# 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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### Covariant Derivative and energy momentum tensor

In this reference https://arxiv.org/abs/hep-th/0307199 pag.60, it is said that it is possible to find an infinitesimal spacetime diffeomorphism (a vector field) $X_{\nu}$ independently to its ...
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### Why is the magnetic field dot producted in the integral version of Ampere's circuital law?

you know amperes circuital law? Well in that equation there's a dot product between the magnetic field B and a length element dl...why is that? I mean its not like the magnetic field can be at an ...
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### Killing vectors and isometry

Let $X=x\partial_{t}+t\partial_{x}$ and $Y=y\partial_{t}+t\partial_{y}$ be Killing vectors on Minkowski $(-,+,+,+)$. It can be shown that $[X,Y]$ is also Killing. I get the following: \begin{equation} ...
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### Name this Vector Calculus Theorem

There is an important theorem in vector calculus that says $\boldsymbol{\nabla}\boldsymbol{\cdot}\mathbf{G}\boldsymbol{=}0$ (where $\mathbf{G}$ is some differentiable vector function) implies and is ...
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### What's the index (or topological charge) of this vector field image?

I am doing some research in a condensed matter system, and found this Berry curvature / vector field configuration that is unusual. I cannot find another example of something similar, either from ...
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### David Tong's passive transformation of the fields is wrong

David Tong's definition of active transformation is clear. Under active transformation coordinates (basis vectors) are not changed but rather the field is. I denote the old and new fields as $\phi$ ...
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### What is the magnitude of the curl of a electric field? and in general from a vectorial field?

I found the following result in a book: but I do not understand what is the meaning of the magnitude of the curl of a vectorial field? and how is this related with the amount of spatial oscillations.
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### If a cosine wave satisfies Maxwell's equations then how does a sine satisfy the equations as well?

Say the real part of $$\tilde{\mathbf{E}}(z, t)=\tilde{\mathbf{E}}_{0} e^{i(k z-\omega t)}$$ satisfies all Maxwell's equations. Then how can we say the imaginary part satisfies the equations as well? ...
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### Killing vectors in Minkowsky Metric

So I was in the process to find the Killing vectors for the Minkowsky Metric and I stumbbled into a material that does a different procedure at the very end of the process, in comparisson to usual ...
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### How is potential difference defined across a resistor with time varying current

From this discussion How can we define a potential for a moving charge? we know that we cannot define a scalar potential (as in electrostatics) in the case of moving charges as described by G. Smith: ...
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### How can we define a potential for a moving charge?

Say a charge is moving in space. Ignoring relativistic effects, how can we define a scalar potential for its electric field ? My thoughts are that we can define the potential in exactly the same way ...
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### Will any solution to the wave equation be a wave in reality?

In the mathematical sense, a wave is any function that moves. In that sense we can consider that any function that complies with the wave equation (let's consider in one dimension to simplify things) ...
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### How to find acceleration from velocity, coefficient of kinetic friction and radius of curvature [closed]

I've been going through the various sections of my Engineering Dynamics HW and I've been struggling to solve this problem for a while: A car is travelling at a speed of 30 m/s at the top of a hill at ...
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### Why does Gauss's law apply to any shape of a closed surface?

What seems to incredibly bother me is why Gauss's law applies to any shape of a closed surface. Moreover, the fact that the electric flux is proportional to the enclosed charge is by many sources ...
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### Why can a force field only be conservative if it is spherically symmetric?

I saw in my textbook that a field can only be conservative if it happens to be spherically symmetric. Why is this so? Is there a good proof for this?
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### How to find the Taylor expansion of $\vec{r}/r^3$?

I want to show that the Taylor expansion of $\frac{R\vec{e_1}-\vec{y}}{|| R\vec{e_1}-\vec{y} ||^3}$ at $\vec{y}=0$ is equal to $\frac {\vec{e_1}}{R^2}+\frac{3y_1 \vec{e_1}-\vec{y}}{R^3} + O(y^2)$. I ...
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### Contradiction in calculating electric field outside of a dielectric material embedded with free charges

Let's say we have a region filled with a linear homogeneous dielectric, filled with some free charge density $\rho_{f}$ such that outside of this region the electric field is zero. Then we can write ...
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### Navier Stokes: $(u⋅∇)u$ vs $u⋅∇u$

I can find this term stated both ways in different literature. Are they equivalent? It's weird because the dot is a dot product in (u⋅∇), but ∇u being a gradient of a vector field, would (presumably) ...