Advanced calculus

These notes were prepared for the honors course in Advanced Calculus, Mathematics 303 304, Princeton University. The standard treatises on this subject, at any rate those available in English, tend to be omnibus collections of seemingly unrelated topics. The presentation of vector analysis often degen- erates into a list of formulas and mantpulative exercises, and the student is not brought to grips with the underlying mathematical. ideas. In these notes a unity is achieved by beginning with an abstract treatment of vector spaces and linear transformations. This enables us to introduce a single basic derivative (the Frechet derivative) in an invariant form. All other derivatives (gradient, divergence, curl and exterior derivative) are obtained from it by specialization. The corresponding theory of integration is like- wise unified, and the various multiple integral theorems of advanced calculus appear as special cases of a general Stokes' formula concerning the integration of exterior forms. In a final chapter these concepts are applied to analytic functions of com- plex variables.