This unique compendium deals with the differentiation and integration of vector functions. It examines critical effects and extracts important features using powerful tools of differentiation and integration. Techniques and codes for computing the divergence, curl, and gradients of a given field function, which reveal the mathematical behavior of the vector field, are discussed. Green's theorem, Stokes's theorem, and Gauss's formula, along with their novel extensions, are presented in detail with applications such as the smoothed gradient method.

Written in Jupyter notebook format, the book offers a unified environment for theory description, code execution, and real-time interaction, making it ideal for reading, practicing, and further exploration.

About the Author

Gui-Rong Liu received his PhD from Tohoku University, Japan, in 1991. He was a Postdoctoral Fellow at Northwestern University, USA, from 1991–1993. He was a Professor at the National University of Singapore until 2010. He is currently a Professor at the Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, USA. He was the Founder of the Association for Computational Mechanics (Singapore) (SACM) and served as the President of SACM until 2010. He served as the President of the Asia-Pacific Association for Computational Mechanics (APACM) (2010–2013) and an Executive Council Member of the International Association for Computational Mechanics (IACM) (2005–2010; 2020–2026). He authored a large number of journal papers and books including two bestsellers: Mesh Free Method: Moving Beyond the Finite Element Method and Smoothed Particle Hydrodynamics: A Meshfree Particle Methods. He is the Editor-in-Chief of the International Journal of Computational Methods and served as an Associate Editor for IPSE and MANO. He is the recipient of numerous awards, including the Singapore Defence Technology Prize, NUS Outstanding University Researcher Award, NUS Best Teacher Award, APACM Computational Mechanics Award, JSME Computational Mechanics Award, ASME Ted Belytschko Applied Mechanics Award, Zienkiewicz Medal from APACM, the AJCM Computational Mechanics Award, and the Humboldt Research Award. SACM Medal from the Association of Computational Mechanics (Singapore). He has been listed as one among the world's top 1% most influential scientists (Highly Cited Researchers) by Thomson Reuters for a number of years.

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