Vector and Complex Calculus

Abstract

Vector and complex calculus are essential for applications to electromagnetism, fluid and solid mechanics, and the differential geometry of surfaces. The standard multivariable calculus courses are largely limited to `xyz’ calculus, but vector calculus is about geometric concepts invariant under coordinate transformations. This textbook takes the students from the geometry and algebra of vectors, to the key concepts and tools of vector calculus, including differential geometry of curves and surfaces, curvilinear coordinates, and capping off with a study of the essential elements of the calculus of functions of one complex variable. The book develops vector and index notation and is richly illustrated to help the student develop a solid visual understanding of the material. Every chapter is accompanied with many exercises that lead the student to practice key concepts and explore further interesting results. The development of this textbook was informed by the author’s teaching of many upper level applied mathematics and engineering physics courses at MIT and UW-Madison.