Quoted By:
Differentials can be well-defined, as the poster a few posts above me has done. It is simply that neither Leibniz nor introductory calculus students had this particular precise definition and indeed had only a fuzzy understanding of the concept based on physical ideas.
After I learned about this conception of differentials in a course on Differential Geometry, it made my life in various advanced Physics classes much easier, as now I had a concrete idea of, for example if u = f(x,y), what precisely du meant and how it could be stated in terms of f, dx, and dy. No longer did I have to rely on physical intuition, but instead I could simply perform systematic manipulations according to given rules and formulas.
To give one relatively simple definition: If f(v) is a function of some vector v, then df is the (linear) function which takes a vector v into g'(0) where g(t) = f(tv).
With that said, just because differentials *can* be well-defined does not mean that it is valid to use these results naively without first doing the groundwork, especially when proving important bases of calculus that are used to define these differentials in the first place! Especially by a student learning calculus for the first time, who would only be confused by a term such as "linear functional".
