> but those operations over what usually appears at the denominator of the derivative purely for notation purpose has always looked to me as complete magical garbage.
You will be delighted to discover that they are in fact not magical or garbage.
> What object is "dx" ? is it a number ? a limit ? is it zero ? can i divide another number by it ?
dx is a differential one-form. You can think of it as a generalisation of a gradient, if you like. These are very important in Differential Geometry.
You can use differential forms to do all sorts of things, but one example you may be familiar with is to compute area or volume forms over arbitrary manifolds. It gets a bit hard to define things on HN without TeX support, but using differential one-forms and the related exterior derivative, you can define a generalised Stokes' theorem that works for any smooth, oriented manifold.
I used this in my PhD, and implemented it directly in a numerical method, so this has very practical engineering uses also.
"Elementary Differential Geometry" by Barrett O'Neill is a pretty beginner-friendly introduction to some of these topics if you're interested, though there are many other good texts also.
You will be delighted to discover that they are in fact not magical or garbage.
> What object is "dx" ? is it a number ? a limit ? is it zero ? can i divide another number by it ?
dx is a differential one-form. You can think of it as a generalisation of a gradient, if you like. These are very important in Differential Geometry.
You can use differential forms to do all sorts of things, but one example you may be familiar with is to compute area or volume forms over arbitrary manifolds. It gets a bit hard to define things on HN without TeX support, but using differential one-forms and the related exterior derivative, you can define a generalised Stokes' theorem that works for any smooth, oriented manifold.
I used this in my PhD, and implemented it directly in a numerical method, so this has very practical engineering uses also.
"Elementary Differential Geometry" by Barrett O'Neill is a pretty beginner-friendly introduction to some of these topics if you're interested, though there are many other good texts also.