Introduction to Chapter 4: Calculus on a Surface
This chapter begins with the definition of a surface in
R3 and with some standard ways to construct surfaces.
Although this concept is a more-or-less familiar one, it is not as widely
known as it should be that each surface has a differential and
integral calculus strictly comparable to the usual calculus on the
Euclidean plane R2.
The elements of this calculus--functions, vector fields, differential
forms, mappings--belong only to the surface and not to the
Euclidean space R3 in which the surface is located.
Indeed, we shall see in the final section that this calculus survives
undamaged when R3 is removed leaving just the
surface and nothing more.
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