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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Contravariance and covariance of vectors

My main source of confusion is the following. Suppose I have a scalar potential $V(x,y,z)$. The electrostatic field for this potential is $ -\vec{E} =\vec{\nabla}V = \frac{\partial{V}}{\partial{x}}\...
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Magnetic field lines are closed lines?

How is it proven that the magnetic lines are closed lines? Because the divergence of $\mathbf{B}$ is zero does not sound convincing, since fields like $\mathbf{F} = a\mathbf i +b\mathbf j+c\mathbf k$ (...
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Can the electric field have closed field lines?

In electrostatics, we know that $\vec{\nabla}\times\vec{E} = 0$ and so, the field lines can't form loops. But when we have time-dependant magnetic fields, there's the Faraday-Lenz law which tells us ...
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Exercise 32 from Bernard Schutz [closed]

The exercise In Bernard Schutz's 'General Relativity, there is an exercise, I don't understand how to start with. To develop a TT Gauge, we should use the given transformation. But somehow I am ...
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How to calculate the electric field of a polarization density?

By polarization density here I just assume I have a "blob" of free positive and negative charges, and instead of describing the system with a charge density $\rho(\pmb{r})$ I want to use the ...
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Calculating work done when the lower bound of integral is greater than the upper bound

In this video, Dr. Peter Dourmashkin explained friction as an example of a force by which the work done is not path independent. In $2$$:$$50$ min of the video, when we're coming back, he said, $d\...
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2 answers
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Are there types of spacetime that have no symmetries?

We derive the most basic laws of physics from several fundamental symmetries (those from Noether's theorems, gauge symmetries, Lorentz symmetry...). But are there any types of spacetime where no ...
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Time derivative of unit velocity vector?

Let's say I have some parametric curve describing the evolution of a particle $\mathbf{r}(t)$. The velocity is $\mathbf{v}(t) = d\mathbf{r}/dt$ of course. I am trying to understand what the expression ...
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Conservation and potential with non-cartesian forces

I understand how to determine if a force is conservative from \begin{equation} \nabla\times \mathbf{F}=0 \implies \mathbf{F}\text{ is conservative} \end{equation} When $F$ is in cartesian coordinates. ...
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2 votes
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Spherical harmonics expansion: from scalars to tensors

It is well known that a scalar field on the unit sphere can be expanded in spherical harmonics, see e.g. this. I am wondering if there is a related concept for vector fields and, in general, for any ...
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How do I expand: $\langle u , \nabla\rangle u$?

I'm studying the Landau (VI) and when he "introduces" the material derivative (he is building up the continuity equation), something like this appears: $(u,\operatorname{grad})u$ (sometimes ...
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About the equation $\frac {d^2} {dt^2}\vec x(t) = \nabla \times \vec F(x(t))$. Motion in a curl vector field

I was wondering if there is a physical interpretation of ODEs of the form $$\frac d{dt}\vec x(t)=\vec y(t)$$ $$ \frac d{dt} \vec y(t) = \nabla \times \vec F(x(t))$$ (or equivalently $\frac {d^2} {dt^2}...
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Calculating divergence and flux of geodesic word lines

Given a family of neighbouring geodesic word lines, is there a way of calculating properties such as their divergence or flux? maybe by converting the tangent vectors of the world lines to a vector ...
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Electric field lines intersection with common tangent [duplicate]

We say that electric field lines do not intersect, because that cannot be two directions of electric field. Then can electric field line intersect in the way shown below? This image shows that two ...
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4 votes
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Generalizing Fermi-Walker Derivative/Transport to General Vector Bundles

The usual definition of the Fermi derivative (eg as given in Hawking and Ellis) is to consider a Lorentzian manifold $(M,g)$ and a unit timelike curve $\gamma$, a smooth vector field $X$ along $\gamma$...
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2 answers
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Meaning of potentials in Helmholtz decomposition

I am trying to understand the meaning of the potentials $U$ and $A$ arising in the Helmholtz decomposition of a vector field $\vec{V}$: $\vec{V} = \nabla U + \nabla \times A$ Let's focus on the curl-...
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Will a surface perpendicular to the electric field always be equipotential

In the figure attached there are four different electric field vectors with different magnitudes, point away from A. My question is will C be an equipotential surface? i.e. will the potential ...
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Killing field with a time dependent metric (including $g_{00}$)

Let's suppose that (in cartesian coordinates) $$g_{\mu\nu}=diag(-f(t)^2, g(t)^2, g(t)^2, g(t)^2).$$ So that all of the components of the metric are dependent on coordinate time. If we produce a ...
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Rotation and Killing vectors in Minkowski spacetime

There are $3$ Killing vectors in the Minkowski spacetime related to the conservation of angular momentum. Sometimes it is mentioned that it is related to the rotational symmetry of the spacetime. But ...
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1 answer
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I'm confused about the number of Killing vectors in Schwarzschild metric

I'm trying to perform a calculation to derive the Killing vectors of a spherically symmetric metric (so I use the Schwarzschild metric without loss of generality because the Birkhoff theorem tells me ...
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Can Field lines touch? [duplicate]

A doubt about field lines Can electric field lines TOUCH somewhere before infinity? I have learnt that it cannot intersect but can it be like $x$-axis for $y=x^3$ since at zero the direction of fields ...
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Working with decomposition of fields

I'm trying to follow a text I found online. The author decomposes EM fields such $$ \mathbf{E} = \sum_{lm}\left(f_l(r) \mathbf{Y}_{lm} - i \frac{l(l+1)}{r} g_l(r) \mathbf{\Psi}_{lm} - i\left(\frac{d ...
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Basis Vectors as Partial Derivatives Issues

I have been introduced a number of times to people defining vectors as derivatives of a curve, with basis vectors as partial derivatives, but I have several issues with this that make this formalism ...
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1 answer
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Gravitational field strength between equipotential lines

Is the gravitational field strength between two equipotential lines the same at all distances? For example, in the image, does point P experience the same gravitational field strength as a point ...
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1 answer
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What happens to the electric field lines in a high resistance conductor?

When a conductor is connected to a potential difference, an electric field occurs inside the conductor. We know in a high resistance conductor (for example a 20m long thin carbon rod connected to a 1....
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Applying Divergence to query moving points

Problem statement :- There is a moving source($s$) and other moving points ($p_{1}.... p_{n}$). There are fixed obstacles and a fixed destination point($d$). In each time step I have to query "Is ...
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How to Identify divergence and curl graphically

I came across this solution to a problem in Griffith's Introduction to Electrodynamics where we had to construct a non uniform field whose curl and divergence are zero. The picture is the equation of ...
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6 votes
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String theory: Conformal invariance and Conformal Killing Vectors

I am confused by the relation between the invariance of the Polyakov action under conformal transformations and the Conformal Killing Vectors (CKVs) appearing during the process of quantization. Let ...
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Why does a ${\bf B}$-field follow a high $\mu$ core in an electromagnet?

I'm having a hard time trying to understand why ${\bf B}$-field lines tend to follow the path of a high $\mu$ material. Below is what actually happens when you apply a coil with some current around a ...
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How to differentiate the electric field on the surface of a conductor?

Given a conductor, let $n$ be the unit normal field of the surface and $E$ the magnitude of the electric field. How to understand the expression $$\frac{\partial E}{\partial n}$$ on the surface of the ...
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How is the divergence of a unit timelike vector related to expanding space?

Just for fun I'm trying to understand spacetime with an expanding universe from a 3+1 point of view (yes, similarly to the ADM formalism). If I consider our spatial 3-surface $\Sigma$ as embedded ...
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Maxwell's eq-meaning of del's cross and dot product?

In maxwell's eq there is del whose cross and dot products exist. So what is del in cross vs dot product. What's the difference when it's just a partial differential operator.
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7 votes
3 answers
582 views

Vector Potential that vanishes outside infinite solenoid

Consider the magnetic field $\vec{B}$ generated by an infinite solenoid on the $z$-axis with radius $R$. Then $$\vec{B}(r)=\begin{cases} B_z \hat{z} & \text{ if }r<R, \\ 0 & \text{ ...
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Unique solutions to divergence equation?

A very common problem in physics is to search for a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $$ \nabla \cdot f = g $$ for some given source density $g: \mathbb{R}^n \rightarrow \...
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Does the space between field lines inside the conductor increase with distance and eventually disappear?

In practical scenarios where resistance is present, what happens to the electric field lines within a conductor? Practically, we say there is no flow of current in a high resistance conductor with a ...
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Determinating the position and charges of particles from electric potential

Special configuration of charges gives us electrical field with scalar potential described by equation $$U(\vec{r})=q\ln\sqrt{\frac{r-\vec{a}\cdot\vec{r}}{r+\vec{a}\cdot\vec{r}}}$$ where $q$ is ...
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5 votes
3 answers
328 views

Why do the electric field lines not originate from a positive charge in the following situation?

Consider two fixed positive point charges, each of magnitude $Q$ placed at a finite distance apart. Let point $O$ be the midpoint of the two charges. We can see that the electric field at $O$ is zero, ...
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3 answers
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Do $\boldsymbol{\nabla}\cdot \mathbf E=0$ and $\boldsymbol{\nabla}\times\mathbf E=\boldsymbol 0$ imply uniform $\mathbf{E}$?

Do the equations $$\:\boldsymbol{\nabla}\cdot\mathbf{E}=0 \qquad \boldsymbol{\nabla}\times\mathbf{E}=\boldsymbol 0\:$$ imply that the vector field $\mathbf{E}$ is uniform? I think yes, since by ...
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1 answer
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Question regarding vector calculus [closed]

Question regarding vector calculus: Mathematically, what properties does this vector field have, in order for its line integral to be equal to the area enclosed by that curve?
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Condition on particle momentum crossing the Kerr horizon

Around the equation (7.143) in [Carroll] Lecture Notes on General Relativity, the note talks about momentum $p$ of a particle entering the ergosphere and crossing the outer horizon. The claim is that ...
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Ways to represent the metric tensor using a vector field

I was wondering if there were any ways of representing the metric tensor with a vector or scalar field and started calculating some potential ways. I recently stumbled across the equation $$\widetilde{...
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How is the parity transformation defined? Especially for vector or tensor fields?

If I have a scalar field \begin{align} f: \mathbb{R}^3 &\rightarrow \mathbb{C}\\ (x, y, z) &\mapsto f(x, y, z) \end{align} We can define an operator $P$ that takes a function like $f$ and ...
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How to derive the commutation coefficient from coordinate basis (GR)?

Given two vectors $U$ and $V$ \begin{align} [U, V] &= [U^\mu e_{(\mu)}, V^\nu e_{(\nu)}] \tag{1} \label{eq1} \\ &= [U^\mu e_{(\mu)}(V^\nu) - V^\mu e_{(\mu)}(U^\nu)]e_{(\nu)} + U^...
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2 answers
48 views

Tensor gradient order of terms

I am working on some algebraic manipulation with the compressible Navier-Stokes equations, specifically this form, screenshotted from Wikipedia: I'm confused by the gradient operator being applied to ...
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0 answers
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Spin, Orbital and Total Angular Momentum For Classical Vector Fields

I ask this question motivated by trying to understand vector spherical harmonics and find or come up with an elegant abstract derivation of their form. Suppose we have a 3D field of 3D vectors: $\...
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In Griffiths' Electrodynamics, Appendix A on Curl and Stokes' Theorem, is the books depiction of an infinitesimal loop correct?

In Griffiths' Electrodynamics, there is a section in Appendix A where he sketches a proof of Stokes's theorem. Consider a vector function $\mathbf{A}=A_u\mathbf{\hat{u}}+A_v\mathbf{\hat{v}}+A_w\mathbf{...
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In Griffiths' Electrodynamics, how does he calculate the contribution of two opposing faces of infinitesimal rectangular volume to a surface integral?

In appendix A of Griffiths' Electrodynamics, he sketches proofs of certain theorems of vector calculus. My question is related to this question, indeed the latter concerns the exact passage of the ...
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1 vote
1 answer
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Justifying that $B$ can always be represented as a curl of a vector field [duplicate]

I'm trying to justify a claim from Feynman's 14th lecture, In electrostatics we saw that (because the curl of $E$ was always zero) it was possible to represent $E$ as the gradient of a scalar field ϕ....
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2 votes
1 answer
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Laplacian of a scalar can be derived using definitions of dot product and gradient, can the Laplacian of a vector be derived or is it a definition?

In the first chapter of Griffiths' Electrodynamics, he introduces some math that will be used. There is a section on second derivatives. One type of second derivative is $\nabla \cdot(\nabla T)$, ...
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How to prove practically that something is not a vector field?

I have some intuitive understanding about the definition of a vector field, eg. that a vector field has to be invariant under coordinate rotations. I can also find the following short-hand equation ...
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