This textbook is suitable as a continuation course in analysis following the introductory lectures in the first year of study. Vector analysis is a classic subfield of mathematics with numerous applications, for example in physics. The book introduces students to the world of differential forms and analysis on submanifolds of Rn. Parts of the book can also be very well used for lectures in differential geometry or mathematical physics. The text includes many detailed examples with complete solutions, which are helpful for practice. Numerous illustrations clarify the text. At the end of each chapter, there are additional exercises. The authors offer a special service: any student who encounters difficulties while solving the exercises can directly contact the authors for assistance via email.