Advanced calculus II-1

Advanced calculus occupies the most fundamental position in mathematics training. It is an essential path from elementary numerical calculation to higher-level abstract thinking. Advanced calculus is a two-semester course, four credits per semester, 200 minutes of lectures a week, plus 2 hours of recitation, weekly homework, and four exams in a semester in the mathematics department of National Tsing Hua University of Taiwan. It can be said that it is the most loaded undergraduate course. Many students are quite afraid of it. The author is trying to write some books that may help students in understanding the materials. In these books, all proofs are explained in detail, easy to understand and complete, with many graphics and colors. Also, they have to be easy to read on mobile phones. With these ideas in mind, the author produces a series of textbooks: Advanced Calculus I-1, I-2 and Advanced Calculus II-1, II-2. These books stemmed from lecture notes for courses of advanced calculus that the author taught in the mathematics department of National Tsing Hua University of Taiwan. A dim hope for these books is that students will be more receptive and more willing to spend time in this course. Mathematical knowledge and the ability of abstract thinking become more and more important in modern sciences and technology industries. Mathematics is indispensable from automated processes and big data processing. Knowledge of calculus is not enough for applied sciences. This may be a reason that regardless of its heavy loading, it attracts students from electrical engineering, computer science, financial engineering, management and medical school to take. In order to facilitate the use on mobile phones, the author needs to make the files of the books small. The content of Advanced Calculus II is divided into two books: Advanced calculus II-1 and Advanced Calculus II-2. Each contains two midterm exams. Exams and practice exams are all attached to the books. Each section is accompanied by exercises. These books can be regarded as self-complete and suitable for self-study.