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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 do my books introduce the equation $\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_0}$ without showing partial derivatives of $\mathbf{E}$ exist?

In electromagnetism (electrostatics), we often come across the equation $\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_0}$. In order for this equation to be meaningful, $\mathbf{E}$ must be a ...
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Is velocity of a fluid the gradient of something physically significant?

For incompressible flow, $$\nabla\cdot \mathbf v=0.$$ That means $\mathbf v$ got to be the gradient of some scalar field. How can I find the scalar field? Is it physically important?
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Please clarify a doubt in the article: Reflections in Maxwell's treatise

While going through an article titled "Reflections in Maxwell's treatise" a misunderstanding popped out at page 227 and 228. Consider the following equations $(23\ a)$ and $(23\ c)$ in the article (...
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Does $Curl(E) = 0 $ along an equipotential sphere require that the radial component of $E =$ constant on the sphere?

I'm arriving at the conclusion that "$\nabla \times \vec{E} = 0$ on the surface of an equipotential sphere ($E_\theta = E_\phi = 0$) (as the field must be normal to an equipotential/conductor) implies ...
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Critical points of vector field with zeros in the magnitude

I am studying a vector field which has critical points (sources, sinks, saddle points and centers). The magnitude of the vector field goes to zero smoothly in these points, however. Contrast that to ...
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Confused about scalar fields

A scalar field is one which is unchanged under rotation. But how do we decide whether a given field (e.g., the temperature in a room) is unchanged under rotation or not? We need to measure the ...
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Inconsistencies in finding magnetic vector potentials

Recently I've been studying for my electromagnetism finals and I reached a question about magnetic vector potentials. If I have a wire with constant current distribution, what is the magnetic vector ...
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Divergence and rotation of ideal dipole field at d --> 0

I am working on an assignment in vector calculus where we are supposed to find the divergence and the rotation of the electric field caused by an ideal dipole when the distance d between the charges ...
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What is meant by surface divergence of a vector function?

My book says: If there is a surface discontinuity in a vector field $\vec{E}$, we enclose it in a thin transitional layer (of width $h$) and apply divergence theorem. If $\hat{n}_1$ and $\hat{n}_2$ ...
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Why are fields which do not transform in a certain way not fundamental?

When I was first exposed to Math physics textbooks and textbooks on vector calculus, I found: Temperature distribution in a room $T(x,y,z,t)$ or the density variation in a fluid $\rho(x,y,z,t)$ etc ...
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Can I write a 2-dimensional electric field as an analytic function on the complex plane? [closed]

Let's consider a two-dimensional electric field $\textbf{E}=\textbf{E}(\mathbf x)$, where $\mathbf x\in \mathbb R^2$, and $\textbf{E}$ is a vector representing the direction and strength of the field ...
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How do you determine the path of a particle placed in a vector field?

I have recently found a way of expressing newtonian gravity as a vector feild. First the Equation $$F=\frac{Gm_1m_2}{r^2}$$ I only want to know the accelleration in the equation since I only want to ...
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Rigid body in a vector field

If I let a particle with coordinates $x(t)$ move in a vector field $F$ the equation I have to solve is $x'(t) = F(x(t))$, right? But if, instead of a particle, I have a rigid body, which equation I ...
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How to derive a force expression for dipole-dipole interaction

Background I recently asked a question about how to derive a force expression between a point charge and a dipole. This got me wondering whether I could extend this solution to dipole-dipole ...
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How to cast dipole point charge force expression from cylindrical to Cartesian coordinates

Background I am currently building simulations of molecular dynamics and one thing I want to model is dipole interactions. I recently came across this post about calculating the force between a point ...
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Vector calculus simplification in calculation of generalized force

Consider a system of $N$ particles subject to forces $\vec F_i\ (i=1\dots N)$ that derive from a potential $V$. My lecture notes propose a simple proof that $$Q_j = -\frac{\partial V}{\partial q_j}$$ ...
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Is there a useful relationship between connection on space coordinates and material derivative?

I am referring to an important part of the question Relationship between Connection and Material Derivative. Here is a paste and cut of the relevant part. That is the directional derivative along $...
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Magnitude of vector field [closed]

I think this is more of a mathematical question, but since it's for a physics problem I decided to ask it here. I have this complicated magnetic field in spherical coordinates $(r, \theta,\phi)$, $$ ...
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Spherically Symmetric Spacetimes

I have been studying the Schwarzschild metric $g$ and its derivation. The starting point is to assume the spacetime it describes is spherically symmetric. This means that the algebra of its Killing ...
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A problem of energy conservation in GR

The following discussion with the Minkowskian space-time. It is known that the conservation is described by $P_{\xi}=\int_{\Sigma} T_{a b} n^{a} \xi^{b}$, where $\Sigma$ is a Cauchy surface and $n^a$ ...
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Physical significance of one-form in a velocity field

Still tentatively feeling my way through this stuff, so please go easy. The velocity of a fluid at a point P are the components $V^{a}$ of a contravariant vector:$$v^{x},v^{y},v^{z}\equiv\frac{dx}{dt}...
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Magnetic flux (and flux in general)

The general interpretation of flux as I understand it (and please correct me if I'm wrong) is that it represents how much something is going through another (surface or volume (and perhaps lines?)), I'...
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Expressing Maxwell's equations as scalar equations involving differentials in Euclidean space

I am trying to convert Maxwell's equations from the well known differential form (found on Wikipedia https://en.wikipedia.org/wiki/Maxwell%27s_equations) into scalar equations involving partial ...
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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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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 ...