I just started trying to get into relativity using Schutz's book but ive been getting a little confused with some of the tensor stuff. For starters, what does it mean by a one form maps covectors to real numbers? is it an operation or a transformation? and when it maps to a real number does that mean it will loose information about its direction? Also i don't understand some of the notation for example i get that A with a superscript alpha is the alpha component of the vector but then why are the basis vectors written with a subscript ( is this the whole contrivariant covariant part) also two subscripts(alpha beta) next to one another, i understand is just a matrix at the alpha row beta column. But i dont get it when there is a something with an alpha beta component one on top and one on the bottom (and they arent even in a vertical line one is shifted from the other. i tried looking up some of these question in penroses the road to reality book but i got confused at the part where he give a coordinate change where X= x , Y= y+x but then the partial with respect to X= the partial with respect to x minus the partial with respect to y while the partial of Y= the partial with respect to y (on page 189). sorry for the lack of math symbols i dont really know how to add them in.
-thanks for all the help anyone can offer.
this isn't HW just something im doing on my own but i figured it was the most appropriate place to put the question.