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 & include plenty of examples & intermediate steps within proofs while providing an invitation to more excursive applications & 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 & engaging Over 300 color illustrations bring the mathematics to life instantly clarifying concepts in ways that grayscale could not Green-boxed definitions & violet-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 & equiareal functions is grounded in its application to cartography Evolutes involutes & 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 & 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 & visualizations are tools that motivate & clarify the rigorous mathematical content but never replace it