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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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2
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1answer
265 views

Understanding scalar and vector fields

I know the definitions of vector and scalar field but I don't know what is meant by them. Suppose a scalar field is given by $\phi(x,y,z) = 3xyz$. Then what does it mean? What is the relationship ...
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2answers
240 views

Difference between physicist's vector and mathematician's vector

Mathematically a vector is defined as an element of vector space which obeys certain properties. While reading about the special theory of relativity, I came to know about another definition of ...
3
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1answer
378 views

How is dot or cross product possible using the del operator?

Yesterday in class my teacher told me that the del operator has a direction but no value of its own (as its an operator). So it can't be called exactly a vector. But in vector calculus we see that div ...
3
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3answers
321 views

Why is $ \frac{\vec{r}}{r^3} = \frac{1}{r^2} $?

I know it's surely a beginner's question but I don't see why you can write \begin{align} \frac{\vec{r}}{r^3} = \frac{1}{r^2}\cdot \frac{|\vec{r}|}{r} \end{align} Could someone explain it please? It ...
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1answer
78 views

How electric field lines are defined?

I am having confusion understanding that how for finitely many fields lines in the space, the intensity of a field is proportional to the number of field lines passing through a surface area? Also, ...
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2answers
119 views

Problem with the Landau gauge

I'm having a very simple problem which probably has an equally simple answer. I'm following the wikipedia article: https://en.wikipedia.org/wiki/Landau_quantization We have a uniform magnetic field ...
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1answer
118 views

Derivation of the Raychaudhuri equation following Wald

I have a quick question, in the derivation of the Raychaudhuri equation in Wald: General Relativity on page 218, equation 9.2.10, an identity for $\xi^a \nabla_a B_{ab}$ is derived, where $B_{ab} = \...
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1answer
101 views

Finding the divergence in spherical co-ordinates using the metric tensor

I need to find the divergence in spherical co-ordinates using the expression $$ \nabla \cdot \vec{v} = \frac{1}{\sqrt{g}} \frac{\partial}{\partial u^{j}} (\sqrt{g} v^{j})$$ I'...
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2answers
246 views

Will electric field lines begin or end at a point other than a charge? Feynman Lectures Vol II Chapter 1

In The Feynman Lectures on Physics Vol. II Ch. 1: Electromagnetism, the following is stated: There have been various inventions to help the mind visualize the behavior of fields. The most correct ...
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0answers
33 views

Defining potential energy for a non-conservative field with a global attractor

Consider a vector field $\vec{F}(\vec{x})$ that is not necessarily conservative (meaning that the line integral of this field need not be path-independent). If we now describe the motion of a particle ...
2
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1answer
914 views

Magnetic field lines vs. Electric field lines

What's the difference between magnetic field and electric field lines? Does $d\vec B$ point in the direction you would experience a force if you were a moving charged particle at that point? I know ...
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2answers
245 views

Hamiltonian equation in Cartesian coordinates

My Lagrangian equation is $$L = \dfrac{1}{2}m\dot{q}^{2} \tag{1},$$ where $q=(x,y)$. Performing the Legendre transformation I get the Hamiltonian equation, \begin{equation} H(p,q) =p\dot{q}-\dfrac{1}{...
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1answer
245 views

Understanding Conservation of Momentum for Fluids

In Fluid Dynamics, the equation $$\sum F = \dfrac{d}{dt} \int_{CV} \mathbf{U} \rho dV + \int_{CS} \mathbf{U} \rho \mathbf{U} . d\mathbf{A}$$ was introduced, where is $\mathbf{U}$ and $V$ is volume. ...
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3answers
259 views

Boundary conditions in E&M

While deriving boundary condition for $B$ and $D$ we take a pill, box but for $E$ and $H$ we take a rectangular loop, why?
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1answer
190 views

Simple 2D Curl example with units?

I'm studying Calculus, not physics, but was curious how units work when we consider curl over a 2D velocity field. Given $\mathbf{F} = M(x,y)\mathbf{i} + N(x,y)\mathbf{j}$ the curl is defined to be $(\...
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1answer
89 views

Curl and divergence [duplicate]

I am trying to understand curl and divergences in a more intuitive manner, especially the curl. And is curl a surface phenomenon, if yes then how?
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1answer
82 views

Vector Identity using Coulomb Gauge in Stancil and Prabhakar's 'Spin Waves'

I'm working through Stancil and Prabhakar's 'Spin Waves', and am stuck with a vector identity which I am not sure how the authors have justified. On page 34, we adopt the use of a scalar potential $\...
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1answer
52 views

Given several flow lines, find a suitable vector field equation?

So for this problem there's a number of curves in a bounded region that represents a fluid flow. The green point is a source and the fusian point is a sink. I need the fluid to flow exactly along ...
4
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1answer
155 views

Marginally Bound Vector Field

I was reading about congruences of timelike geodesics from "A Relativist's Toolkit - Eric Poisson". There is a solved example of a timelike geodesic congruence for Schwarzschild spacetime. The kind ...
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1answer
66 views

What is the gradient of the inverse of the magnitude of a difference vector?

This is a follow-up to a question I recently asked on Math.SE which was about the math in particular: Proof of the gradient of the inverse magnitude of a vector? - Mathematics Stack Exchange. It is ...
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0answers
65 views

Can field lines be ever parallel?

I know that this question is not really relevant , but it's still troubling me and that's why I am asking it out of curiosity. I wonder whether it is theoretically possible for field lines to be ...
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1answer
83 views

If its given that laplacian of a scalar field is zero, what we can tell about that scalar field? [closed]

What can be deduced about scalar field if its laplacian is zero.
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1answer
228 views

Can any symplectomorphism (1 Definition of canonical transformation) be represented by the flow of a vectorfield?

For this question I will use the definition that a canonical transformation is a map $T(q,p)$ from the phase space onto itself, which leaves the symplectic 2-form invariant (which is the definition of ...
4
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2answers
244 views

Difference between a vector field and a force field

In mathematics while learning about vector fields, we define a "vector field" as "a function of space whose value at each point is a vector quantity". That is, at each point in space there is a vector ...
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3answers
5k views

Can electric field be negative?

According to the equation , $$E = kQ/r^2$$ If the source charge is negative electric field produced by the charge must also be negative. My teacher said electric field can never be ...
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1answer
55 views

Special relativity- magnetic field tensor

We have a tensor associated with a magnetic field in special relativity defined as $B_{jk} = \partial _j A_k - \partial _k A_j$ and two related equations: $B^i = \frac {1}{2} \epsilon ^{ijk} B_{jk}$ $\...
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0answers
99 views

Uniqueness of A Field From Its Divergence and Curl

In my EM textbook, the author says that a vector field can be determined if we know both its divergence and curl (Griffith's Electrodynamics, 3rd edition, page 273, section 6.3.2). For example, for ...
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0answers
83 views

Question about Covariant Derivatives in General Relativity

I'm following the differential approach of Schutz' book where vectores are geometrical objects written as \begin{equation} \vec{V}=V^a\ \vec{E}_a \end{equation} Where $V^a$ are the components of the ...
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0answers
68 views

About the convective constancy of Clebsch potentials

For a fluid with non-zero vorticity, the following representation for the velocity in terms of scalar variables is well-known: $$ \vec{v}=\vec{\nabla}\phi +\beta\vec{\nabla}{\gamma} $$ It is called ...
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0answers
102 views

Vector calculus and principle of reciprocity MRI

I'm studying the principle of reciprocity for the MRI. At some point during a calculation the book states that: $$\oint d\vec{l} \cdot \left[\int d^3r' \frac{\vec{\nabla'}\times\vec{M}(\vec{r}')}{\...
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1answer
268 views

heat generation due to viscosity in a 3D fluid flow

Consider an arbitrary 3D fluid flow: $$\vec{\nu}=\vec{V}\left( \vec{x} ,t \right) \tag{1}$$ where velocity at each point $\vec{\nu}$ is a function $\vec{V}$ of position $\vec{x}$ and time $t$ (non-...
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1answer
80 views

Divergence of inverse cube law

My intuition tells me that the divergence of the vector field $$\vec{E} = \dfrac{\hat{r}}{r^3} $$ should be zero everywhere except at the origin. So I think it should be $$ \vec{\nabla}\cdot \vec{...
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1answer
220 views

Help with the Interpretation of the Closed Line Integral of Current Density Resulting from an Electrostatic Field

The line intergral of a static electric field around a closed loop is: $$ \unicode{x222E} \mathbf{E} \cdot d\mathbf{l}=0 $$ For an ohmic material, this is equivalent to: $$ \unicode{x222E} \frac{1}{\...
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1answer
141 views

How does Laplace's equation $\nabla^2U = 0$ indicate saddle points?

When I learned about saddle points I had this expression 'rt-(s^2)', where r=Dx, t=Dy, s=Dxy=Dyx. And the intuition behind why it is so was also clear. In an electric/magnetic field, in Earnshaw' ...
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1answer
58 views

If $\vec{F}=-\nabla V$ and $V ∝ 1/r^2$, then shouldn't objects fly up instead of fall down?

If the apple moves from the higher potential to lower potential $(\vec{F}=-\nabla V)$ and the closer the distance the higher potential $(V ∝ 1/r^2)$, then shouldn't the apple fly up instead of fall ...
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1answer
2k views

Vorticity of a velocity field in cylindrical coordinates [closed]

I'm currently working on a obtaining the vorticity of my velocity field $u_r, u_\theta, u_x$. I know that this is equal to the curl of the velocity field $\nabla \times u$: $$\nabla \times u = \frac{...
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1answer
89 views

Some doubts about Field Lines [duplicate]

Take electric field lines as example. In EM textbooks it is stated that The tangent of a field line gives the direction of the electric field The density of field lines is proportional to the ...
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1answer
286 views

Vector field of a simple pendulum [closed]

Classical Mechanics by John Taylor walks through an example of a skateboard on a frictionless half-pipe of radius $R=5.0$m. This is equivalent to a frictionless pendulum, I believe. The example goes ...
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1answer
264 views

The Killing vector $\chi=\partial_t+\Omega_H\partial_\phi$ doesn't look normal to the Killing horizon for a Kerr BH

As mentioned in Carroll's Spacetime and Geometry p. 244, a Killing vector is normal to its Killing horizon. With some help from the other forum, I could check this is true. (FYI, here the Killing ...
2
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1answer
486 views

Getting an intuition of a “Vector Field” in General relativity

In GR the way we define a vector field is : $$ v^a=\sum_\mu v^\mu (\partial_\mu)^a $$ The a in the superscript is the abstract index notation. I understand this attachment of a vector field if ...
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1answer
105 views

Calculus of Variations: Refractive Index Problem [closed]

The problem is as follows: "Given that the refractive index $µ(r)$ of some material equals $|∇f|$ for some function $f(r)$, show that the optical path length $\int_A^B \mu(r) dl$ between points A ...
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1answer
97 views

Approximate Killing vector field in general relativity

In this paper the authors consider an approximate Killing field $\chi$. It vanishes on a given 2 surface and its first order part is given.They say that if it obeys the Killing equation $\chi_{a;b}+\...
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1answer
127 views

Quantum particle on a ring with magnetic flux through it: Can we gauge away the magnetic field?

Consider a quantum particle on a ring and a non-zero homogeneous magnetic field perpendicular to the disk that the ring defines and is non-zero only in the inside of the perimeter of the ring. Let $\...
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2answers
2k views

Why are cubes considered Gaussian surfaces?

I've been trying to wrap my head around Walter Lewin's lecture on Gauss's law and electric flux and I can't go on without thoroughly getting this first. I think I've understood the electric flux part, ...
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0answers
150 views

Has the Helmholtz decomposition of the $\mathbf{E}$ field from the Liénard–Wiechert potentials been worked out?

If you look at Maxwell's equations for $\mathbf{E}(\mathbf{x},t)$ they split neatly into two categories. They are: \begin{align} \nabla\cdot\mathbf{E}(\mathbf{x},t)&=\frac{\rho(\mathbf{x},t)}{\...
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2answers
61 views

$B$-fields where field lines are parallel have zero magnetic tension?

The magnetic tension of a magnetic field $\vec{B}$ is given by $(\vec{B} \cdot \nabla )\vec{B}$. It is easy to see that a uniform field has zero magnetic tension. My textbook says the magnetic ...
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1answer
34 views

Finding the Changing Shape of a Moving Fluid

In this question, we won't give two hoots about forces or acceleration in a fluid. We'll just focus on velocity. We have a given continuous velocity vector field that changes with time. However, the ...
4
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2answers
305 views

Is Helmholtz decomposition inherently a non-local operation?

Helmholtz decomposition, the process for splitting a vector field into parts which have vanishing divergence and curl, plays a central role in our ability to quantize the electromagnetic field because ...
3
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0answers
168 views

Curl operator in Schwarzschild metric

I'm trying to write down the curl operator explicitly for a Schwarzschild metric in cylindrical coordinates. I am trying to use the general expression of the curl operator in orthogonal curvilinear ...
0
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1answer
79 views

Mathematical form of the magnetic force - a vector field or not?

Is the magnetic force a vector or a vector field? The magnetic force is written without arguments: $$ \mathbf F=q \mathbf u \times \mathbf B \tag{1} $$ Does it mean that $\mathbf r=(x,y,z)\in\mathbb ...