In the book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by David A. Cox, John Little, Donal O'Shea, the rational mapping is defined as follows.
In the book "Algebraic Geometry: A First Course" by Joe Harris, the rational mapping is defined as follows.
What are differences between these two definitions? What are the regular maps in Definition 4? Thank you very much.