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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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3
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2answers
196 views

On the method described by Purcell for finding the magnetic field by measuring the force on a test particle

The following text is a method for finding the magnetic field as described in Purcell's Electricity and Magnetism (page $151$, the top part). Measure the force on the particle when its velocity is $...
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1answer
38 views

Applying Ampere's law in situation with non-physical E-field?

On an exam I was given this question: Suppose an electric field in a region with no current $(\textbf{J}=\textbf{0}$) is given by $\textbf{E}(t,x,y,z) = \sin(\omega t)\hat{\textbf{k}}$ and $C$ is ...
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1answer
74 views

How do we know that the actual universe has no Killing vector fields?

This article states the following: The infinitude of conserved energies constructed via Noether’s theorem suffers a startling reversal as soon as Special Relativity is superseded by General ...
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0answers
21 views

Do pseudo forces create a field?

Suppose i am travelling in an accelerated car (with constant acceleration), would there be a vector field in the space i view due to the pseudo forces? If yes what would be its nature and would it be ...
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0answers
28 views

Time-like null-like and space-like killing vector fields

If the Lie derivative of a metric tensor with respect to some vector field is zero then the vector is called killing vector (KV). The KV can be time-like, null-like and space-like. What is the ...
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2answers
194 views

What is gradient with respect to components of a position vector?

I am reading "Classical mechanics" by Goldstein, Poole and Safko, Third edition. Kindly please refer to page no 10, last paragraph. They write the subscript $i$ on the del operator indicates that ...
5
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1answer
88 views

Deriving $\nabla_\mu \nabla_\sigma \mathcal{K}^\rho=R^\rho_{\sigma\mu\nu}\mathcal{K}^\nu$

I want to derive this equation from Carroll's book. $$\nabla_\mu \nabla_\sigma \mathcal{K}^\rho=R^\rho_{\sigma\mu\nu}\mathcal{K}^\nu$$ We know that $\mathcal{K}^\nu$ is a killing vector and ...
4
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2answers
1k views

What are the generators of spherical symmetry?

The title says it all. I think this should be a pretty simple question but I just couldn't find the answer. Ok -- I'll give a bit more context to my question. I'm encountering this in the context of ...
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0answers
42 views

Vector potential, magnetic field and dipole moment due to a rotating cylinder

I've been struggling with the following problem: Consider a cylinder with height h and radius a with a homogeneous surface charge density $\sigma$ rotating about its symmetry axis with constant ...
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0answers
62 views

How to further expand $\text{grad} \left( \vec{a} \cdot\vec{b} \right ) = \vec{\nabla} \left (\vec{a} \cdot\vec{b} \right )$? [migrated]

With $\vec{a}, \vec{b}: \mathbb{R}^3 \to \mathbb{R}^3$ vector fields: I want to expand $\text{grad} \left( \vec{a} \cdot \vec{b} \right ) = \vec{\nabla} \left (\vec{a} \cdot \vec{b} \right )$. So I ...
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1answer
342 views

Transforming flat space to retarded Bondi coordinates and expanding around future null infinity

I am trying to derive my way through the paper https://arxiv.org/abs/1603.07706 and I am struggling to understand Eq.(3), where they expand the Bondi metric around Scri+. Firstly, flat Minkowski ...
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0answers
45 views

Writing del, divergence, and curl in generalized coordinates [migrated]

In three dimensional Cartesian coordinates the Hamilton operator, del, is written as $\nabla= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} ...
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3answers
49 views

Conceptual Understanding of Zero Curl in Ampere's Law

I understand that Ampere's law tells us that the current density times $\mu_0$ at some location must be equal to the curl of $\mathbf{B}$ at that location. However, conceptually this is troubling me. ...
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1answer
27 views

Torque experienced by a coplanar loop of current in a uniform magnetic field

There are a lot of posts on this already, but apparent all of them just consider some special case. I am now struggling with this more general case. Let there be a magnetic field with strength $B$. ...
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2answers
29 views

How to determine the constant of integration when solving a DE given by Ampere's law?

A cylindrical conductor with axis along the z-axis carries a current density $J\mathbf k$. From my understanding, $$ \mathbf B=\frac12 \mu_0 Jr\hat{\boldsymbol {\phi}}. $$ However, if we consider an ...
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1answer
193 views

Converting from magnetic scalar potential to magnetic vector potential

Suppose that I have a situation where I can use the magnetic scalar potential. Given the magnetic scalar potential, is there a simple way to calculate the magnetic vector potential from the magnetic ...
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0answers
35 views

Vector Calculus Problem on Gradient Cross Product [migrated]

$Problem:$ If a vector function $V=V(x,y,z)$ is not irrotational, show that if there exists a scalar function $g=g(x,y,z)$ such that $gV$ is irrotational, then $$V\cdot (\nabla \times V )=0$$ ...
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2answers
131 views

Gauss divergence theorem (GDT) in physics

Some of the statements for $GDT$ which I find in modern textbooks (both electromagnetism and multivariable calculus textbooks) are: (1) Calculus: Several variables Adams Let $D$ be a regular, ...
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0answers
48 views

How do I show that the two definitions of the curl of a vector field equal each other? [migrated]

The curl of a 3D vector field is a 3D vector itself and has two definitions - one in integral form and one in differential form. Definition 1: $$ \operatorname{curl}\vec{F}(x,y,z) \, \cdot \, \hat{n} ...
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0answers
45 views

The flux of a vector field

This is probably a basic question. I'm actually taking a class that introduces me to Maxwell's equations. I am currently trying to make sense of the Gauss's law and have some difficulty understanding ...
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1answer
1k 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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0answers
49 views

The two definitions of the divergence of a vector field? [migrated]

Now, I am aware that the divergence of a vector field, $\vec{F}$, can be defined in two ways. What I don't understand is why do these equal each other formally? Definition 1: $$\text{div}\vec{F} = \...
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0answers
15 views

Problem with relative reference frames [closed]

I came across this problem a while ago and I'd like to solve it, but I'm not even sure where to start. How would I start this problem and what formulas do I need? I'm not asking for a solution, but ...
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0answers
47 views

How does vectorization affect $\nabla$?

The homework and exercise was to prove $\nabla \times {A}$ transform as a vector, and I've solved it thorough hard algebra. However, something occurred to my mind and I have a hard time to resolve it....
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3answers
111 views

Proof for Vector Identity

I am currently studying electrodynamic and came across the following vectoridentity, but I am unable to prove it: $$ \vec{f} \times ( \nabla \times \vec{f} ) -\vec{f}(\nabla\cdot\vec{f}) = \nabla \...
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1answer
43 views

Why is the strain rate tensor 1/2(gradV+gradV^T)

$$\ \\1/2\left( \nabla\mathbf{u} + (\nabla\mathbf{u}) ^\mathrm{T} \right) \ $$ Why is the strain rate tensor the equation above?
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2answers
76 views

Why is the stress tensor simplifying like this

\begin{eqnarray} \nabla \cdot \boldsymbol \tau &=& 2 \mu \nabla \cdot \boldsymbol \varepsilon\\ &=& \mu \nabla \cdot \left( \nabla\mathbf{u} + (\nabla\mathbf{u}) ^\mathrm{T} \right)\\ &...
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1answer
67 views

Divergence of a vector multiplied by dot product [closed]

If I am correct, then $\operatorname{div} [(\vec A\cdot \vec B)\vec C] = (\vec A \cdot \vec B) \operatorname{div} \vec C + \vec C \cdot \nabla (\vec A\cdot\vec B)= (\vec A \cdot \vec B) \operatorname{...
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1answer
102 views

What was the real need of the operators of divergence and curl? [closed]

As I'm advancing my study in Electromagnetism I'm getting introduced to more mathematical operators which are exclusively used in Electromagnetism and Fluid Dynamics only. Let me try to explain ...
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1answer
52 views

Coordinate-free proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic

I would like to know if the following proof of the constancy of the scalar product of a Killing vector with the geodesic tangent vector along the geodesic is correct. I already found a coordinate-...
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2answers
98 views

In electrostatics, $\nabla\times\mathbf E=0$. Why is this?

I can understand why this is mathematically but I do not understand the actual why, like in words why this is.
3
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2answers
59 views

Proof for condition of a Force being conservative

While studying Mechanics, I came to know about a way to test whether a force is conservative. Check whether the expression for the Work done is solvable without the path of the object that is $\int \...
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1answer
466 views

Gauge fixing with vector potential: Coulomb gauge

There is something I would like to clarify with gauge fixing. In E.M, we can introduce the potential vector. As $div(\vec{B})=0$ we know that we can write $\vec{B}=\vec{curl}(\vec{A})$. But as $\...
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1answer
29 views

The flux of a vector field through a cylinder [closed]

The question is by using Gauss’ Theorem calculate the flux of the vector field $$\overrightarrow{F} = x \hat{i} + y \hat{j}+ z \hat{k}$$ through the surface of a cylinder of radius $A$ and ...
0
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1answer
264 views

Lie derivative - Problem 8.5 from General Relativity by Hughston & Tod

I was trying to solve the problem 8.5 from the textbook An Introduction to General Relativity by Hughston & Tod. Given a Riemannian manifold and a vector field $V^a$ such that $\mathcal{L}_Vg_{ij}=...
3
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1answer
504 views

What does charge-free region mean?

In my book it has been given that In a charge-free region, electric field lines can be taken to be continuous curves without any breaks what does actually the word charge free region means? And ...
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2answers
259 views

Hamiltonian with ext. vector potential - complex kinetic energy

in a given (TD)DFT code with an atomic basis set, i.e. $$ \psi(\mathbf{r},t) = \sum_i c_i(t) \phi_i(\mathbf{r}) $$ (where the non-on-site basis functions $\phi_i$ aren't necessarily orthogonal), the ...
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5answers
770 views

Intuitive methods for representation of Cartesian Coordinates in terms of Spherical Coordinates as basis [closed]

I was going through Griffith's Electrodynamics and came upon an example, where he used that, $$\cos\theta \ \hat{r} - \sin\theta \ \hat{\theta} = \hat{z} $$ Now I admit I was confused for a while ...
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1answer
147 views

Stokes Theorem with Ampere's Law

Whilst researching Maxwell's Equations (here), I found (effectively) the following pieces of logic: $$\int_S \left(\nabla \times \boldsymbol{H}\right) \cdot d\boldsymbol{S} = \oint \boldsymbol{H} \...
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3answers
1k views

Example of two linearly independent, nowhere vanishing vector fields in $\mathbb{R}^{2}$

I knew that two linearly independent and nowhere-vanishing vector fields provide a basis for the tangent space at each point in $\mathbb{R}^{2}$. Is it necessary that these two vector fields commute? ...
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2answers
54 views

Electromagnetic wave equation: can we ignore the constant of integration?

Suppose we obtain a solution for each of $\mathbf B$, $\mathbf E$ of maxwell equations in the vacuum ($\rho=0$). Clearly, for any constant vector $\mathbf k, \mathbf m$, $\mathbf {B+k}$ and $\mathbf{E+...
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1answer
23 views

Deriving potential of Continuous charge distribution using Vector Calculus

I was reading Classical Electrodynamics by J.D Jackson and stuck at a point. He considers the potential due to a dipole with charge density $\sigma$ and distance between then d such that: $$\lim_{n\to\...
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0answers
39 views

What is the meaning of a vector integral over another vector?

Reading Portis's Electromagnetic Fields: Sources and Media I came across this expression for the stored electric energy in a volume in a general medium: $$ U = \int dV \int \mathbf{E}\cdot d\mathbf{D} ...
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1answer
730 views

Counting the number of linearly independent Killing vectors for some $N$-dimensional geometry of space

In symmetric spaces(for spacetimes of Einsteinian General Relativity) we would like to find the vector space of Killing vectors($\xi^{(n)}_\mu(x)$) for the given metric tensor($g_{\mu\nu})$ at some ...
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1answer
72 views

Show that $\partial_i A_j - \partial_j A_i = \epsilon_{ijk}B_k$

Let us start from $\textbf{B}=\nabla \times \textbf{A}$ and write its components $B_k=\epsilon_{ijk}\partial_i A_j$. I want to show that $\partial_i A_j - \partial_j A_i = \epsilon_{ijk}B_k$. I can ...
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0answers
45 views

Holonomic basis

Is the following definition correct? Given a differentiable manifold $M$ and an ordered basis $\{e_j^m\}$ of the tangent space $T_m M$ with $m\in M$ (they are vectors and not vector fields). An ...
2
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1answer
487 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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6answers
8k views

Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis $i$, $j$, $k$, so by invariance nature of vectors, component of ...
2
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2answers
262 views

A simple calculation about surface gravity in classical GR

I am reading An Introduction to General Relativity Spacetime and Geometry by Sean Carroll, but simple calculations stop me. At page 245, a formula for the surface gravity is given $$\kappa^2=-\frac{1}...
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0answers
56 views

Taylor expansions giving different answers before and after applying Euler-Lagrange equation

I have the Lagrangian $$\alpha(\boldsymbol{\dot{r}} -\boldsymbol{v}(\boldsymbol{r}))^{2} + \beta \nabla \cdot \boldsymbol{v}(\boldsymbol{r}),\tag{1}$$ where $\boldsymbol{r}$ is the position and $\...