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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Coordinate-free proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic

I would like to know if the following proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic is correct. I already found a coordinate-...
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Divergence of a Vector Field - Surprising Result [duplicate]

I'm following the text Introduction to Electrodynamics by Griffiths, and I came across the following in an in-text problem: Sketch the vector function v = $\frac{\boldsymbol{\hat{\mathbf{r}}}}{r^2}$...
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Trace of second-order tensor and its invariance under coordinate transformation

Let's consider an arbitrary scalar field. If I act twice on the scalar field with a gradient operator, I will obtain second-order tensor. If I will take a trace of this tensor, I will obtain another ...
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Can the flow be irrotational if the viscous forces act on fluid?

I tried to answer the question only using the definitions and the Navier-Stokes equation: $$\rho \frac{Dv}{Dt} = -\nabla P +\rho g -\mu[\nabla \times(\nabla \times v)] $$ In my opinion if the ...
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Is there a generalization of ${{\partial }_{\alpha }}\left( \sqrt{-g}{{V}^{a}} \right)=\sqrt{-g}{{\nabla }_{a}}{{V}^{a}}$ for arbitrary connection?

I am studying general relativity and I am trying to understand how to perform variation of the Einstein–Hilbert action with respect to the metric ${{g}_{\mu \nu }}$ and an arbitrary connection ${{\...
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Physical meaning of the convection term in the momentum equation of acoustic wave

In deriving the acoustic wave equation, the momentum equation is used. $$\frac{\partial \mathbf{u}}{\partial t}+ (\mathbf{u}\nabla)\mathbf{u}=-\frac{1}{\rho} \nabla p$$ Intuitively, the convection ...
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Electric field line density : Theory vs Reality

I've already went through this post. Yet, I still can't understand the meaning of "density" of electric field lines whose number is, in reality, infinite. One of the answers , for instance, states ...
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How to prove that $\vec{E}$ is an intensive property?

In my homework I had a question to sort a few variables into intensive properties and extensive properties. I wrote that $\vec{E}$ (electric field) is an extensive property, thinking of a situation ...
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Globally constant vector field in a curved spacetime

Is it possible to define a globally constant vector field in a curved spacetime, that is a vector field for which the covariant derivative vanishes along every world line? The vector field $V^{\mu}=0$ ...
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Magnetic fields and closed loop

It is well known that there don't appear to be magnetic poles. In Maxwell's equations this has the implication $$ \nabla \cdot \mathbf{B} = 0 $$ and results in the statement "the magnetic field forms ...
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Surface Tension of Floating Object

I was reading this thesis about surface tension and its role in floating bodies. I couldn't quite understand at page 10 how the author applied the 2D divergence theorem outside the region bounded by ...
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How can surface integral's contribution goes to zero but volume's does not in this particular derivation?

When we derive the formula for energy of a continuous charge distribution $\rho$ using this equation $$W = \frac{1}{2}\int\rho V \text d\tau$$ with $V$ being the electric potential, we get this ...
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Covariant derivative with respect to commutator

I have some confusion with the notion of $\nabla_{[A, B]}\bf{v}$, that expression, with a commutator of vector fields as the subindex of the connection appears for instance in the definition of the ...
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Magnetic Field Lines Vs Magnetic Vector field

I am studying electromagnetic theory and when I started researching the history of conventions used in magnetic interactions I could not get them. The basics of how they modelled the magnetic ...
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1answer
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Understanding of Gauss law using vector fields

I was going through the conventions and terminologies followed to describe the magnetic interactions. I understood that the field lines are just a simpler representation of the magnetic interaction ...
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What is the gravity on a “partial” ringworld?

This was inspired by https://worldbuilding.stackexchange.com/questions/149706/life-on-the-broken-ring-an-issue-of-size. Let's say I have a part of a Ringworld (see link for specifications). ...
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1answer
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Maxwell Laws Summary Diagram - Suggestions that I am missing? [closed]

I have been going through a summary book of Maxwell's equations and hope I have organised this correctly but I think perhaps I am missing things important prompts that I could add? Image below Thanks ...
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Interpretation of surface integral of vector field over surface

Is it correct to interpret the surface integral of a vector function $\mathbf{v}$ over four sides of a cube as the rate of flow of fluid (in mass per unit time) that would flow out of the cube when ...
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71 views

Partial Integration and the Levi-Civita Symbol

I'm currently working through the book Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized ...
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Killing vectors in an static space-time

How can I show that a given space-time is static, i.e. exists a time-like Killing vector $\xi = \partial_0$ that $\partial_0 g_{\mu \nu} = 0$ (Killing eq.) and $g_{0i}=0$, if and only if the relation $...
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What is electric field vector? [closed]

What is electric field vector? How to find out the Electric Field vector at a point on a equipotential surface. Please explain by giving an example.
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Is the Lie derivative along the normal well-defined?

This question is cross-posted at https://math.stackexchange.com/q/3274757/247251 Let $(\Sigma, q)$ be a non-degenerate submanifold of a Lorentzian manifold $(M,g)$. Let $N$ be the section of $T\Sigma ...
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Divergence of $\frac{ \hat {\bf r}}{r^2}$ , what is the 'paradox'?

I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2}$ , now from the book, Griffiths says: Now what is the paradox, exactly? ...
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Are there cases where $\nabla\cdot\iiint\frac{\mathbf{J}(\mathbf{x}')}{\left|\mathbf{x}-\mathbf{x'}\right|}\mathrm{d}V' \neq 0$?

In Jackson's Classical Electrodynamics, Section 5.4 (Vector Potential), the author seems to assume that because $\nabla\cdot\mathbf{J} = 0$, the following holds for the current density (where the ...
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longitudinal and transverse components in higher dimensions

I am familiar with the Helmholz decomposition of a vector field in three dimensions: $$\vec{V}=\vec{\nabla}\wedge\vec{A}+\vec{\nabla}\phi$$ But I am interested to show that something similar can be ...
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Killing Tensor of Friedman-Robertson-Walker Metric

I would like help showing that the tensor, $$K_{\mu\nu}=a^2(g_{\mu\nu}+u_\mu u_\nu),$$ where $u^\mu =(1,0,0,0)$, is a Killing tensor of the spatially flat FRW metric, $$ds^2=-dt^2+a(t)^2\left(dr^2+d\...
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Deriving Jefimenko's equations in Fourier space

From the Fourier-transformed Maxwell equations we have, with some algebraic manipulation, $$\mathbf{E}=\frac{1}{|k|^2}\left[\mathbf{k}\frac{\rho}{\epsilon_0}-\mathbf{k}\times k_0\mathbf{B}\right]$$ $$\...
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Quantum mechanics in phase space - what are coordinate components?

I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum ...
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Calculating $[H,\vec r]$ [closed]

I'm doing Griffiths' Introduction to Quantum Mechanics. In a question, it introduces the modified Schrodinger equation in which the Hamiltonian, $$ H~=~-\frac{\hbar^2}{2m}\nabla^2 + V $$is replaced by ...
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Misunderstanding of lowering indexes using Euclidian metric

One may define a vector field in $R$ and see how its components transform under a basis transformation. $ v= v^{u}\partial _{u} $ In principle, the components transform as contravariant such that ...
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Change of variables in gradient

Take two coordinates with $\mathbf r$ and $\mathbf r'$ and take a function $f(|\mathbf r - \mathbf{r'}|)$. In many electromagnetism derivations I see a conversion like this $$ \nabla_r f(|\mathbf r - \...
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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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68 views

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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1answer
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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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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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1answer
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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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1answer
63 views

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 $n^a$ ...