Mappings of a plane into itself

As stated in the introduction, our purpose in Part II is to prove an existence theorem for solutions of pairs of simultaneous equations. This theorem is stated in Section 18, and its proof is completed in Section 26. Sections 27 through 36 apply the theorem to questions about fixed points of mappings, singularities of vector fields, and zeros of polynomials. To formulate the main theorem, we must develop two-dimensional analogs of the one-dimensional concepts of Part I. The crucial concept needed is that of the winding number of a closed curve in a plane about some point of the plane not on the curve. We shall give first an intuitive definition of this notion together with an intuitive proof of the main theorem (Sections 17, 18). In Sections 19–26, the definition is made precise and the proof rigorous.

Recall that the main theorem of Part I deals with a mapping f: [a, b] → R of a segment into a line, and gives conditions on a point y ∈ R under which it could be asserted that y is in the image f[a, b] (e.g. fa ≤ y ≤ fb). The main theorem of Part II will deal with a mapping f: D → P of a portion D of a plane P(= R2) into P, and will give conditions on a point y ∈ P under which it can be asserted that y is in the image fD.