Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers’ transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting a sound approach to the study. The traditional theorems of elementary differential and integral calculus are rigorously established, presenting the foundations of calculus in a way that reorients thinking toward modern analysis. After the invention of calculus there was a tremendous flowering of new results in mathematics and of applications in science and technology. Of particular note is Fourier’s work on the propagation of heat. The need to make sense of Fourier’s wonderful ideas, and to answer an increasing number of objections and logical problems with the immensely successful methods of calculus led to an increasing interest in providing a rigorous foundation for calculus. Cauchy (1789-1857) tried to formulate the notion of the derivative in terms of the concept of limit. He was not completely successful because he lacked a rigorous notion of limit. The present book is written to meet the requirements of undergraduate and postgraduate students of several Indian Universities. The subject matter both as regards the arrangement of chapters as well the contents of each chapter has been so graded proceeding from simple to more difficult topics that a student may follow the course with great ease and interest.
