"Differential and Integral Calculus, Volume 2" "Unlike modern mathematicians who pursue their research apart from engineering or physical applications, Richard Courant was adverse to abstract theories and vague theorems. The topics covered in this set will provide the reader with a solid background to understanding the mathematics of heat conduction, electricity and magnetism, fluid dynamics and elasticity." -Amazon Review This book includes not only calculational techniques, but also an introduction to real analysis, good mathematical reasoning, and proof techniques. Courant leads the way straight to useful knowledge, and aims at making the subject easier to grasp, not only by giving proofs step by step, but also by throwing light on the interconnexions and purposes of the whole.
The Fundamentals of Mathematical Analysis, Volume 2 is a continuation of the discussion of the fundamentals of mathematical analysis, specifically on the subject of curvilinear and surface integrals, with emphasis on the difference between the curvilinear and surface ""integrals of first kind"" and ""integrals of second kind."" The discussions in the book start with an introduction to the elementary concepts of series of numbers, infinite sequences and their limits, and the continuity of the sum of a series. The definition of improper integrals of unbounded functions and that of uniform convergence of integrals are explained. Curvilinear integrals of the first and second kinds are analyzed mathematically. The book then notes the application of surface integrals, through a parametric representation of a surface, and the calculation of the mass of a solid. The text also highlights that Green's formula, which connects a double integral over a plane domain with curvilinear integral along the contour of the domain, has an analogue in Ostrogradski's formula. The periodic values and harmonic analysis such as that found in the operation of a steam engine are analyzed. The volume ends with a note of further developments in mathematical analysis, which is a chronological presentation of important milestones in the history of analysis. The book is an ideal reference for mathematicians, students, and professors of calculus and advanced mathematics.
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