# 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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### The Curvature of Electric Field Lines

I have been practicing many questions regarding electrical field lines. However, I can't seem to understand when electrical field lines remain straight and when they start to curve. Does it depend on ...
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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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### 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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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 $f : \mathbb{R}^4 \... • 1,669 0 votes 1 answer 74 views ### What does the notation$(k \cdot \nabla ) v$mean? [duplicate] I am reading a paper and it uses a notation I am not too familiar about. Although I saw it used elsewhere, I don't remember the meaning of it and I don't want to misinterpret it and realize after ... • 165 1 vote 1 answer 77 views ### Magnetic vector potential in 1+1 spacetime dimensions In the theory of electromagnetism in 1+1 spacetime dimensions (one temporal and one spatial coordinate), one can define the 2-potential vector (analogous to the 4-potential vector in 3+1 spacetime ... 1 vote 2 answers 85 views ### What is Dirac's reasoning when saying parallel displacement creates vector field with vanishing covariant derivative? Section 12 of Dirac's book "General Theory of Relativity" is called "The condition for flat space", and he is proving that a space is flat if and only if the curvature tensor$R_{\...
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 ...