In geometric calculus, manifolds can be thought of as embedded in n-dimensional Euclidean space, and it is often convenient to take advantage of that ambient space.
Now all serious treatments of manifolds with which I am familiar develop the properties of manifolds in an intrinsic way which requires no ambient Euclidean space. Yet it seems to me that there is sometimes a tendency in the literature of geometric calculus to slide back and forth between the intrinsic topological setting and the non-intrinsic setting in Euclidean space without noting the fact and to assume (without examination) that any argument or tool that works in one setting must work in the other one.
Therefore I have written a little paper on Elementary Analysis on Manifolds in n-Dimensional Euclidean Space which can be accessed at
The idea is to establish the tacitly assumed equivalences noted above, at least for the simplest manifold concepts.
The paper is meant to be fairly rigorous and has a lot of epsilon-delta analysis in it, however it assumes only the grounding available in an advanced calculus class.
Source: Michael D. Taylor, taylorseries_AT_embarqmail.com, 12 Oct. 2017