The use of differential forms, and especially of orthonormal bases, as presented in this book, represents a radical change in my own thinking. The relativity community consists primarily of physicists, yet they mostly learned differential geometry as I did, from mathematicians, in a coordinate basis. This gap is reminiscent of the one between the vector calculus taught by mathematicians, exclusively in rectangular coordinates, and the vector calculus used by physicists, mostly in curvilinear coordinates, and most definitely using orthonormal bases. Together with my wife and colleague, Corinne Manogue, I have had the pleasure of working for more than a decade to try to bridge this latter gap between mathematics and physics. Our joint efforts to make $d\rr$ the key concept in vector calculus also led to my redesigning my differential geometry and relativity courses around the same idea.

My debt to Corinne is beyond words. She opened my eyes to the narrowness of my own vision of vector calculus, and, as a result, of differential geometry. Like any convert, I have perhaps become an extremist, for which only I am to blame. But the original push came from Corinne, to whom I am forever grateful.

I am also grateful for the extensive support provided by the National Science Foundation for our work in vector calculus. Although this book is not directly related to those projects, there is no question that it was greatly influenced by my NSF-supported work. The interested reader is encouraged to browse the project websites for the %* Paradigms in Physics Project and the Vector Calculus Bridge Project, as well as our online vector calculus text.

Finally, I am grateful for the support and interest of numerous students over many years.