If you have a function, f, in cylindrical coordinates, can you normalize the function (in a rigorous sense)?
You can find the magnitude by
but can you truly normalize the function and apply that to a mathematical formula which requires the normalized vector. Or do you have to transform back to rectangular coordinates before normalizing? I have looked through several textbooks and the internet and cannot find anything about normalizing a vector in the cylindrical system.
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
What do you mean by "in a rigorous sense"? I would be interested to see what others think, but what you are describing seems okay to me:
The norm of a vector, in the case of a 3-D vector, like here, is its length. To normalize a vector, you divide the vector by its norm, creating a unit vector with the same direction as the original vector. So to normalize a 3-D vector, you would divide the vector by its length. Thus it seems that
would be the normalized function, as you have suggested. Again, I do not know if this is "rigorous" enough.