field from a dipole
While deriving the electric field from a dipole source, from the notes I am following I am required to process the following vector operation:
$$
\nabla \left(\frac{e^{jkr}}{r}\mathbf n\cdot \mathbf p\...
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Validity of assuming the integral of a vanishing integrand over an infinitely large surface is equal to zero
In Griffiths' Electrodynamics, Chapter 8, Griffiths claims that if we compute the surface integral of a vector field that vanishes at infinity over an infinitely large surface, the result will be zero....
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Why are magnetic fields that shape?
When you place iron filings around a bar magnet, they take an arrangement which ends up looking like this:
But I'm having trouble understanding where the strange shape of these field lines comes from;...
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Connection between magnetic field $\bf B$ and vector potential $\bf A$ in Griffiths
In Griffiths' Electrodynamics (3rd Ed.) p. 240 there is a triangle of relations between $\bf{A}, B$ and $\bf J$ with an indication of the vector operations that link these quantities in both ...
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Do Killing fields give conservation laws for momentum of photons?
Any Killing field gives a conserved quantity
$$ K_\mu u^\mu $$
And in the case of massive particles one can multiply by the rest mass and obtain that
$$ K_\mu p^\mu $$
is a conserved quantity.
However ...
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The Lagrangian of a moving electric and magnetic dipole
I am reading Chapter 24 of Modern Electrodynamics by Zangwill. I have a question about example 24.2 (a) even after reading the solution.
The statement of the problem:
A neutral and point-like particle ...
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Why there is no defined quantity for line integral in Ampère's law, like flux in Gauss's law?
I am a high school student that was unable to find a convincing response regarding my following question.
In electrostatics, we studied Gauss's law:
$$\Phi_E = \oint \vec{E} \cdot \mathrm{d}\vec{A} = ...
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Can spacetime be considered as a field like the tetrad vector field? [closed]
i saw a preprint paper https://doi.org/10.5281/zenodo.15815535 and it give an idea to consider the spacetime as a field through tetrad formalism and if so can we quantize the tetrad vector field and ...
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Stokes' theorem in non-Riemannian geometry [duplicate]
Does Stokes's theorem in curved space, usually written in the context of GR:
$$\int_V \nabla_\mu A^\mu \sqrt{|g|}d^nx = \int_{\partial V}A^\mu n_\mu \sqrt{|\gamma|}d^{n-1}x$$
(Where $\nabla$ is the ...
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Space/Time decomposition at a point and particle dynamics
I am reading Topics in the Foundations of General Relativity and Newtonian Gravitation Theory by David B. Malament. Chapter 2. and I am stuck in two equations.
Both $\xi_a$ $\xi_b$ are future directed....
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Special Relativity and Rotating Reference frames for calculating Magnetic fields
There is the well-known textbook problem of determining the magnetic field/ vector potential inside a hollow sphere of radius $R$ with constant surface charge density $\sigma$ which is spinning with ...
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Why integral curves of a null vector must be null geodesics?
In many General Relativity literature, the following question takes as a very obvious fact but I don't understand why or how it holds.
Why integral curves of a null vector on a null hypersurface is ...
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Gyroscope transport
Say I'm inside a closed box and I spin an ideal gyroscope, its spinning axis is perpendicular to the ground and it spins CCW.
I go to sleep inside the closed box and I wake up unknowingly in the ...
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Can the Klling vector on Killing horizon be globally null vector?
I am reading Geometry of Killing horizons and applications to black hole physics 2 Killing horizons by Eric Gourgoulhon. Here is a link
https://relativite.obspm.fr/blackholes/pdf/ihp24/lecture2.pdf ...
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Understanding condition of Hypersurface orthogonality
Wald at page#436 under the heading B.3.2 wrote
The dual formulation of Frobenius's theorem gives a useful criterion for when a vector field $\mathcal{E}^a$ is hypersurface orthogonal. Letting $T^*$ ...
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How to derive these variations of Stokes' theorem and divergence? [closed]
The book seems to expect me accept these results without building up the "why". I lack some operator formalism to understand what's going on here. Also, divergence theorem is:
$$\int_{S} \...
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Fermi-Walker transport of proper acceleration vector field along timeline congruence's worldlines
If we take a generic irrotational/zero vorticity timelike congruence, do the 4-velocity and the direction of proper acceleration $($i.e. the vector in that direction at each point with norm $1$$)$ ...
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When is the Lamb vector the gradient of a function?
There is something I am not seeing in this derivation of advanced Bernoulli's principle: https://open.oregonstate.education/intermediate-fluid-mechanics/chapter/bernoulli-equation/
The Lamb vector is ...
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Double Null Foliation Schwarzschild Metric
Thinking two null rays coming from a timelike geodesic of , say , a star of mass $m$, one future pointing $l_a = \nabla_a u$ and one past pointing $l'_a = \nabla_a u'$ $ [l_al^a = l'_al'^a = 0] $ and ...
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Calculate normal vector to future light cone
Consider a future light cone in Minkowski spacetime $(-,+,+,+)$ defined by $u(t,x,y,z)=t - \sqrt{x^2+y^2+z^2}$ and $t>0$. Derivative of $u$ is $du=dt-\frac{x}{r}dx-\frac{y}{r}dy-\frac{z}{r}dz$ with ...
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Reason why magnetic field lines never intersect [closed]
Assertion (A): Magnetic field lines around a bar magnet never intersect each other.
Reason (R): Magnetic field produced by a bar magnet is a quantity that has both magnitude and direction
Is the ...
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Prove that tangent and deviation vectors are orthogonal to each other
I am reading geodesic deviation equation from Caroll page # 144 but I have some confusion
Let $T^\mu=\partial x^\mu / \partial t$ and $S^\mu=\partial x^\mu / \partial s$ are tangent vector fields and ...
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