Mathematical Analysis: An Introduction (Undergraduate Texts in Mathematics) Book

This is a textbook suitable for a year-long course in analysis at the advanced undergraduate or beginning graduate level. It is intended for students with a strong background in calculus and linear algebra. The first semester of this course is the basic introductory course in analysis, introducing the words "compact", "complete" " connected", "continuous", "convergent", etc. Among traditional purposes of such a course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. The topics covered in the second semester, and the second half of this book, are differentiation (of vector-valued functions of several variables), integration, and the connection between these concepts which is displayed in the theorem of Stokes, in its general form. Also included are some beautiful applications of the theory such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions.