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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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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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 $\frac{ \hat {\bf r}}{r^2} \equiv \frac{{\bf r}}{r^3}$, what is the 'paradox'?
I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2} \equiv \frac{{\bf r}}{r^3}$, now from the book, Griffiths says:
Now what is the ...
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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,
$$ds^2=-dt^2+a(t)^2\left(dr^2+d\...
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Can electric field lines form close loops in EM?
I was reading on wikipedia about electric field lines
In the Precise definition part it says it can form close loops but Kelvin–Stokes theorem it's written it cannot be closed loops. (My teacher too ...
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Quantum mechanics in phase space - what are coordinate components?
I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum ...
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Calculating $[H,\vec r]$ [closed]
I'm doing Griffiths' Introduction to Quantum Mechanics. In a question, it introduces the modified Schrodinger equation in which the Hamiltonian, $$ 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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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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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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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 Maxwell's equations) into scalar equations involving partial derivatives (more than four equations)....
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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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