I wanted to know of book suggestions that can help me overcome my fear of indices

Question

I want to go deeper into General Relativity and Tensor Analysis. However, manipulating the indices always seems to overwhelm me. I wanted to know if there is a good book that covers up that and also lists all the important results related to index manipulation. The main focus would be index notation. The best example would be how generalized coordinate transformations are done, the algebra of covariant and contravariant tensors and all such examples that contain the summation symbol and the analysis.

Answer 1

This is not something I really know that much about, but for what it's worth, here are some possibly relevant books from my bookshelves.

Vectors, Tensors and the Basic Equations of Fluid Mechanics by Rutherford Aris

I had a reading course out of this book back in 1982 from an engineering faculty member, with the goal of getting better acquainted with the physics/engineering notation for vector calculus and tensors, a goal that didn't really materialize because I didn't devote enough time to the material.

Introduction to Vector Analysis by Harry F. Davis

I've had this book since the early 1980s also, and probably should have quickly worked through parts of it (especially the sections at the end where tensors are introduced) before trying to plow through the first half of the book by Aris.

Tensor Geometry by C. T. J. Dodson and T. Poston

I've had this book since the paperback edition appeared in 1979, and for certain people this book could be extremely useful. It doesn't get involved in the stuff you're looking for, but the fact that it tries to bridge the gap between such computations and the more abstract viewpoints seen in mathematics classes WITHOUT requiring much background in "modern mathematics" makes this book rather unique. I'm also mentioning it because it has a lot of applications to relativity.

Introduction to Vector and Tensor Analysis by Robert C. Wrede

Another book I've had for a long time, since the late 1970s. I've never really read much in this book, but when I was younger I always toyed with the idea of this being the book where I'd really learn all this stuff, because there's so much in it and classical notation is used. The last section of the book gives an introduction to general relativity.

Vector Analysis and an Introduction to Tensor Analysis by Murray R. Spiegel (Schaum's Outline Series)

This might be the book I'd recommend most highly for you, given what I know. This is sufficiently well known that I probably don't need to say anything.

Applications of Tensor Analysis by A. J. McConnell

I got this (the Dover edition) in the early 1970s when I had hardly yet learned algebra because I had read that the mathematics used in relativity was tensor analysis, and I wanted to learn it. As I got a few years older I realized this book was rather old-fashioned, but still I assumed that one day I would understand most things in it. Nope. Anyway, of all the books I've mentioned, this one is probably steeped the most in classical tensor notation.

A Brief on Tensor Analysis by James G. Simmonds

I don't have a copy of this book, but I've looked through library copies on several occasions. It seems well worth looking at, although I don't know whether it has sufficient emphasis on classical notation and symbolical manipulation of tensor notation for you.