anirudh215 said:
See, what I find out of place is that after reading Linear Algebra, Analysis etc. from math textbooks, I find the treatment given in most physics books quite odd. Instead of simply calling vectors as part of some vector space, they have all these round-about definitions like "a vector is something that transforms properly". Why go into linear transformations and other mappings just to define the same thing??
I agree. In fact, I don't think anyone hates that "definition" as passionately as I do. It's been about 15 years since I took classes where that definition was used, and I still get angry when I think about it. It's not just that it's a stupid and obsolete definition. It's also that the books I had to read back then as well as all the teachers I had always stated the definition in a way that doesn't make sense. Would it have killed them to say e.g. "an assignment of four functions v^\mu:M\rightarrow\mathbb R to each coordinate system..." instead of "
something"??
But I have some good news for you. Schutz explains tensors really well, if I remember correctly. He defines them as multilinear functions, defines their components in a basis for the vector space (and it's dual space), and derives the formula for how the components associated with one basis are related to the components associated with another basis, i.e. how the components "transform".
So I still recommend that you read that part. When you get to the GR part, where he starts talking about differential geometry (manifolds, tangent spaces, and tensor
fields), you will probably want to study a math text instead. (I know I did).
This one is probably the best.
anirudh215 said:
They have all these weird connotations. A simple 4-D space that you might encounter all the time in an Algebra book is given some funny hokey name, "spacetime". Yeesh.
This is actually very natural. I mean, it's the mathematical structure that we use to represent real-world concepts "space"
and "time", so I think the name is
very appropriate. Also, it's not just "a simple 4-D space", because even though the vector space structure is defined exactly the way we would do it for a Euclidean space, it doesn't have an inner product, and is equipped with a bilinear form that isn't positive definite instead.
anirudh215 said:
They'll add "spacetime is curved" to sound more fancy. It's just a different metric, dammit! I'd find it so much easier if I could avoid all this weird stuff. This is why I'm looking for a book written on the math side.
There's a very good reason why the word "curved" is used, and you'll find the same terminology as well as an explanation of the terms in the best math books. (
This one has to be the best).
