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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46 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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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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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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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
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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
114 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
100 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
100 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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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
32 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
797 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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59 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
26 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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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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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
78 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
63 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
121 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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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
94 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
234 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
73 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
37 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
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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
139 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
85 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 $V^{\mu}=0$ ...
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2answers
145 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
64 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 ...
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1answer
60 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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65 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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30 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
55 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 ...
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1answer
57 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
41 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
36 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
83 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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76 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
61 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
100 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
482 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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219 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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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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1answer
140 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
69 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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50 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 ...
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2answers
86 views

Misunderstanding of lowering indexes using Euclidian metric

One may define a vector field in $R$ and see how its components transform under a basis transformation. $ v= v^{u}\partial _{u} $ In principle, the components transform as contravariant such that ...
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1answer
132 views

Change of variables in gradient

Take two coordinates with $\mathbf r$ and $\mathbf r'$ and take a function $f(|\mathbf r - \mathbf{r'}|)$. In many electromagnetism derivations I see a conversion like this $$ \nabla_r f(|\mathbf r - \...
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3answers
188 views

Why do my books introduce the equation $\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_0}$ without showing partial derivatives of $\mathbf{E}$ exist?

In electromagnetism (electrostatics), we often come across the equation $\nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_0}$. In order for this equation to be meaningful, $\mathbf{E}$ must be a ...
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1answer
70 views

Is velocity of a fluid the gradient of something physically significant?

For incompressible flow, $$\nabla\cdot \mathbf v=0.$$ That means $\mathbf v$ got to be the gradient of some scalar field. How can I find the scalar field? Is it physically important?

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