I am familiar with the basic concept of the Helmholtz decomposition and I have read a number of materials on it (they all follow structure similar to that on Wikipedia page).

However, I am not able to find one single example of taking some simple vector field and actually showing how it is decomposed according to the Helmholtz decomposition.

Does anyone know some link with some simple (or more complicated) examples or if it is actually trivial to do could someone show an example here? For example some simple vector field in 2D?

Thank you! :)

  • $\begingroup$ You can build examples by adding together one irrotational vector field, and one solenoidal vector field. (And the point is that every example is built this way.) Do you know how to construct examples of those kind of fields? Or were you looking for a procedure on how to decompose a given $1/r$-like decaying smooth vector field? The Wikipedia page gives the formula for how to do so. $\endgroup$
    – Elle Najt
    Jan 9, 2016 at 17:42
  • $\begingroup$ I do not have experience with constructing such field myself. Regarding the decomposition, I saw the formulas based on integrating on Wikipedia, but I still wanted to see an example done by someone else, to make sure I got it right. Its incredible that there are 393 000 search results on Google and not one example. $\endgroup$
    – nevermind
    Jan 10, 2016 at 9:13


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