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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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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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3answers
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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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1answer
48 views

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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1answer
74 views

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
33 views

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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1answer
69 views

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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1answer
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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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59 views

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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2answers
441 views

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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1answer
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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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1answer
55 views

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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1answer
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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? ...
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Basics of Tensor theory

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 ...
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Line element to polar coordinates [closed]

I'm calculating the effective metric for a vortex in polar coordinates. The velocity and the potential is: \begin{equation} \mathbf{v}=\frac{A}{r} \hat{r} + \frac{B}{r}\hat{\theta} \end{equation} So:...
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1answer
69 views

Why aren't coordinates induced vector fields always Killing fields?

We have that $$ L_K g_{\mu\nu}=\nabla_\mu K_\nu + \nabla_\nu K_\mu$$ A vector field $K$ is a Killing field if $ L_K g_{\mu\nu}=0$, but consider the coordinate induced vector field $\partial_\alpha$, ...
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2answers
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Killing vectors - Schwarzschild metric

Given the Schwarzschild metric, $$\mathrm{d}s^{2}=-\left(1-\frac{R_s}{r}\right)\mathrm{d}t^{2}+\left(1-\frac{R_s}{r}\right)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\theta^{2}+ r^2 \sin^{2}\theta\mathrm{d}\...
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1answer
37 views

Convert magnetic field from cylindrical to cartesian coordinates

It's been a while since I had to convert cylindrical to cartesian unit vectors, and even though I have the transformation rules, I can't seem to grasp how to go about the following: How would I (what ...
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0answers
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Where can I find an interactive vector field?

Imagine a low-pressure system as a vacuum that sucks all the surrounding air straight towards it, creating many vectors of wind that all focus on one spot. Because of the Coriolis effect, each of ...
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3answers
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How do we deduce the vector potential for a constant magnetic field?

How do we show that for a constant magnetic field $\vec B = const$, the vector potential is $\vec A = \frac12 \vec r \times \vec B$?
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Meaning of flux 2-integral

Can someone please explain the meaning of flux 2-integral in this sentence: Mass is evaluated as a flux 2-integral at the asymptotic infinity. For asymptotic infinity, I believe it is as ...
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0answers
37 views

Why Gauss divergence theorem isn't working? [duplicate]

$\vec{E}$ is electric field $r$ is distance between source and field points $\hat{r}$ is a unit vector from source point to field point $x,y,z$ are Cartesian coordinates of field point ...
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1answer
42 views

Will the flux through an arbitrary closed surface be finite or infinite when a plane charge intersects the Gaussian surface?

Let's consider a closed Gaussian surface (in red). The white line and the white shaded part lies inside the Gaussian surface and the black line and the portion above it lies outside the Gaussian ...
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1answer
52 views

Visually and physically reasonable analytic approximation to the field of a bar magnet

I want to produce publication-quality drawings of the field of a bar magnet, in both the field line representation and as a "sea of arrows." I've found some open-source software that looks like it ...
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1answer
31 views

Curl of magnetic field produced by current carrying wires with infinitesimal small area

Can Magnetic fields produced by thin current carrying wires with infinitesimal area have curl with a delta function in it ?? As area is Zero current density J definitely becomes infinite at where ...
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1answer
73 views

What is the physical meaning of irrotational flow?

I am studying fluid dynamics and while searching on the internet, found many videos explaining the mathematical aspect of rotational and irrotational flow, but i need more clarity on what actually it ...
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2answers
126 views

How is Maxwell's second equation true here?

$\vec{B}=\nabla \times \vec{A}\tag1$ This is true because at every point $\nabla\cdot\vec{B}=0 \tag2$ In free space points, $\displaystyle \vec{B}=\dfrac{\mu_0}{4 \pi}\int_C \dfrac{I\ dl \times\hat{...
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1answer
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Is flux a function of (relative) velocity?

Consider a 3D space, where there is a field (electric, magnetic, whatever). The field is $-c_1\hat k$, where $a$ is a positive constant, such that the field is constant everywhere towards the ...
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1answer
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How should MTW's derivation of the Maxwell-Faraday formula be interpreted?

In the following derivation, I am not sure exactly how the components of the final vector equation are established. I suspect this is a situation analogous to the vector addition of infinitesimal ...
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3answers
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What actual difference does it make whether electric or magnetic field lines are open or closed?

If both electric and magnetic fields attenuate with distance by the inverse square law, what difference does it make that the latter's field lines are 'closed'? And if a magnetically-induced electric ...
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1answer
70 views

Relation between curl and gradient [duplicate]

I need to prove the following relation (with vector $\mathbf{V}$) : $$(\mathbf{V} \cdot \nabla)\mathbf{V} = \frac{1}{2}\nabla (\mathbf{V}^2)+(\nabla \times \mathbf{V}) \times \mathbf{V}\quad\quad\...
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2answers
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How does Gauss's Law work with this charge density setup?

My friend and I are self-studying electrodynamics. In Griffiths, Introduction to Electrodynamics (1999), the concept of divergence is introduced mathematically and the following vector field is drawn. ...
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1answer
109 views

Physical significance of divergence

In my textbook They considered a parallelopiped $ABCDEFGH$ with sides $dx,dy,dz$ parallel to $x,y,z$ axis respectively $\vec V$ represents the vector velocity of the fluid at the centre $P$ of f ...
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1answer
56 views

No clue about a term [closed]

$\int_S\int \vec{A}\cdot\hat{n}dS= \int_S\int A cos(\theta)dS= \int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$ I have no clue about the term $$\int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$$ ...
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1answer
39 views

Where is the error in this calculation of net curl for simple magnetic field?

I wasn't sure whether to post this on MSE, but PSE seems more appropriate. Let B be a static magnetic field in spherical coordinates, defined as $B=r\hat{\theta}$. Then, it's curl is $$\nabla \times ...
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0answers
26 views

Why are bulk plasmons longitudinal waves and how do we excite them if we can't use transverse electromagnetic waves?

For bulk plasmons we have the dispersion relation starting at some non-zero value of frequency, the plasmon frequency $\omega_p$ $$ \omega= \sqrt{\omega^2_p + (ck)^2} $$ This means $k=0$ for $\...
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1answer
33 views

Notion of flux and field lines

I am curious about the origin of electric flux and field lines. -Flux- I am aware of the fact that flux is a mathematical concept, but how did it find its way into physics? Was it just introduced ...
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2answers
59 views

What is the tangent vector representing rotation?

I am reading Mathematics for physics: A guided tour for graduate students by Michael Stone. On the page 379, the book says The surface of the unit sphere is a manifold...We may label its points ...
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2answers
33 views

Closed field lines in case of a Bar magnet

Field lines in case of charges go from +ve to -ve but incase of magnet, they dont start or stop anywhere. They form closed loops. Is this consequence of the fact that single poles dont exist or ...
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2answers
322 views

Puzzle concerning the Divergence Theorem

Something is puzzling me concerning the divergence theorem. Usually, one writes the divergence theorem as \begin{equation} \int_\mathcal{M} d^4x \sqrt{-g} \nabla_\mu v^\mu=\int_{\partial \mathcal{M}} ...
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1answer
145 views

Vortex anti-vortex?

i'm studying Kosterlitz THouless transition and i have a doubt: what is a vortex anti-vortex configuration? Is this thing? or this one I think that they are quite different !
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1answer
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I'm confused about what streamlines are in electromagnetism

In the below picture, what exactly is the streamline? Is it the three blue curves or is it the E vector drawn? Also, to calculate the equation of the streamline, we are using the ratio of the ...
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0answers
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Relation between Lie algebra of conformal Killing vector fields and conformal algebra

I'm new to conformal transformations and I have a question. Following the book of Barrett O'Neill "Semi-Riemannian geometry with applications to relativity", there is a Lie anti-isomorphism between ...
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1answer
51 views

Proof regarding curl of a vector [closed]

I'm stuck in this problem where we need to prove that the curl of the velocity vector is twice the angular velocity of a rigid body in circular motion. How do I prove it? I am very new to the concepts ...
4
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1answer
109 views

3D Stream function in fluid mechanics

The 3D stream function ${\bf \Psi}$ for a steady flow field can be defined as: $\rho{\bf u}={\bf \nabla}\times {\bf \Psi}$. Where, ${\bf u}$ = velocity, $\rho$ = density. Now, this ${\bf \Psi}$ can ...
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2answers
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Do meaningful bifurcation diagrams exist for systems described by vector fields on circles?

I've been reading about the vector field on a circle, and how it's been used to describe stable points for periodic motion. I have also read about how bifurcation diagrams describe changes in ...