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3

Let us begin with $$\mathrm{div} \left[\mathrm{curl} \mathbf{A}(\mathbf{x})\right]=0$$ Consider the electrostatic field, it has sources to emerge from (positive charges) and sinks to go into (negative charges). Such a field has a non-zero divergence. When we say that the divergence of $\mathrm{curl} \mathbf{A}(\mathbf{x})$ is equal to zero, this means ...

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A bit of 1, a bit of 3... The technical name is flow velocity, as correctly stated in the Wikipedia article about NS equations. But one could ask what "flow velocity" means. From the Wikipedia article: flow velocity [...] is a vector field which is used to mathematically describe the motion of a continuum. Although correct, this definition is ...

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You are correct, it is the velocity of a small volume of fluid centered at the point, that is a macroscopic motion, but it is also the result of the average velocity of the particles in that volume.

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You need to look up the Helmholtz Theorem and similar results that will basically give you ACuriousMind's Answer. But a way I like to visualize this is through the Fourier transform; in Fourier space the curl $X\mapsto\nabla\times X$ and divergence $X\mapsto \nabla\cdot X$ become simply the cross $\tilde{X}\mapsto k\times\tilde{X}$ and scalar$\tilde{X}\... 4 The converse is not a "theorem" in that you can't prove that it is true, even assuming Maxwell's equations as axioms. It is simply a "hunch" that$|\vec{S}|$represents the power intensity and$U=\frac{1}{2}\,\epsilon\,|\vec{E}|^2+\frac{1}{2}\,\mu\,|\vec{H}|^2$the energy density. What you can prove (and what you already understand) from Maxwell's equations ... 1 Consider some path$\gamma^{\mu}(\tau)$, and some vector$x^{\mu}$. Parallel transport is the condition when$\gamma^{a}\nabla_{a}x^{b} = 0$Torsion is present if, for two paths$\gamma^{a}$and$\delta^{a}$that satisfy$\partial_{a}\gamma^{b} = \partial_{a}\delta^{b} = 0$, it is the case that parallel transport of$\gamma$along$\delta$produces a ... 1 To some extent Your answered the question already! Look at the basic postulates of cosmology, these are homogeneity and isotropy of space-time. Isotropy implies three Killing vectors (SO(3)) and homogeneity gives another three killing vectors (for translation in three spatial direction). Therefore altogether six Killing vectors. Remember we not considering ... 0 The evolution of systems in the Hamiltonian formalism is called a flow, not because it can be described by a mapping, but because it is described by a particular mapping: one whose evolution in (q,p)-space resembles fluid flow. This resemblance gives rise to Liouville's theorem, where the Hamiltonian flow, like certain fluid flows, is shown to be ... 3 The result is almost correct, the divergence has the term $$-\frac{ b\exp(-br)}{r^2}$$ which looks simple but it is not the behavior of the Yukawa potential which only has$1/r$, not$1/r^2$, in 3+1 dimensions. Near$r=0$, the most singular term with the$\exp(-br)\sim 1$behaves as$E_r=1/r^2$which is the same as in the Coulomb potential$V\sim -1/r\$ ...

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Qualitative drawing:___________________ Sorry, I've mistaken. it should be:

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Draw it like segmented sun rays with longer ray segments at farther distances, arrows pointing inward.

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