# Tagged Questions

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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### Flux - Scalar Multiplication in Integral?

No textbook and website seems to answer this so here is my question: When we have a scalar flux: I understand that you take the scalar product of the vectors. And I understand the need for using an ...
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### Divergence-free vector field on a non-simply connected domain

We know that divergence-free vector fields are themselves curls of vector fields on simply connected domains. I want to construct a counterexample in the case the domain is not simply connected. So ...
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### EM waves and fields

According to wikipedia, electromagnetic waves are "synchronized oscillations of electric and magnetic fields that propagate at the speed of light". I understand what it means in theory. But in ...
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### How can I mathematically describe the parallel transport in the Roman soldiers example?

I've been trying to understand parallel transport. Many of the descriptions present a mathematical version: $\nabla_V X = 0$. And/or they present an example involving soldiers (usually Roman) ...
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### Parallel Transported Orthonormal Basis

The following argument results in a conclusion that I find strange, and makes me suspect there is something wrong with the reasoning. First, consider a timelike geodesic $\gamma$ with normalized ...
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### Electric field in a hollow object

I am currently visiting a course about electrodynamics. In my last lecture it was said that if a hollow sphere is inside of a bigger sphere, but only in the bigger sphere there are charges, the ...
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### Are all maximally symmetric spacetimes constant curvature spacetimes?

A $d$ dimensional maximally symmetric spacetime is a spacetime with the maximum allowed number of Killing vectors. This number is $\frac{d(d+1)}{2}$. Constant curvature spacetimes are spacetimes ...
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### Coulomb gauge and vector identites

consider a coulomb gauge and the following volume integration: $$\int d^3x{\dot{A}.\nabla A}$$ How can we show that this is zero in coulomb gauge? (A is a vector potential) this is my attempt at ...
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### What are the mathematical models for force, acceleration and velocity?

In mechanics, the space can be described as a Riemann manifold. Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector ...
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### How to Model Differential Gravitational Field from Vectors?

The tides are caused by differences in the gravitational field of the moon on the near side and far side of the earth. If I set this gravitational field as a vector field using Newton's Law of ...
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### Tangent Vectors as Infinitesimal Displacements

I'm reading Wald's General Relativity, and I'm stuck on something that is stated very early on in the book. For an abstract manifold $M$, he goes through the usual definition of a tangent vector at ...
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### Preservation of a scalar along geodesic trajectory

Let $u^\mu$ be the velocity of a particle , and $\xi^\mu$ be a killing vector. would taking a contravariant derivative of to scalar product $\xi_\mu u^\mu$ , and showing that it equals to 0 shows that ...
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### Null vector fields given Bondi metric

I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric $g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$ with $d\Omega$-standard metric ...
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### Wick rotating vector fields in QFT

Suppose we perform a Wick rotation $t \rightarrow i\tau$ from Lorentzian to Euclidean signature. Consider its action on the generator of boosts in the 01-plane: $$\zeta = t\partial_x + x\partial_t.$$ ...
As we know, force is product of pressure and area as below:$$d\vec{F}=Pd\vec{A}$$ P is a scalar and orientations of the small force and small area are same. I have three questions (they are ...