ABSTRACT
Let us return to the set $ ( T ) situated in the unit sphere S 2 . The vector-valued function n( t l , t2) is the position vector of the points of $(T) . At the same time, n(t l , tZ) is a unit normal vector to s2 for any tl , t2. Hence, the area element dA of s2 at the points of $ ( T ) is
where (*, *, *) denotes the mixed product. Next, we will compute the "oriented" area of $(T) . For this purpose, we decompose T into domains T, where (n, n,, , n,,) are of constant signs. Then we sum the areas of domains T, where (n,n,,,n,,) is positive, and subtract the areas of domains T, where (n, n,, , n,,) is negative; we call the obtained number the "oriented" area of $(T) . Denoting the element of the "oriented" area of $ ( T ) by the same notation dA, we get
We have
Therefore,
To every continuous map p of a closed surface into the sphere one can assign some integer called the degree of cp; the pre-image of any point of the sphere consists of n points and the degree of p is equal to n by definition. It is known that the degree of cp is equal to the oriented area divided by 4~ of the image under cp. The degree of the map $, which we have constructed starting from the curves y,, we denote by I(yl,y2). From (29.1) it follows that
Thc integral on the right-hand side is called the Gauss integral. Since I(yl, y2) is an integer, it does not vary under continuous deformations of disjoint y,.
