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.

1 - Find your bearings
2 - Vector spaces
3 - Dot product
4 - Cross Product
5 - Cartesian index notation
6 - Determinant
7 - Points, lines, planes, etc.
8 - Orthogonal transformations
9 - Matrices and tensors
10 - Kinematics
11 - Dynamics
12 - Curves
13 - Surfaces
14 - Curves on surfaces
15 - Curvilinear coordinates
16 - Fields
17 - Electromagnetism
18 - Complex algebra and geometry
19 - Elementary complex functions
20 - Functions of a complex variable
21 - Complex integration

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 held positions as a Postdoctoral Fellow at Stanford University/NASA Ames, Center for Turbulence Research, Assistant Professor of Applied Mathematics at MIT, and has taught numerous courses at all levels.