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This book is a radical departure from all previous concepts of advanced calculus, declared the Bulletin of the American Mathematics Society, and the nature of this departure merits serious study of the book by everyone interested in undergraduate education in mathematics. Classroom-tested in a Princeton University honors course, it offers students a unified introduction to advanced calculus. Starting with an abstract treatment of vector spaces and linear transforms, the authors introduce a single basic derivative in an invariant form. All other derivatives gradient, divergent, curl, and exterior are obtained from it by specialization. The corresponding theory of integration is likewise unified, and the various multiple integral theorems of advanced calculus appear as special cases of a general Stokes formula. The text concludes by applying these concepts to analytic functions of complex variables.