# 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.

1,075 questions
Filter by
Sorted by
Tagged with
141 views

238 views

### 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 ...
38 views

### 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 ...
21 views

### 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. ...
52 views

### 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 ...
1 vote
45 views

### 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 ...
58 views

46 views

### 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 ...
18 views

### 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 ...
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, ...
105 views

### 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 ...
36 views

### 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?
26 views

### 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 ...
114 views

### 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{...
65 views

### 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 ...
60 views

### 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^...
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 ...
28 views

39 views

### 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 ...
1 vote
48 views

### 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....
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 $\nabla \cdot(\nabla T)$, ... 