The course in advanced calculus contained in this book has for many years been given by the author to students in the Massachusetts Institute of Technology. The choice of the subject matter and the arrangement of the material are the result of the experience thus gained. The students to whom the course has been given have been chiefly interested in the applications of the calculus and have felt the need of a more extensive knowledge than that gained in the elementary courses, but they have not been primarily concerned with theoretical questions. Hence there is no attempt to make this course one in analysis. However, some knowledge of theory is certainly necessary if correct use is to be made of the science; therefore the author has endeavored to introduce the students to theoretical questions and possibly to incite in some a desire for more thorough study. As an example of the method used, a proof of the existence of the definite integral in one variable has been given; for the multiple integral the proof has been omitted and simply the result stated. The student who has mastered the simpler case is in a position to rvad the more difficult case in easily accessible texts.