Advanced Calculus

Calculus is a very multipurpose and valuable tool. It is one of the greatest inventions of modern science. There is just so much it can offer. The success of calculus has been extended over time into various other important topics in mathematics. Some are: differential equations, vector calculus, calculus of variations, complex analysis and differential topology. Calculus is intensely integrated in every branch of the physical sciences, such as physics and biology. It is found in computer science, statistics, and engineering; in economics, business, and medicine. Modern developments such as architecture, aviation, and other technologies all make use of what calculus can offer. This book "Advanced Calculus" is intended to present coverage on harmonic analysis, integral transforms and equations, ordinary and partial differential equations, measure and integration, sequences and series, functional analysis, functions of real and complex variables, approximation and expansions. It gives an exposition of methods for approximating integrals, including those that are multidimensional. Applications of some of these methods in statistics are discussed. This subject area has recently gained prominence in several fields of science and engineering, and, in particular, Bayesian statistics. The material should be helpful to readers who may be interested in pursuing further studies in this area. Advanced calculus has had a fundamental and seminal role in the development of the basic theory underlying statistical methodology. With the rapid growth of statistics as a discipline, particularly in the last three decades, knowledge of advanced calculus has become imperative for understanding the recent advances in this field. Students as well as research workers in statistics are expected to have a certain level of mathematical sophistication in order to cope with the intricacies necessitated by the emerging of new statistical methodologies. This book is unique also in its special attention to the needs of applications and due to its unusually elaborate motivations coming from the history of mathematics and applications. This book is designed for advanced graduate students, practitioners as well as for those who are novice researchers in this field.