The Sixth Edition was completely redesigned, but retains and improves on the balance between theory, applications, optional material, and historical notes that was present in earlier editions. Content changes include: Trimming some of the historical material, making it more relevant to the mathematics under discussion. Moving some of the more difficult discussions in the Ffth Edition to the Book Companion Web Site (including Conservation Laws, the derivation of Euler's Equation of a Perfect Fluid, and a discussion of the Heat Equation). Revision highlights: New Design The modern redesign emphasizes the pedagogical features, making the text more concise, student-friendly, and accessible. More Exercises The new edition adds 15-20% new easy-to-moderate exercises graded from less difficult to more difficult, providing extra practice for students to master key concepts. New Examples The new examples were shaped by feedback from reviewers and UC-Santa Cruz teaching assistants teaching the course. Enhanced Three Dimensional Figures The quality of the artwork has been significantly improved, with crucial three-dimensional figures better reflecting key concepts to students. Definitions/Theorems/Proofs/Corollaries/Key Concepts These pedagogical features these are now more clearly boxed to better highlight them for students and differentiate from the main text.