Vector Lapacian conceptual question

Discussion in 'Physics & Math' started by Secret, Sep 7, 2014.

  1. Secret Registered Senior Member

    Messages:
    299
    So in order to better understand about the vector lapacian (which wikipedia said it measure the average change in vector field at a point, thus if using fluids as an analogy, measuring the average change in flow), am wondering what happens if the vector lapacian of a point is zero

    Using the two identities that has no name (they are corrolary from the stoke and gausss theorems), I get the following
    http://en.wikipedia.org/wiki/Vector_calculus_identities

    \(\nabla^2\bold{F}=\nabla(\nabla \cdot \bold{F})-\nabla \times(\nabla \times \bold{F}) \\When \hspace{1mm} \nabla^2\bold{F}=\bold{0} \\\nabla(\nabla \cdot \bold{F})=\nabla \times(\nabla \times \bold{F}) \\Integrate \hspace{1mm} wrt \hspace{1mm} volume \hspace{1mm} V \\\iiint_V \nabla(\nabla \cdot \bold{F})dV=\iiint_V \nabla \times(\nabla \times \bold{F})dV \\ \oint_{\partial S} (\nabla \cdot \bold{F})d\bold{S}=\oint_{\partial S} \bold{\hat{n}}\times(\nabla \times \bold{F})dS\)

    However, as interpreted from the result, why it must be true that
    "total number of sources/sinks on a closed surface"="Total vector (perpendicular to the vector that measures how much the vector field whirls at that point) flux of said closed surface"?

    Or going back to line 3
    Why when vector lapacian=0 means
    "Rate of change in the number of source/sinks"="rate of change of the whrling of the vector field in the neighbourhood of the point"?

    Isn't it possible that since sources and sink radiate/absorb vectors in all directions, then it is possible to have more sources/sinks but no change in the amunt of whirling of the vector field (since the vectors are radiaitng out instead of circulating around a point?

    Or more concisely, how is it guarantee that if the change in the whirling is equal to the change in the number of source/sinks, then the average change in flow at that point must be zero (since the vectors can point in any direction they like (and for vector fields on a 2D plane, the curl vector is perpendicular to the vector that represents the gradient of the divergence, so they cannot cancel, can they?), thus the vectors from the whirling may not necessary cancel out with those radiating from the source/sinks?)
     
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  3. danshawen Valued Senior Member

    Messages:
    3,951
    This sounds like the vector field equivalent of the infamous three body problem in physics, which only has a discrete number of stable configurations derived from computer modeling.

    Could not a whirling vortex in a vector field also be created from sources / sinks arranged along a slightly offset, diminishing circular or even a spiral path? Is this not how some flush toilets circulate the water before exiting through a u-shaped basin at the bottom?

    Contrary to what some on these forums think, I do enjoy such math. But if you wish to do convincing real world physics with vector fields like this, you'll be needing some bindings to the real world. I don't think we have quite enough of them yet, but this is not a reason to stop considering the possibilities. Otherwise, when a solution presents, we may not recognize it.
     
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