Lecture 7 Throughout the following discussion, f : R → R2 is a T -periodic smooth function (f (t + T ) = f (t) for all t), such that �f (t)� = 1 for all t. Lemma 7.1. One can write f (t) = (cos θ(t), sin θ(t)), where θ : R → R is a smooth function, unique up to adding constant integer multiples of 2π. Speciﬁcally, all such functions are of the form � t θ(t) = θ0 + det(f (τ ), f � (τ )) dτ. t0 where (cos θ0 , sin θ0 ) = f (t0 ). Definition 7.2. The degree of f is deg(f ) = 1 1 (θ(T ) − θ(0)) = 2π 2π � T det(f (τ ), f � (τ )) dτ ∈ Z. 0 Instead of [0, T ], one can take any other interval [t0 , t0 + T ]. Lemma 7.3. If deg(f ) �= 0, f is a surjective (onto) map to the unit circle. Proposition 7.4. Let �p� = 1 be a point on the circle with the following properties: (i) There are only ﬁnitely many 0 ≤ t1 < t2 < · · · < tm < T for which f (tk ) = p; (ii) each such tk satisﬁes f � (tk ) �= 0. In that case, m � deg(f ) = sign det(p, f � (tk )). k=1 Here is a popular application of degrees. Let f be more generally a T periodic function R → R2 , and q ∈ R2 a point not on its image. The winding number of f around p is the degree of the map f (t) − q/�f (t) − q�.