This is the only text that introduces differential geometry by combining an intuitive geometric foundation, a rigorous connection with the standard formalisms, computer exercises with Maple, and a problems-based approach. Text has running theme of intrinsic vs. extrinsic ways of looking at curves and surfaces. Starting with basic geometric ideas and proceeding to the analytic and algebraic formalisms, this text provides a common and accessible foundation on which all of the various formalisms of differential geometry can be based and from which they can be assessed.