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Gradient, Divergence and Curl

Gradient, divergence and curl are frequently used in physics. The geometries, however, are not always well explained, for which reason I expect these meanings would become clear as long as I finish through this post.


Curl means curl, which is explicitly shown by this word.

The curl of a singular point doesn’t always show the singularity. One of the examples is the magnetic field generated by dipoles, say, magnetic dipoles, which should be

where the vector potential is

The reason for the extra Dirac delta is that vector is singular at point 0 meanwhile the curl of such a function does’t really show the singularities of the field. We need to calculate the integral without calculating the curl directly, i.e.,

in which we used the trick similar to divergence theorem.

By OctoMiao

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