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The outline of the book is as follows. Chapter 1 reviews some basic facts about smooth functions from IRn to IRm, as well as the basic facts about vector spaces, basis, and algebras. Chapter 2 introduces tangent vectors and vector fields in IRn using the standard two approaches with curves and derivations. Chapter 3 reviews linear transformations and their matrix representation so that in Chapter 4 the push-forward as an abstract linear transformation can be defined and its matrix representation as the Jacobian can be derived. As an application, the change of variable formula for vector fields is derived in Chapter 4. Chapter 5 develops the linear algebra of the dual space and the space of bi-linear functions and demonstrates how these concepts are used in defining differential one-forms and metric tensor fields. Chapter 6 introduces the pullback map on one-forms and metric tensors from which the important concept of isometries is then defined. Chapter 7 investigates hyper-surfaces in IRn, using patches and defines the induced metric tensor from Euclidean space. The change of coordinate formula on overlaps is then derived. Chapter 8 returns to IRn to define a flow and investigates the relationship between a flow and its infinitesimal generator. The theory of flow invariants is then investigated both infinitesimally and from the flow point of view with the goal of proving the rectification theorem for vector fields. Chapter 9 investigates the Lie bracket of vector-fields and Killing vectors for a metric. Chapter 10 generalizes chapter 8 and introduces the general notion of a group action with the goal of providing examples of metric tensors with a large number of Killing vectors. It also introduces a special family of Lie groups which I've called multi-parameter groups. These are Lie groups whose domain is an open set in IRn. The infinitesimal generators for these groups are used to construct the left and right invariant vector-fields on the group, as well as the Killing vectors for some special invariant metric tensors on the groups.
tangent vectors, vector fields, metric tensor fields, theory of flow invariants
Fels, Mark E., "An Introduction to Differential Geometry through Computation" (2016). Textbooks. 2.