My goal is to develop an intuitive understanding of the math underlying general relativity and ultimately be able to take a book like Wald or Carroll and, as someone on these forums commented once, “be able to casually read it while sipping my morning coffee and listening to the news.” :) So where am I now? Although I have an eng degree, I completed it years ago and I've been brushing up. Math-wise, I’ve worked through vector calculus (Stewart) and linear algebra (Poole.) Physics-wise, I've also brushed up on mechanics (Taylor), EM (Griffiths) and done some light conceptual reading on SR. So now the question is where to go next? What I'd like to do is come up with a list of textbooks that would makes sense when used together. ie. not too much overlap and a progression towards grad-level texts that is reasonable for self-study. Also, rather than go with the math-for-physicists route, I’m interested in a broader understanding of the associated math. My hope is to find a path that offers the holy trinity of intuition, motivation AND rigour. That said, I am self-studying, so if a text is particularly good at developing intuition and offering motivation at the expense of rigour I would be fine with that. I’ve done some skimming of book previews at amazon and recommendations here and come up with the list of topics below that seem important to my goals but I really don’t understand these terms well enough to know if my list makes any sense, or in many cases how these topics relate to one another. (For example, how do differential forms, differential geometry and manifolds relate?) Thus you can imagine my difficulty navigating a course through these topics and picking out associated textbooks for self-study. covariant/contravariant vectors tensors differential forms - I heard Lovelock’s book is pretty good, also Hubbard https://www.amazon.com/Tensors-Differential-Variational-Principles-Mathematics/dp/0486658406 https://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/0971576688 metrics coordinate free geometry differential geometry - I heard Keyszig is good but not “coordinate free” whatever that means https://www.amazon.com/Differential-Geometry-Dover-Books-Mathematics/dp/0486667219 topology, manifolds- I hear John Lee’s series is very good, also Mendelson https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395 https://www.amazon.com/Introduction-Topology-Third-Edition-Mathematics/dp/0486663523 Riemannian manifolds - also Lee? I have no particular love for the texts listed above, they're just my best guess right now. Anyways, I would really welcome suggestions from this group - thanks!