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This proof from Griffiths book introduction to electrodynamics Consider the vector function $$\vec{a}=\frac{1}{r^2}\hat{r}$$ At every location $\vec{a}$ is directed radially outward ; if ever there was a function that ought to have a large positive divergence, this is it. and yet, when you actually calculate the divergence, you will get ...

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Indeed the answer is not zero but $-4\pi\delta(r)$ (Dirac delta function). The formula of divergence can be found in any standard textbook on mathematical physics, for example chapter 2 of Mathematical methods for physicists by Arfken. But since this function is singular at $r=0$ we must be careful. At any other points is easy to calculate it. It is ...

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As it looks like another question I've supplied an answer to might be duplicated here (and hence closed), I am going to provide a similar but not identical answer here. In words - divergence is the flux of something into or out of a closed volume, per unit volume. The best visual picture I have of this is a fluid flow. Imagine water spewing out of a tap - ...

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Divergence can be thought of as the flux of a vector field per unit volume. It is positive if there is a net flux out of a small volume and negative if there is a net flux inwards. When you say "its diagram" - of course there are different ways of plotting vector fields. Perhaps the most common way is using field lines. In which case it can be ...

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Such things like as homogeneity or isotropy are important because they are assumed as the basis for the principle of relativity. This principle says you can choose any coordinate system, doesn't matter where it is (since the space is homogeneous) or how is directed (since it is isotropic), or it is still or moves with constant velocity. So, resolving a ...

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Is important to remember that curvature does not imply curl. Here is an analogy from fluid mechanics which deals with lack of vorticity (curl) in curved flow. This may seem counter-intuitive at the start, but hopefully you can see the similarities between the velocity field in the flow and the gravitational field. ...

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The paragraph is implying that the force is dictating the position function r(t). (...) The force field does not necessarily move the particle all the way through the curve because that would simply be impossible if the motion of the particle only comes from the force field, which is what the definition is implying. I think that you're confused about ...

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