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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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### Understanding scalar and vector fields

I know the definitions of vector and scalar field but I don't know what is meant by them. Suppose a scalar field is given by $\phi(x,y,z) = 3xyz$. Then what does it mean? What is the relationship ...
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### Difference between physicist's vector and mathematician's vector

Mathematically a vector is defined as an element of vector space which obeys certain properties. While reading about the special theory of relativity, I came to know about another definition of ...
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### How is dot or cross product possible using the del operator?

Yesterday in class my teacher told me that the del operator has a direction but no value of its own (as its an operator). So it can't be called exactly a vector. But in vector calculus we see that div ...
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### Why is $\frac{\vec{r}}{r^3} = \frac{1}{r^2}$?

I know it's surely a beginner's question but I don't see why you can write \begin{align} \frac{\vec{r}}{r^3} = \frac{1}{r^2}\cdot \frac{|\vec{r}|}{r} \end{align} Could someone explain it please? It ...
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### How electric field lines are defined?

I am having confusion understanding that how for finitely many fields lines in the space, the intensity of a field is proportional to the number of field lines passing through a surface area? Also, ...
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### Problem with the Landau gauge

I'm having a very simple problem which probably has an equally simple answer. I'm following the wikipedia article: https://en.wikipedia.org/wiki/Landau_quantization We have a uniform magnetic field ...
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### Curl and divergence [duplicate]

I am trying to understand curl and divergences in a more intuitive manner, especially the curl. And is curl a surface phenomenon, if yes then how?
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### Uniqueness of A Field From Its Divergence and Curl

In my EM textbook, the author says that a vector field can be determined if we know both its divergence and curl (Griffith's Electrodynamics, 3rd edition, page 273, section 6.3.2). For example, for ...
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### Question about Covariant Derivatives in General Relativity

I'm following the differential approach of Schutz' book where vectores are geometrical objects written as \begin{equation} \vec{V}=V^a\ \vec{E}_a \end{equation} Where $V^a$ are the components of the ...
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### About the convective constancy of Clebsch potentials

For a fluid with non-zero vorticity, the following representation for the velocity in terms of scalar variables is well-known: $$\vec{v}=\vec{\nabla}\phi +\beta\vec{\nabla}{\gamma}$$ It is called ...
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### Some doubts about Field Lines [duplicate]

Take electric field lines as example. In EM textbooks it is stated that The tangent of a field line gives the direction of the electric field The density of field lines is proportional to the ...
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### Vector field of a simple pendulum [closed]

Classical Mechanics by John Taylor walks through an example of a skateboard on a frictionless half-pipe of radius $R=5.0$m. This is equivalent to a frictionless pendulum, I believe. The example goes ...
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### The Killing vector $\chi=\partial_t+\Omega_H\partial_\phi$ doesn't look normal to the Killing horizon for a Kerr BH

As mentioned in Carroll's Spacetime and Geometry p. 244, a Killing vector is normal to its Killing horizon. With some help from the other forum, I could check this is true. (FYI, here the Killing ...
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### 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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### Calculus of Variations: Refractive Index Problem [closed]

The problem is as follows: "Given that the refractive index $µ(r)$ of some material equals $|∇f|$ for some function $f(r)$, show that the optical path length $\int_A^B \mu(r) dl$ between points A ...
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### Why are cubes considered Gaussian surfaces?

I've been trying to wrap my head around Walter Lewin's lecture on Gauss's law and electric flux and I can't go on without thoroughly getting this first. I think I've understood the electric flux part, ...
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### Has the Helmholtz decomposition of the $\mathbf{E}$ field from the Liénard–Wiechert potentials been worked out?

If you look at Maxwell's equations for $\mathbf{E}(\mathbf{x},t)$ they split neatly into two categories. They are: \begin{align} \nabla\cdot\mathbf{E}(\mathbf{x},t)&=\frac{\rho(\mathbf{x},t)}{\...
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### $B$-fields where field lines are parallel have zero magnetic tension?

The magnetic tension of a magnetic field $\vec{B}$ is given by $(\vec{B} \cdot \nabla )\vec{B}$. It is easy to see that a uniform field has zero magnetic tension. My textbook says the magnetic ...
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### Finding the Changing Shape of a Moving Fluid

In this question, we won't give two hoots about forces or acceleration in a fluid. We'll just focus on velocity. We have a given continuous velocity vector field that changes with time. However, the ...
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### Is Helmholtz decomposition inherently a non-local operation?

Helmholtz decomposition, the process for splitting a vector field into parts which have vanishing divergence and curl, plays a central role in our ability to quantize the electromagnetic field because ...
Is the magnetic force a vector or a vector field? The magnetic force is written without arguments: $$\mathbf F=q \mathbf u \times \mathbf B \tag{1}$$ Does it mean that \$\mathbf r=(x,y,z)\in\mathbb ...