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In my electrodynamics book. Griffiths 3rd. p223. I see these equations.

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well I think it kind of makes a little sense now. but I don't get why there are all of these dots and crosses. why after divergence is there an integral of a del dot of a cross product. to me that is very confusing.

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Well at least I think I see the math now. Finally, you know sometimes you have write it down and really look at it for it to make sense.
But still why is the biot savart law some integral of a cross product. And how did we turn an integral into a non-integral? How did they do that?
The Biot-Savart law was mostly pieced together from experimental observation. The cross product is a result of the fact that the direction of the magnetic field is perpendicular to the piece of current that creates it.

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