I'm currently studying physics and more precisely theoretical mechanics. During that course we use knowledge from calculus courses but also some manipulations that are not quite covered in this courses like virtual displacements and infinitesimal distances. I asked for some help on where I can find more information about that and was directioned into reading some noncalculus books but after some research in internet I found that in this manipulations differential geometry is also used. So my equation is for what each of them is used and also some quick but meaningful introductions into these subjects with proof.
Some applications to physics are dealt with, as well.
This approach to true infinitesimals is based on the traditional ZFC set-theoretic framework with background logic being classical logic. This is unlike the SIA framework mentioned in the comments, which relies on category-theoretic foundations with background logic being intuitionistic logic.