حساب دیفرانسیل و انتگرال پیشرفته با کاربردهایی در فیزیک، شیمی و فراتر از آن بررسی شده است

Are you a student that is often puzzled about how to translate your math background to applications in your science and engineering classes?Are you a mathematical enthusiast who loves challenging calculus problems?In this example-focused book, Hamza Alsamraee presents a plethora of techniques and methods to aid the regular student develop their skills to transition to more advanced mathematics.In Advanced Calculus Explored, you will findAdvanced mathematical and scientific concepts broken down for the beginnerVery important special functions such as the gamma, beta, polygamma, and zeta functions introduced, extensively discussed, and used in real-world applicationsDozens of proofs of famous equations due to well-known mathematicians such as Euler, Gauss, etc.Applications of the methods discussed in topics ranging from astrophysics to pharmaceutical engineeringA variety of illustrations to strengthen both mathematical and scientific understandingMany exercise problems accompanying each chapter, with answers!Advanced Calculus Explored equips readers with the tools they need for success in their STEM courses and provides a multitude of nonstandard problems to engage the mathematically curious reader, enabling the reader to achieve their academic goals. Advanced Calculus Explored helps build a stronger, more intuitive understanding of calculus that will serve the reader in advanced mathematics and science courses. About the Author Preface I Introductory Chapters Differential Calculus The Limit L'Hopital's Rule More Advanced Limits The Derivative Product Rule Quotient Rule Chain Rule Exercise Problems Basic Integration Riemann Integral Lebesgue Integral The u-substitution Other Problems Exercise Problems Feynman's Trick Introduction Direct Approach Indirect Approach Exercise Problems Sums of Simple Series Introduction Arithmetic and Geometric Series Arithmetic-Geometric Series Summation by Parts Telescoping Series Trigonometric Series Exercise Problems II Series and Calculus Prerequisites Introduction Ways to Prove Convergence The Comparison Test The Ratio Test The Integral Test The Root Test Dirichlet's Test Interchanging Summation and Integration Evaluating Series Introduction Some Problems Harmonic Numbers Exercise Problems Series and Integrals Introduction Some Problems Exercise Problems Fractional Part Integrals Introduction Some Problems Open Problems Exercise Problems III A Study in the Special Functions Gamma Function Definition Special Values Properties and Representations Some Problems Exercise Problems Polygamma Functions Definition Special Values Properties and Representations Some Problems Exercise Problems Beta Function Definition Special Values Properties and Representations Some Problems Exercise Problems Zeta Function Definition Special Values Properties and Representations Some Problems Exercise Problems IV Applications in the Mathematical Sciences and Beyond The Big Picture Introduction Goal of the Part Classical Mechanics Introduction The Lagrange Equations The Falling Chain The Pendulum Point Mass in a Force Field Physical Chemistry Introduction Sodium Chloride's Madelung Constant The Riemann Series Theorem in Action Pharmaceutical Connections The Debye Model Statistical Mechanics Introduction Equations of State Virial Expansion Lennard-Jones Potential Blackbody Radiation Fermi-Dirac (F-D) Statistics Miscellaneous Volume of a Hypersphere of Dimension N Spherical Coordinates Calculation Discussion Applications Mathematical Connections V Appendices Appendix A Appendix B Acknowledgements Answers Integral Table Trigonometric Identities Alphabetical Index