The impetus for writing these notes came from my attempt to understand General Relativity. My original attempts at understanding the subjects had been greatly impeded because I did not understand what a tensor was. There were largely two kinds of explanations; the first was meant for physicists with little to no background in higher mathematics and thus involved rather ungainly explanations as to how to think of tensors. These explanantions mostly involved of thinking of physical fields and motivating why one would think of them as tensors. The official defintion then involved telling the reader as to how they transformed. Of course all this was largely un-satisfactory because one never got an invariant and precise way of saying un-equivocally what a tensor was i.e one merely got impressions as to how think about what it was. On other hand, was the precise definition meant for mathematicians. But since mathematicians properly defined what tensors were after they had learned alot else, the discussion was almost invariably couched in what I thought was highly technical and un-nessary baggage. Thus one was faced with a decision: either give up or embark on an incredibly daunting journey to understand the baggage that came along with the discussion. Finally, I found a happy middle ground and laid the basis for learning more deeply what differential geometry was all about.
Why would a physicist bother to put in the effort to carefully consider differential geometry? The most important benefit is to free oneself of a co-ordinate dependent way of deriving formulae. Co-ordinate free expressions have a way of revealing the underlying structure and thus making seemingnly disparate concepts unified. From a physics point of view there is a philosophical point. We would like an-invariant way of talking about the physics i.e one that is indpendent of reference frame so it seems rather odd when for example a tensor is defined by how it transforms under a coordinate change. After all such a concept requires providing an atlas for the manifold and inherently making the discussion local. An extra step is then needed to ensure the reader that despite the local nature of the discussion nothing actually depends on the local structure thereby leaving a serious reader wondering why if the local structure contributes nothing, why does it seem so necessary in the whole discussion? These notes were meant to still my troubled heart. If any mistakes are found please contact me at henriamaa@gmail.com