In geometric calculus, manifolds can be thought of as embedded in ,  and it is often convenient to take advantage of that ambient Euclidean space.  Sometimes derivations in the literature are carried out which appear to appeal to that ambient space.  However all serious treatments of manifolds with which I am familiar develop the properties of manifolds in an intrinsic way which requires no ambient .  Therefore I have written a little paper on Elementary Analysis on Manifolds in which can be accessed by clicking on the paper’s title.  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.