Wishing to go on to Tensor Analysis

[James2]

Hello, I know all my algebra, trig, and I'm still fine tuning calculus and I've solved ODEs using the Laplace transform. Now, my question is... what else must I know to study Tensor calculus/Tensor Analysis? I really want to know so that I have a true understanding of relativity(the famous einstein field equations especially) I am a high school senior, in AP Calculus, and I'm looking to the future. What classes will I need to take in college to truly understand enough so that I don't get frustrated when I try to deal with vector analysis/calculus and tensors? I'm reading about it, but having trouble grasping some of it, something is missing in my background most certainly. Can anybody tell me what I need to know to understand these concepts?

 

[dx]

Linear algebra.

 

[Fredrik]

Linear algebra is very important, because tensors are "linear" (actually multilinear) transformations that take many vectors as input. So study linear algebra, preferably from a book that introduces linear transformations early, like Axler, or Friedberg, Insel & Spence. Make sure that you understand matrix multiplication and the relationship between linear transformations and bases. Then you won't have any problems understanding the basics of tensors, if you read about it in a book like "A first course in general relativity" by Schutz. Chapter 3, titled "Tensor analysis in special relativity" is a nice introduction. This book is a great place to learn special relativity and the basics of tensors.

To move beyond that, you're going to have to study differential geometry, either from a GR book, or from a differential geometry book. I think the best one is "Introduction to smooth manifolds" by Lee. If you want to get a rough idea what sort of things you will learn there, check out this post, the one I linked to in it, and the ones I linked to in the one I linked to.

Every book on differential geometry assumes that you know calculus and some real analysis and topology (limits, continuity, etc). This will make it hard for you do all of this now. I think you can study the basics of linear algebra and the first three chapters of Schutz now, but you may not be ready for Lee until you have taken a course in real analysis.