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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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1answer
33 views

Vector Calculus Problem on Gradient Cross Product

$Problem:$ 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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43 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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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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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
45 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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1answer
35 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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3answers
107 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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2answers
74 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
99 views

What was the real need of the operators of divergence and curl?

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
60 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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2answers
97 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.
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2answers
58 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
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 ...
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5answers
764 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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2answers
51 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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42 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 ...
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1answer
71 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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1answer
50 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

Divergence of a Vector Field - Surprising Result [duplicate]

I'm following the text Introduction to Electrodynamics by Griffiths, and I came across the following in an in-text problem: Sketch the vector function v = $\frac{\boldsymbol{\hat{\mathbf{r}}}}{r^2}$...
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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 $\...
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1answer
52 views

Trace of second-order tensor and its invariance under coordinate transformation

Let's consider an arbitrary scalar field. If I act twice on the scalar field with a gradient operator, I will obtain second-order tensor. If I will take a trace of this tensor, I will obtain another ...
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3answers
115 views

Can the flow be irrotational if the viscous forces act on fluid?

I tried to answer the question only using the definitions and the Navier-Stokes equation: $$\rho \frac{Dv}{Dt} = -\nabla P +\rho g -\mu[\nabla \times(\nabla \times v)] $$ In my opinion if the ...
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1answer
68 views

Is there a generalization of ${{\partial }_{\alpha }}\left( \sqrt{-g}{{V}^{a}} \right)=\sqrt{-g}{{\nabla }_{a}}{{V}^{a}}$ for arbitrary connection?

I am studying general relativity and I am trying to understand how to perform variation of the Einstein–Hilbert action with respect to the metric ${{g}_{\mu \nu }}$ and an arbitrary connection ${{\...
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1answer
29 views

Physical meaning of the convection term in the momentum equation of acoustic wave

In deriving the acoustic wave equation, the momentum equation is used. $$\frac{\partial \mathbf{u}}{\partial t}+ (\mathbf{u}\nabla)\mathbf{u}=-\frac{1}{\rho} \nabla p$$ Intuitively, the convection ...
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4answers
59 views

Electric field line density : Theory vs Reality

I've already went through this post. Yet, I still can't understand the meaning of "density" of electric field lines whose number is, in reality, infinite. One of the answers , for instance, states ...
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1answer
71 views

How to prove that $\vec{E}$ is an intensive property?

In my homework I had a question to sort a few variables into intensive properties and extensive properties. I wrote that $\vec{E}$ (electric field) is an extensive property, thinking of a situation ...
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2answers
64 views

Globally constant vector field in a curved spacetime

Is it possible to define a globally constant vector field in a curved spacetime, that is a vector field for which the covariant derivative vanishes along every world line? The vector field $V^{\mu}=0$ ...
2
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2answers
64 views

Magnetic fields and closed loop

It is well known that there don't appear to be magnetic poles. In Maxwell's equations this has the implication $$ \nabla \cdot \mathbf{B} = 0 $$ and results in the statement "the magnetic field forms ...
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1answer
48 views

Surface Tension of Floating Object

I was reading this thesis about surface tension and its role in floating bodies. I couldn't quite understand at page 10 how the author applied the 2D divergence theorem outside the region bounded by ...
2
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1answer
54 views

How can surface integral's contribution goes to zero but volume's does not in this particular derivation?

When we derive the formula for energy of a continuous charge distribution $\rho$ using this equation $$W = \frac{1}{2}\int\rho V \text d\tau$$ with $V$ being the electric potential, we get this ...
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0answers
44 views

Covariant derivative with respect to commutator

I have some confusion with the notion of $\nabla_{[A, B]}\bf{v}$, that expression, with a commutator of vector fields as the subindex of the connection appears for instance in the definition of the ...
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0answers
24 views

Magnetic Field Lines Vs Magnetic Vector field

I am studying electromagnetic theory and when I started researching the history of conventions used in magnetic interactions I could not get them. The basics of how they modelled the magnetic ...
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1answer
50 views

Understanding of Gauss law using vector fields

I was going through the conventions and terminologies followed to describe the magnetic interactions. I understood that the field lines are just a simpler representation of the magnetic interaction ...
2
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1answer
53 views

What is the gravity on a “partial” ringworld?

This was inspired by https://worldbuilding.stackexchange.com/questions/149706/life-on-the-broken-ring-an-issue-of-size. Let's say I have a part of a Ringworld (see link for specifications). ...
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1answer
39 views

Maxwell Laws Summary Diagram - Suggestions that I am missing? [closed]

I have been going through a summary book of Maxwell's equations and hope I have organised this correctly but I think perhaps I am missing things important prompts that I could add? Image below Thanks ...
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1answer
27 views

Interpretation of surface integral of vector field over surface

Is it correct to interpret the surface integral of a vector function $\mathbf{v}$ over four sides of a cube as the rate of flow of fluid (in mass per unit time) that would flow out of the cube when ...
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1answer
73 views

Partial Integration and the Levi-Civita Symbol

I'm currently working through the book Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized ...
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0answers
66 views

Killing vectors in an static space-time

How can I show that a given space-time is static, i.e. exists a time-like Killing vector $\xi = \partial_0$ that $\partial_0 g_{\mu \nu} = 0$ (Killing eq.) and $g_{0i}=0$, if and only if the relation $...
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1answer
57 views

What is electric field vector? [closed]

What is electric field vector? How to find out the Electric Field vector at a point on a equipotential surface. Please explain by giving an example.
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1answer
82 views

Is the Lie derivative along the normal well-defined?

This question is cross-posted at https://math.stackexchange.com/q/3274757/247251 Let $(\Sigma, q)$ be a non-degenerate submanifold of a Lorentzian manifold $(M,g)$. Let $N$ be the section of $T\Sigma ...
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4answers
247 views

Divergence of $\frac{ \hat {\bf r}}{r^2}$ , what is the 'paradox'?

I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2}$ , now from the book, Griffiths says: Now what is the paradox, exactly? ...
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3answers
207 views

Are there cases where $\nabla\cdot\iiint\frac{\mathbf{J}(\mathbf{x}')}{\left|\mathbf{x}-\mathbf{x'}\right|}\mathrm{d}V' \neq 0$?

In Jackson's Classical Electrodynamics, Section 5.4 (Vector Potential), the author seems to assume that because $\nabla\cdot\mathbf{J} = 0$, the following holds for the current density (where the ...
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0answers
45 views

longitudinal and transverse components in higher dimensions

I am familiar with the Helmholz decomposition of a vector field in three dimensions: $$\vec{V}=\vec{\nabla}\wedge\vec{A}+\vec{\nabla}\phi$$ But I am interested to show that something similar can be ...
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0answers
34 views

Killing Tensor of Friedman-Robertson-Walker Metric

I would like help showing that the tensor, $$K_{\mu\nu}=a^2(g_{\mu\nu}+u_\mu u_\nu),$$ where $u^\mu =(1,0,0,0)$, is a Killing tensor of the spatially flat FRW metric, $$ds^2=-dt^2+a(t)^2\left(dr^2+d\...
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1answer
58 views

Deriving Jefimenko's equations in Fourier space

From the Fourier-transformed Maxwell equations we have, with some algebraic manipulation, $$\mathbf{E}=\frac{1}{|k|^2}\left[\mathbf{k}\frac{\rho}{\epsilon_0}-\mathbf{k}\times k_0\mathbf{B}\right]$$ $$\...
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0answers
49 views

Quantum mechanics in phase space - what are coordinate components?

I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum ...
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1answer
66 views

Calculating $[H,\vec r]$ [closed]

I'm doing Griffiths' Introduction to Quantum Mechanics. In a question, it introduces the modified Schrodinger equation in which the Hamiltonian, $$ H~=~-\frac{\hbar^2}{2m}\nabla^2 + V $$is replaced by ...