Vector and Complex Calculus A Textbook for Students of the Physical Sciences

Vector and complex calculus are essential for applications to electromagnetism, fluid and solid mechanics, and the differential geometry of surfaces. Moving beyond the limits of standard multivariable calculus courses, this comprehensive textbook takes students from the geometry and algebra of vectors, through to the key concepts and tools of vector calculus. Topics explored include the differential geometry of curves and surfaces, curvilinear coordinates, ending with a study of the essential elements of the calculus of functions of one complex variable. Vector and Complex Calculus is richly illustrated to help students develop a solid visual understanding of the material, and the tools and concepts explored are foundational for upper-level engineering and physics courses. Each chapter includes a section of exercises which lead the student to practice key concepts and explore further interesting results.

Fabian Waleffe, Professor of Mathematics, University of Wisconsin-Madison

Professor Fabian Waleffe is a Professor in the Department of Mathematics, Dept. of Mechanical Engineering, and the Dept. of Engineering Physics at the University of Wisconsin-Madison. He has previously help positions as a Postdoctoral Fellow at Stanford University/NASA Ames, Center for Turbulence Research, MIT Applied Math, and has taught numerous courses at all levels.

Part I: Vector Geometry and Algebra1. Find your bearings2. Vector spaces3. Dot product4. Cross Product5. Cartesian index notation6. Determinant7 Points, lines, planes, etc.8. Orthogonal transformations9. Matrices and tensorsPart II: Vector Calculus10. Kinematics11. Dynamics12. Curves13. Surfaces14. Curves on surfaces15. Curvilinear coordinates16. Fields17. ElectromagnetismPart III: Complex calculus18. Complex algebra geometry19. Elementary complex functions20. Functions of a complex variable21. Complex integrationIndex