(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

So this is part of a problem set in which I have to show that a vector field is divergence free but not the curl of any vector field.

Let[tex]F =\frac{<x,y,z>}{(x^2 + y^2 + z^2)^{3/2}}[/tex]

Then F is smooth at every point of R3 except the origin, where it is not defined. (This vector field is identical, up to a constant multiple, to the electric field generated by a point charge at the origin.) Let E be the region [tex]1 < x^2 + y^2 + z^2 < 9[/tex] that is, the region between two concentric spheres of radii 1, 3 centered at the origin.

Let S be the surface [tex]x^2 +y^2 +z^2 = 4,z ≤ a[/tex] where a is a number slightly smaller than 2. (S is the sphere of radius 2 with its top sliced off.) Let C be the boundary of this surface. Show that if F = ∇ × G for some vector field G defined on E, then

[tex]\int _C G • dr = 4π[/tex] as [tex]a → 2^-[/tex]

3. The attempt at a solutionSo I have confirmed that ∇ • F = 0. I have also found that the region E is not a simply connected since any sphere contained in E will form the boundary of a region that contains points not in E. Therefore, such a sphere could not be shrunk to a point without leaving E. Establishing these two points were the first two parts of the problem.

This third part, however, seems much more difficult. First, just a bit of clarification, what does [tex]a → 2^-[/tex] mean? Is it like "as a approaches 2 from the negative direction?"

I think I probably need to define C in some way. Supposing I let z=a, then

[tex]x^2 +y^2 = 4-a^2[/tex]Could C then be parametrized as [tex]C = r(t) = <(4-a^2)cost,(4-a^2)sint>[/tex]Anyway, I'm not entirely sure how to begin.

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# Divergence Free But Not the Curl of Any Vector

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