Steven W. answered • 08/07/16
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Did you try looking at the Wikipedia article on the theorem? There is a derivation there that looks pretty good, and shows how, with the given conditions, to write a vector field as the sum of divergence and curl terms. I cannot post the link here, but if you search Google "Helmholtz decomposition theorem," it is one of the first items that comes up. We can use that as a point of reference if you have more questions, which we can look at in more detail, if you like.
Steven W.
tutor
That is not something that is derived; it is defined. It is a function that equals some value (often normalized to 1) when the argument is 0, and equals zero otherwise. In this case, it is the three-dimensional version. What it means is that is it one is r = r' (as a vector) and zero if r is anything else. In other words, it mathematically specifies that we are looking at a location specified by the vector r' at any point in our calculation.
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08/07/16
Steven W.
tutor
Somehow I moved right past the fact that it is called the delta function.
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08/07/16
Steven W.
tutor
Vector calculus is not my specialty, but I believe the delta function is usually used (as it is here) under integration, so that every point were r ≠ r' contributes a value of zero to the integral, leaving the only nonzero contribution to be when r = r'.
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08/07/16
William Z.
Thank you very much! That was an excellent explanation! It cleared things up tremendously.
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08/07/16
William Z.
08/07/16