Book that Develops the Theory of Tangent Space whilst Defining Tangent Vectors as Equivalence Classes of Curves

Question

Currently reading Lee's Introduction to Smooth Manifolds. At the end of chapter $3$ he mentions that tangent vectors may be defined in terms of equivalence classes of curves, but by that time he has developed the theory of the tangent space using Derivations, and thus does not delve further.

Is there a book that develops the theory of tangent spaces whilst definiting tangent vectors as equivalence classes of curves?

Answer 1

• As mentioned in the comments, Foundations of Mechanics by Abraham and Marsden. This is a pretty abstract text (but detailed).
• Related is Manifolds, Tensor Analysis, and Applications by Abraham, Marsden Ratiu. Also pretty abstract (but detailed).
• Lectures on Differential Geometry, by Shlomo Sternberg, section 2.5.
• Advanced Calculus, by Loomis and Sternberg. See section 9.4 for tangent spaces (they actually introduce both curves and directional derivative operators).
• A Course in Mathematics for Students of Physics (Vol I) by Bamberg and Sternberg. This is a great introductory text (mathematically, you can think of it as a kiddo version of Loomis and Sternberg’s text), covering lots of interesting stuff, and using those interesting applications to motivate the use and definitions in differential geometry (exterior derivative, tangent vectors, pullbacks, chains etc). They don’t do abstract manifolds if I recall correctly, but they still introduce the idea of tangent vectors as equivalence classes of curves (section 5.5).
• A Comprehensive Introduction to Differential Geometry, by Spivak. He starts off intuitively, then gives various descriptions of the tangent bundle, including curves.
• Treatise on Analysis, Vol III, by Dieudonne. He mostly deals with things using equivalence classes of curves (chapter 16) due to the geometric nature, and only later, chapter 17, does he discuss the relation to the more algebraic aspects with differential operators. This is a nice book (for me anyway), but I wouldn’t recommend it for a first reading. It’s abstract (no pictures, besides commutative diagrams) and Dieudonne is merciless if you don’t have the necessary prerequisites, which are his previous volumes, even if only to see the particular theorem he’s referencing (though for reading much of this volume in particular, you only need his Volume I (which by itself is a wonderful text)). However, if you’re willing to stick through it, the first few sections 16.1-16.8, where he covers the basics (i.e manifolds, partitions of unity, tangent spaces, immersions/submersions/subimmersions, submanifolds) should be fairly accessible (for moderately high values of ‘accessible’).