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I have always had an issue with understanding the definitions used in mathematics. I need examples before I can start using and reasoning with them. However, with tensor products, I have been completely stuck.

Stillwell's Elements of Algebra was that made abstract algebra "click" for me. However, in the case of algebra, I got the logic behind those definitions; It just took an extra "push" for me to start comfortably using them in proofs. Tensor products, however, I am not even 100% sure I get the underlying logic of.

What I am now hoping to find is the equivalent to Stillwells book for differential geometry. That book does the opposite of what most textbooks do; Textbooks start off with abstract definitions before presenting concrete examples. Stillwell goes through all major results using numbers and polynomials before showing how that logic can be generalized to groups, rings, fields and galois theory.

Of course, I'll try any book that can prepare me for the definitions in differential geometry. Bachman has a good book on differential forms which I am going through right now. However, I obviously need to learn a lot more. Which other books can help me prepare for differential geometry?