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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24 views

Covariant Derivative and energy momentum tensor

In this reference https://arxiv.org/abs/hep-th/0307199 pag.60, it is said that it is possible to find an infinitesimal spacetime diffeomorphism (a vector field) $X_{\nu}$ independently to its ...
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Why is the magnetic field dot producted in the integral version of Ampere's circuital law?

you know amperes circuital law? Well in that equation there's a dot product between the magnetic field B and a length element dl...why is that? I mean its not like the magnetic field can be at an ...
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1answer
47 views

Killing vectors and isometry

Let $X=x\partial_{t}+t\partial_{x}$ and $Y=y\partial_{t}+t\partial_{y}$ be Killing vectors on Minkowski $(-,+,+,+)$. It can be shown that $[X,Y]$ is also Killing. I get the following: \begin{equation} ...
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Name this Vector Calculus Theorem

There is an important theorem in vector calculus that says $\boldsymbol{\nabla}\boldsymbol{\cdot}\mathbf{G}\boldsymbol{=}0$ (where $\mathbf{G}$ is some differentiable vector function) implies and is ...
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Determine the direction of electric and magnetic field in for plane EM wave

A problem states that Measurement of the electric field (E) and the magnetic field (B) in a plane-polarized electromagnetic wave in vacuum led to the following: $$ \begin{array}{ll} \frac{\partial E}{...
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Confused how the expression like $ \nabla \cdot (\rho \textbf{v} \otimes \textbf{v}) \cdot \textbf{v} $ is expanded [migrated]

I'm quite confused to figure out how this expression is expanded: $ \nabla \cdot (\rho \textbf{v} \otimes \textbf{v}) \cdot \textbf{v} \stackrel{?}{=} \nabla \cdot [\rho (\textbf{v} \cdot \textbf{v}) \...
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Tangent lift of a vector field (which is not tangent to a trajectory of a dynamical system)

I'm studying the lifting of the dynamics to a carrier manifold from the configuration manifold. I'm using the tangent bundle as carrier manifold and the canonical lift, which associate a curve in $TQ$ ...
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A question about transuvection in Kerr spacetime

We know there are Killing vectors in Kerr spacetime. I wonder when doing transuvection on Killing vector, like $\left(\frac{\partial}{\partial t}\right)^\mu \bullet (dt)_\mu$ why it equals to $g_{tt}$ ...
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Invariance of inner product under Poincare transformation

The Poincare transformation reads, $$x\rightarrow x^\prime=\Lambda x +a $$ The scalar product is preserved under Lorentz transformation. However I do not see how it is preserved under the more general ...
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Why is the divergence of curl expected to be zero? [migrated]

I was reading the proof of Helmholtz decomposition theorem where I found the relation between the rotational and the irrotational fields are not symmetric. And by that I mean if the divergence of the ...
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Are gravitational field lines really never used and the concept fundamentally un-useful in any context beyond classical Newtonian gravity? [closed]

Since Where do gravitational field lines go exactly? We know where they start, but was closed and further answers thereby prevented, all we have there is that in Newtonian gravity they go to infinity ...
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Is the Divergence of a Vector Field Defined by a (Positive) Point Charge Negative?

I was wondering whether the divergence of a vector field which is defined by a (positive) point charge is positive, zero, or negative everywhere. It is assumed that the charge is at $(0,0,0)$. ...
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Where do gravitational field lines go exactly? We know where they start, but [closed]

When we study electrostatics we have the pleasure of both starting and terminating electric field lines on opposite charges. Termination behavior of gravitational field lines at one end on small ...
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Physical interpretation for Helmholtz's theorem

Maxwell's four equations are enough to specify the fields (electric and magnetic) in a region, once you specified the charge density and current density. Maxwell's four equations basically give you ...
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Behavior of $r$-component of a $\mathbf{B}$-field inside cylindrical magnet

I have a cylindrical permanent magnet with uniform magnetization $\mathbf{M}=\mathbf{a_z}M$, length $L$ and Diameter $D$. The magnet has its center in the origin. So there is a length $L/2$ on each ...
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What is the drawing scheme of the parallel transport of a vector?

Please help me understand (geometrically) how is the parallel transport of a vector performed (along the surface of a sphere along a given path). Consider the parallel transport of a vector from the ...
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What do primed coordinates refer to in Griffith's section on Helmholtz Theorem?

In his section on Helmholtz theorem Griffith uses primed coordinates in the integrals that define $U$ and $\mathbf{W}$. Now, I understand that primed coordinates are used for sources (charges and ...
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Geodesic equation and Lie derivation problem

I have a metric: $$g_{ab}=\text{diag}(-1,a(t)^2,r^2,r^2 \sin^2(\theta))$$ where $$a(t)^2=\frac{3}{B}\cosh^2(\sqrt{\frac{3}{B}}t)$$ where $B$ is constant. Also a tangent vector given: $$k^\mu=(a(t)^{-1}...
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What's the index (or topological charge) of this vector field image?

I am doing some research in a condensed matter system, and found this Berry curvature / vector field configuration that is unusual. I cannot find another example of something similar, either from ...
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176 views

David Tong's passive transformation of the fields is wrong

David Tong's definition of active transformation is clear. Under active transformation coordinates (basis vectors) are not changed but rather the field is. I denote the old and new fields as $\phi$ ...
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What is the magnitude of the curl of a electric field? and in general from a vectorial field?

I found the following result in a book: but I do not understand what is the meaning of the magnitude of the curl of a vectorial field? and how is this related with the amount of spatial oscillations.
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If a cosine wave satisfies Maxwell's equations then how does a sine satisfy the equations as well?

Say the real part of $$\tilde{\mathbf{E}}(z, t)=\tilde{\mathbf{E}}_{0} e^{i(k z-\omega t)}$$ satisfies all Maxwell's equations. Then how can we say the imaginary part satisfies the equations as well? ...
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Killing vectors in Minkowsky Metric

So I was in the process to find the Killing vectors for the Minkowsky Metric and I stumbbled into a material that does a different procedure at the very end of the process, in comparisson to usual ...
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1answer
43 views

How is potential difference defined across a resistor with time varying current

From this discussion How can we define a potential for a moving charge? we know that we cannot define a scalar potential (as in electrostatics) in the case of moving charges as described by G. Smith: ...
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How can we define a potential for a moving charge?

Say a charge is moving in space. Ignoring relativistic effects, how can we define a scalar potential for its electric field ? My thoughts are that we can define the potential in exactly the same way ...
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41 views

Will any solution to the wave equation be a wave in reality?

In the mathematical sense, a wave is any function that moves. In that sense we can consider that any function that complies with the wave equation (let's consider in one dimension to simplify things) ...
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1answer
67 views

How to find acceleration from velocity, coefficient of kinetic friction and radius of curvature [closed]

I've been going through the various sections of my Engineering Dynamics HW and I've been struggling to solve this problem for a while: A car is travelling at a speed of 30 m/s at the top of a hill at ...
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2answers
61 views

Why does Gauss's law apply to any shape of a closed surface?

What seems to incredibly bother me is why Gauss's law applies to any shape of a closed surface. Moreover, the fact that the electric flux is proportional to the enclosed charge is by many sources ...
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3answers
161 views

Why can a force field only be conservative if it is spherically symmetric?

I saw in my textbook that a field can only be conservative if it happens to be spherically symmetric. Why is this so? Is there a good proof for this?
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How to find the Taylor expansion of $\vec{r}/r^3$?

I want to show that the Taylor expansion of $\frac{R\vec{e_1}-\vec{y}}{|| R\vec{e_1}-\vec{y} ||^3}$ at $\vec{y}=0$ is equal to $\frac {\vec{e_1}}{R^2}+\frac{3y_1 \vec{e_1}-\vec{y}}{R^3} + O(y^2)$. I ...
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57 views

Proof that the electric field in a dielectric decreases by a factor

We have a linear homogeneous dielectric material half filling a parallel plate capacitor. It's said that the field inside is reduced by a factor ,so we have $\mathbf {E}=\frac{1}{\epsilon_{r}} \...
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34 views

Killing vector index manipulation

I was doing some problems of the book "Problem Book in Relativity and Gravitation by A. Lightman, R. H. Price" and on problem 10.14 I dont understand why they say: $\xi^{}_{\gamma;\beta}\xi^{...
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Contradiction in calculating electric field outside of a dielectric material embedded with free charges

Let's say we have a region filled with a linear homogeneous dielectric, filled with some free charge density $\rho_{f}$ such that outside of this region the electric field is zero. Then we can write ...
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Similarity between electric displacement and electric field in a linear homogeneous dielectric

Griffiths says ...if the space is entirely filled with a homogeneous $^{10}$ linear dielectric: in this rather special circumstance we have $$ \boldsymbol{\nabla} \cdot \mathbf{D}=\rho_{f} \quad \...
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Can I use the Gauss divergence theorem in a region whose divergence blows to infinity on its surface boundary?

Say we have a vector function $\vec{D}$ defined in some region on whose boundary its divergence goes to infinity and inside we have $\nabla \cdot \vec{D}=\rho$. Then is it valid to use the Gauss ...
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On the continuity of the normal component of the diffusion flux

Is the normal component of diffusion flux is always continuous? I know the continuity at any surface would mean the amount of fluid that is entering through the surface is the same amount that is ...
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1answer
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How to compute the divergence of a measured vector field?

The divergence a 2D vector Field $\mathbf{F}(x,y) = F_x(x,y)\, \hat{i} + F_y(x,y)\, \hat{j}$ is defined as $$\mathrm{div}\,\mathbf{F} = \bigg( \dfrac{\partial F_x}{\partial x} + \dfrac{\partial F_y}{\...
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Navier Stokes: $(u⋅∇)u$ vs $u⋅∇u$

I can find this term stated both ways in different literature. Are they equivalent? It's weird because the dot is a dot product in (u⋅∇), but ∇u being a gradient of a vector field, would (presumably) ...
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49 views

How do i know how to draw these type of graphs? [closed]

For part a and b the answers are fig.(a) and fig(b) respectively. Can anyone explain me the graphs in more details? Like for a why is there lines in graph on the 2nd and 3rd quadrants and why is the ...
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Isn't the following addition wrong on manifold as done in Frankel book?

In ch-$4$ when calculating expression of Lie derivative using Hadamard's Lemma before $(4.4)$ Frankel's do following manipulation: $$\lim_{t\rightarrow0}\frac{\textbf{Y}_{\phi_tx}(f)-\textbf{Y}_x(f)}{...
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45 views

Conformal vector fields in $m$-dimensional Euclidean manifold

A vector field $X=X^\mu\partial_\mu\in\mathfrak{X}(M)$, where $M$ is a (pseudo-)riemannian manifold with a generic metric tensor $g_{\mu\nu}$, is a conformal Killing vector field if the conformal ...
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Why can't an electric field line suddenly break?

My book has the following question in it as an exercise: An electrostatic field line is a continuous curve i.e. a field line cannot have sudden breaks. Why not? I cannot seem to be able to reason ...
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Paths in phase space can never intersect, but why can't they merge?

Page 272 of No-Nonsense Classical Mechanics sketches why paths in phase space can never intersect: Problem: It seems to me this reasoning only implies that paths can never "strictly" ...
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Does electric field obey the triangle law of vector addition and subtraction?

I know that electric field strength is force per unit charge but what I have not yet understood properly is that how electric field can obey the laws of vector addition and subtraction excluding the ...
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21 views

Why is work done in vector fields the integral of the dot product of force and velocity wrt time?

I’ve recently started on a course on engineering calculus and I was wondering why work done is calculated as the integral of the dot product of force vector and derivative of the position vector wrt ...
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98 views

Proof of a vector calculus identity

In https://arxiv.org/abs/hep-ph/0010057 the following vector calculus equality is claimed without proof although in note [4] the cryptic comment is made that "The relation is essentially the ...
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1answer
69 views

Treating expressions of the form $\vec x \cdot \nabla \vec y$: what is the order of operations?

This comes up in the context of a homework assignment. We're given the Euler equations for invisicid fluid flow. The variables at play: $p=p(x,y,z,t)$ is pressure $\rho = \rho(x,y,z,t)$ is mass ...
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1answer
78 views

Calculating the gradient of the dot product of two vectors

I'm trying to calculate $\vec\nabla(\vec k.\vec r)$ where $\vec k =k_x \hat{i}+k_y\hat{j}+k_z\hat{k}$ is a constant vector and $\vec r=x\hat{i}+y\hat{j}+z\hat{k}$ is the position vector. I tried doing ...
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38 views

How do we calculate the directional derivative of a static vector field? (If there is such a thing.)

So, for a static scalar field $T(x,y,z)$, the derivative along $d\vec l$ is given by $$\frac {dT}{|d\vec l|} = |\vec \nabla T| cos\theta$$where $\theta$ is the angle between $\vec \nabla T$ and $d\...
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101 views

Continuity equation in quantum mechanics - Verification Sakurai 2.7.30 [closed]

I am trying to verify Equation 2.7.30 from Sakurai's "Modern Quantum Mechanics" 2ed. The bottom line of my question is: $?? \psi^{*}\vec{A}\cdot\nabla\psi+\psi\vec{A}\cdot\nabla\psi^{*}=0 ?? ...

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