This is a textbook on differential geometry well-suited to a variety of courses on this topic.
For readers seeking an elementary text the prerequisites are minimal and include plenty of
examples and intermediate steps within proofs while providing an invitation to more excursive
applications and advanced topics. For readers bound for graduate school in math or physics
this is a clear concise rigorous development of the topic including the deep global theorems.
For the benefit of all readers the author employs various techniques to render the difficult
abstract ideas herein more understandable and engaging. Over 300 color illustrations bring the
mathematics to life instantly clarifying concepts in ways that grayscale could not.
Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical
content. Color is even used within the text to highlight logical relationships. Applications
abound! The study of conformal and equiareal functions is grounded in its application to
cartography. Evolutes involutes and cycloids are introduced through Christiaan Huygens'
fascinating story: in attempting to solve the famous longitude problem with a
mathematically-improved pendulum clock he invented mathematics that would later be applied to
optics and gears. Clairaut's Theorem is presented as a conservation law for angular momentum.
Green's Theorem makes possible a drafting tool called a planimeter. Foucault's Pendulum helps
one visualize a parallel vector field along a latitude of the earth. Even better a
south-pointing chariot helps one visualize a parallel vector field along any curve in any
surface. In truth the most profound application of differential geometry is to modern physics
which is beyond the scope of this book. The GPS in any car wouldn't work without general
relativity formalized through the language of differential geometry. Throughout this book
applications metaphors and visualizations are tools that motivate and clarify the rigorous
mathematical content but never replace it.
