Advanced calculus, or called mathematical analysis in some universities, is fundamental in mathematical training. It is a two-semester four-credit course in the mathematics department of National Tsing Hua University. This book stems from lecture notes for the classes of advanced calculus I that I taught several times. The goal of this book is to provide rigorous but easy to follow mathematical proofs and a book that is convenient to read on portable digital devices. I try to make this book friendly and, hopefully, readers may find those colorful paragraphs and beautiful pictures of the book attractive. Many students find this course difficult as many abstract concepts are introduced at a rather rapid pace. But being able to think abstractly is probably one of the most important abilities in modern sciences and technologies. Learning mathematics is similar to learning language, we need to have enough vocabulary to express our mathematical ideas and we need to spend enough time on it to get connection of different concepts. I try to cut proofs into small pieces so that readers may verify them easier. Based on some knowledge of basic calculus, this book is self-contained and suitable for self-study. Exercises are provided at the end of each section.

Due to the length and size, materials for advanced calculus I is separated into two books: Advanced Calculus I-1 and Advanced Calculus I-2. We had 3 midterm exams and 1 final exam in the course of advanced calculus I. The book Advanced Calculus I-1 contains materials for midterm exam 1 and 2, and the book Advanced Calculus I-2 contains materials for midterm exam 3 and final exam. Practice exams and exam questions are attached to the books.