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.