These notes were prepared for the honors course inAdvanced Calculus, Mathematics 303304, Princeton University.The standard treatises on this subject, at any rate thoseavailable in English, tend to be omnibus collections of seeminglyunrelated topics. The presentation of vector analysis often degen-erates into a list of formulas and mantpulative exercises, and thestudent is not brought to grips with the underlying mathematical.ideas.In these notes a unity is achieved by beginning with anabstract treatment of vector spaces and linear transformations.This enables us to introduce a single basic derivative (the Frechetderivative) in an invariant form. All other derivatives (gradient,divergence, curl and exterior derivative) are obtained from it byspecialization. The corresponding theory of integration is like-wise unified, and the various multiple integral theorems ofadvanced calculus appear as special cases of a general Stokes'formula concerning the integration of exterior forms. In a finalchapter these concepts are applied to analytic functions of com-plex variables.