# 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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### Why is $\chi_{[\mu}\nabla_\nu \chi_{\sigma ]} = 0$ at the Killing horizon?

Let $\chi$ be a Killing vector field that is null along a Killing horizon $\Sigma$ Why is $\chi_{[\mu}\nabla_\nu \chi_{\sigma ]} = 0$ at $\Sigma$?
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### Curl and circulation of a vector field that is ill-defined at the origin: any interesting physical effects?

In the cylindrical polar $(\rho,\phi,z)$ coordinate, suppose the velocity field in a liquid is given by $$\vec{v}=\frac{K}{\rho}\hat{e}_{\phi}, \qquad K=\text{constant}.\tag{1}$$ It can be easily ...
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### Divergence of $\frac{1}{s}\hat{s}$ in cylindrical coordinates

In Griffiths' electrodynamics, the divergence of $\frac{1}{r^{2}}\hat{r}$ is evaluated in spherical coordinates to be $4\pi\delta(r)$. I encounter the same problem in case of $\frac{1}{s}\hat{s}$ ...
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### Gauss divergence theorem (GDT) in physics

Some of the statements for $GDT$ which I find in modern textbooks (both electromagnetism and multivariable calculus textbooks) are: (1) Calculus: Several variables Adams Let $D$ be a regular, ...
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### Electromagnetic Angular Momentum: Problem with vector integrals

I found in the following reference (p. 10) an interesting decomposition for the electromagnetic angular momentum in terms of an orbital terms $\vec{L_{orb}}$ and an spin term $\vec{L_{spin}}$. However,...
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### What is longitudinal and transverse component of electric field? [closed]

What is longitudinal and transverse component and how are they interpreted?
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### Notation for the divergence of a rank 2 tensor

I am studying advanced fluid mechanics and sometimes you see equations written in index notation like $$Dv_i= \partial_t v_i +v_j\partial_jv_i$$ but sometimes you find this arrow/vector notation (...
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### What is the difference between Closed and Bounded surface?

When I was going through "The Feynman's Lecture on physics" Volume-2 , I found the line "It is useful to speak of the flux not only through a completely closed surface, but through any bounded ...
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### MTW Exercise 4.4: Rotation free 1-forms [closed]

MTW in Exercise 4.4 calls a 1-form $A_\alpha$ a rotation free 1-form if $$\textbf{A}\wedge\textbf{dA}=0.$$ And claims that all such 1-forms may be written as $$\textbf{A}=\phi\,\textbf{d}\psi$$ for ...
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### How to visualize the merge of magnetic field?

Magnetic fields are represented by field lines and it is stated that these lines are closed lines, going through the source (often shown as lines between the two poles of the source). Approaching two ...
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### What electric and magnetic field lines look like in some examples?

I have seen this but couldn't understand so I wrote my own question. We all have learnt in school that electric field lines never intersect. Same is the case for magnetic field lines. But I have a ...
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### Question about vector field on a manifold [closed]

Arnold defined a vector field on a manifold M is a map from M to the tangent space of M (which has all derivations, roughly). In his ODE book, he talks about $\dot{x}(t) = v(x(t))$ for a vector field ...
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### Dependence of electrostatic force on the surrounding medium

Two unlike point charges held at a fixed distance from each other. The force between them is measured. Then a brass rod is placed exactly in the midpoint of the line joining the 2 charges. Will the ...
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### Electrostatic field-Field lines relationship [duplicate]

How is the $\frac{1}{r^2}$ dependence of the electric field intensity due to a stationary point charge consistent with the concept of field lines?
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### What is gradient with respect to components of a position vector?

I am reading "Classical mechanics" by Goldstein, Poole and Safko, Third edition. Kindly please refer to page no 10, last paragraph. They write the subscript $i$ on the del operator indicates that ...
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### When does a vector component keep being a vector, exactly?

English is not my native language, so please forgive my errors. Consider this example: This is a classic: an exercise requiring you to calculate the electric field produced by a charged ring on its ...
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### Why is the concept of electric field lines needed to understand electric fields? [closed]

I am recently learning about electric fields. So I encountered the concept of electric field lines. As they are not real but imaginary lines. Why do we need them to understand electric field? I am ...
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### 2D deSitter conserved charges

For $1+1$-dimensional dS spacetime, the metric takes the form (in comoving coordinates) $$\mathrm{d}s^2=\left(\mathrm{d}x^0\right)^2-e^{2Hx^0}\left(\mathrm{d}x^1\right)^2.$$ I want to find the ...
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### Magnetic field around solenoid and toroid

Solenoid is proving a little bit confusing While getting through solenoid I found that the field outside it is extremely small and is negligible. Also the field at ends is half of that of center. ...
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### Divergence of magnetic field

Consider a point near one of the poles of a bar magnet. The magnetic field lines do appear to spread, but according to Maxwell's equations the divergence of a magnetic field is always zero. So what's ...
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### Can the gravitational field be considered conservative despite the existence of singularities?

Assuming singularities are physical objects as opposed to mathematical artifiacts, can the gravitational field still be considered conservative? And if not, does this open a possibility of breaking ...
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### How do “divergence” and “curl” relate to the three states of matter?

A fluid is said to have divergence (the ability to flow) and curl {the ability to rotate). Do these two characteristics fully define a fluid, or are there other important properties that I am missing? ...
Consider that we have an orthonormal basis $\{e_1, e_2, e_3\}$ We know that $e_2 \times e_3 = \pm e_1$, to show this in terms of tensor notation, from the Continuum Mechanics by Chadwick textbook it ...