I found this thread What is a covector and what is it used for?, in which the top answer states "Many of us got very confused with the notions of tensors in differential geometry not because of its algebraic structure or definition, but because of confusing old notation".
I am currently working through Barret Oniell: Elementary Differential Geometry 2nd Edition and going through all of the problems, it feels rather like fumbling in a dark room for a light switch (at the end of chapter 4). I find myself having to review all sorts of different material using different notations and try to piece everything together in a sensible way on my own. I also have purchased Wolfgang Kuhnel's book on the matter, but before I spend another few months headed down that road, I think it best to ask this advice…
What is a thorough, articulate and reliable (i.e. not riddled with mistakes) book on the subject matter suitable for self study that uses modern (by which I understand to be less confusing) notation? Preferably it would be available for purchase in an electronic format if possible.
Best Answer
Introduction to Smooth Manifolds by John Lee is the best book I know for a modern introduction to Differential Geometry, another good book is Introduction to Manifolds by Tu.
Lee's book is really good good for self study in that it contains a very thorough exposition, a plethora of examples, and many good exercises.