E. Łobos And B. Sikora: Advanced Calculus – Selected Topics

This book includes over 500 most challenging exercises and problems in calculus. Topical problems and exercises are discussed on set theory, numbers, functions, limits and continuity, derivative, integral calculus, Rolle’s theorem, mean value theorem, optimization problems, sequences and series. All the seven chapters recall important definitions, theorems and concepts, making this book immensely valuable to undergraduate students of engineering, mathematics, statistics, computer science and basic sciences.

Carefully dissects key concepts such as limits of sequence, convergence & divergence of monotone sequences, infinite limits, derivatives, integrals, and series of real numbers
Contextualizes subtle, commonly-misunderstood topics such as the notion of an infinite limit, the ε-δ definitions (for a better command of uniform versus pointwise continuity), error in local linear approximations, and integrability criteria

Carefully dissects key concepts such as limits of sequence, convergence & divergence of monotone sequences, infinite limits, derivatives, integrals, and series of real numbers
Contextualizes subtle, commonly-misunderstood topics such as the notion of an infinite limit, the ε-δ definitions (for a better command of uniform versus pointwise continuity), error in local linear approximations, and integrability criteria
Includes more than 120 exercises, with a solution manual available to instructors

My first book had a perilous childhood. With this new edition, I hope it has reached a secure middle age. The book was born in 1969 as an "innovative text­ book"-a breed everyone claims to want but which usu­ ally goes straight to the orphanage. My original plan had been to write a small supplementary textbook on differen­ tial forms, but overly optimistic publishers talked me out of this modest intention and into the wholly unrealistic ob­ jective (especially unrealistic for an unknown 30-year-old author) of writing a full-scale advanced calculus course that would revolutionize the way advanced calculus was taught and sell lots of books in the process.

This book is a student guide to the applications of differential and integral calculus to vectors. Such material is normally covered in the later years of an engineering or applied physical sciences degree course, or the first and second years of a mathematics degree course. The emphasis is on those features of the subject that will appeal to a user of mathematics, rather than the person who is concerned mainly with rigorous proofs. The aim is to assist the reader to acquire good proficiency in algebraic manipulation that can be used in critically assessing the results obtained from using graphics calculators and algebraic software packages.

This book provides a practical guide, complete with accompanying Matlab software, to many different types of polynomial and discrete splines and spline-based wavelets, multiwavelets and wavelet frames in signal and image processing applications.

Selected Topics in Advanced Calculus/Real Analysis with tons of Beautiful Proofs:)