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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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Volume expansion rate

In the paper Backreaction in late-time cosmology by Thomas Buchert and Syksy Rasanen, Annual Review of Nuclear and Particle Science 62 (2012) 57-79, in eq .2.2 the covariant divergence: $$\nabla_\...
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How is the velocity field of a fluid related to conserved quantities?

In multivariate calculus classes you learn a theorem that says that "A vector field is the gradient of a potential function on a domain $D$ if and only if it's curl-free on $D$." When I try to apply ...
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Question about Lie derivative of connection [closed]

This is from Hughston and Tod, Ex 8.5. Please give me an idea how to start with this. If $\mathcal{L}_V g_{ij} = 2\phi g_{ij}$ then prove that $$ \mathcal{L}_V\Gamma^{i}_{jk} = \phi_{,j}\delta^{...
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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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With regard to distributive law of inner product in vector algebra [migrated]

Consider the equality \begin{align*} &\vec{a}\cdot\vec{b}+c=0 \\ \implies &\vec{a}\cdot(\vec{b}+\frac{\vec{a}}{\vec{a}\cdot\vec{a}}c)=0. \end{align*} If the above equation is valid for any $\...
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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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How can we argue $\nabla\cdot v = 0$ when ${\mathscr r} \ne 0$ on this vector function?

I am dealing with the vector field: $$v = \dfrac{\hat{\mathscr r} }{{\mathscr r}^2}$$ And I am studying its divergence. If we compute it we get: $$\nabla\cdot\left(\dfrac{\hat{\mathscr r} }{{\...
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Material Derivative in Fluid Dynamics and Bernoulli Flow

If a fluid flow, such as water, is incompressible then the convective derivative term in the material derivative is equal to zero. How then, in a Bernoulli flow where there is an increase in velocity ...
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Energy of continious charge distribution

In the book of Griffith intro to electrodynamics, on page 94, the energy of continuous charge distribution is derived in the following way: W(total energy) = $\frac{1}{2} \int\rho V d\tau$, where $\...
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Field lines and flux

To my knowledge, flux of a vector field through a given surface is the integral of the dot product of vector field and a unit vector given to a surface element over the entire surface. Field line is ...
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“Killing leaves” in General Relativity?

I now about Killing vector fields in GR but recently stumbled upon the notion of "Killing leaves" and couldn't find any simple explanation of this notion. For example, this paper writes: "integral ...
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Possible maximally symmetric 3D spaces

I was watching Neil Turok's lectures on General Relativity. After introducing the Einstein equation, he tries cosmology and postulates "The space is assumed to be isometric and homogenous." Then he ...
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Why is the flux no infinite around an isolated charge?

I was wondering that the density of electric field lines determine the strength of the electric field .Now let's say you have an isolated charge ; you know the flux through a closed surface which I ...
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Relation between computation of curl and divergence and their formal definitions

both divergence and curl of a vector field have a formal definition, however, we don't use these definitions when we compute the divergence or curl. so can we just derive the computations from the ...
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Liouville's volume theorem in differential forms language

I will cast my question mostly in words. I usually find Liouville's volume theorem cast in two forms: Easy way (without differential forms language): Phase space volume remains preserved under ...
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About “conserved quantities” in a diffeomorphism-invariant theory by Wald and Zoupas

In this work, Wald & Zoupas developed a framework to define the "conserved quantities" in a diffeomorphism-invariant theory using the covariant phase space formalism. Let $\xi^a$ be a vector ...
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Confused about Navier-Stokes equation

Just look at the L.H.S of the compressible navier-stokes equation from wiki $$\rho(\partial_t \vec{u}+\vec{u}\cdot\nabla\vec{u})=...$$ How can I add a vector $\partial_t \vec{u}$ and a scalar $\vec{...
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Lorentz transformation of vector field

Under a Lorentz transformation, a vector field transforms as: $A'_{\mu}(x')=\Lambda^{\nu}_{\mu}A_{\nu}(x)$ My question is, why is the Lorentz transformed vector field evaluated at $x'=\Lambda x$, ...
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Equivalence between Maxwell's equations and vector Helmholtz equations

When are equivalent the Maxwell's harmonic equations: $$ \nabla\times\left(\nabla\times\mathbf{E}\right)=\mu\epsilon\omega^2\mathbf{E} $$ and the vector Helmholtz equations: $$ \nabla^2\mathbf{E}=\mu\...
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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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Divergence of a displacement vector field multiplied by delta function

I'm trying to work out why $$ \boldsymbol{\nabla\cdot[u}\,\delta^3(\mathbf{r})]=0, $$ where $\boldsymbol{u}$ is the displacement field of a source of stress, $\boldsymbol{\nabla\cdot u}\ne 0$, and $\...
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What is the physical meaning of divergence? [duplicate]

I want to visualize the concept of divergence of a vector field. I also have searched the web.Some says it is 1.the amount of flux per unit volume in a region around some point 2.Divergence of ...
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1answer
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Why did we take gradient outside the integral sign in Scalar potential derivation?

I tried to understand the reasoning given in it but I couldn't understand it. It says that "as the gradient operation involves x and not the integration variable x', it can be taken outside the ...
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How would you model a single point outputting fluid in all directions whilst enclosed in a sphere?

The idea is this: There is a point at the center of a sphere. This point is releasing a fluid (say water) in all directions towards the edges of the sphere. The fluid can collide with the sphere's ...
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Is there any special significance of force field in physics?

What is the formal definition of force field? Which is more fundamental force or field? Do field exist in nature (as force do i think as per section 12-1 of Feynman lecture volume 1, and page 8,9 of ...
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Tensorality of the Lie Derivative and $i([X,Y]) = [L(X),i(Y)]$

I'm trying to understand the equation following (2.15) on p.9 of Blau's Symplectic Geometry and Geometric Quantization. For two vector fields $X,Y$ on a symplectic manifold $M$ we are told one ...
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Ambiguous curl for a vector field

I'm trying to work out how to explain why the curl/divergence for the following image is ambiguous without giving the explicit vector function for the following: E has an ambiguous divergence and I ...
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What are the possible magnetic fields with constant magnitude?

A now-deleted answer to this recent question prompted me to wonder about this and I can't find a clear answer in the top layer of google results, so I thought I'd ask here. What are the possible ...
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Is the air velocity and air temperature a vector field?

The air temperature and the velocity of the air have different values at different places in the earth's atmosphere. Is the air velocity a vector field? Why or why not? Is the air temperature a ...
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General coordinate transformations?

Say I have a vector field expressed in Cartesian coordinates: $$\mathbf{A} = \sum_i A_i \mathbf{\hat{e}}_i$$ where the $\hat{\mathbf{e}}_i$ are the generalisation of the unit vectors $\mathbf{\hat i}, ...
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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? ...