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Differential geometry has a long, wonderful history and has found relevance in many areas. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the standard university curriculum to a type of mathematics that is a unified whole, by mixing geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors, but also for students in engineering and the sciences. Into the mix of these ideas comes the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.