By J. M. Aroca, R. Buchweitz, M. Giusti, M. Merle
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Extra info for Algebraic Geometry, la Rabida, Spain 1981: Proceedings
So, each continuous vector field n, defined in E3 including infinity, defines a continuous mapping of S3 onto S2. The inverse statement is also true: each continuous mapping of S3 onto S2 we can put into correspondence with a unit vector field in E3 having a limit value at infinity. To do this, if A E S2 corresponds to P E S3 then define the vector of the field at Q E E3 setting n = OA. The continuous vector fields n, (x) and 02(x) defined in E3 + oo are said to be homotopic if there is a continuous family of vector fields n(x, t), 0 < t < I such that n(x,O) = n, (x), n(x, 1) = n2 (x).
Differentiating b, we find db ds= -n dip dYz =-ncd. VECTOR FIELDS IN THREE-DIMENSIONAL EUCLIDEAN SPACE 21 From the equation dr ds = cos as + sin ab we find dx /ds = cos a. Hence, d b/ds = -nc cos a. We rewrite equation (15) as d2r dS` ,s ) - - sin a cos ac. Therefore, equation(14) of the shortest line has the form d2a _ -c- sin a cos a. Suppose that da/ds 34 0. Multiplying by 2da/ds and integrating, we find dal2 Cds/ =c2cos2a+c1, where cl is a constant. We find the function a(s) from the following equation: r s=J da c2 cos2 a + cl - To find a position vector of the shortest line we use the representation of d r/ds via a and b, which in coordinate form is _ - sin a sin gyp, d-2 = sin a cos gyp,- = cos a, (22) where the dependence of W from s and a is given by dcp= -ccosads= - c cos ada CcoS2a+cl Therefore sin (cp + c2) = -c sin a/ c'- -c1.
If we rotate the coordinate axes by angle a in such a way that the axis Ox coincides with I then we obtain the previous situation. VECTOR FIELDS IN THREE-DIMENSIONAL EUCLIDEAN SPACE 27 FIGURE 7 The new coordinates u, v, w are related to the old coordinates by the following formulas: u= cos ax+sinay, v= -sin ax+cos ay, Let 77j, 7r, 713 be the new components of the vector field n on the sphere which corresponds to the diameter I. 2+1,2 V 173= K,2+V2 The old components of n have the following expression in terms of the new ones: I =ri1 cosa-772 sin a, = 77I sina+7h cos a, 3=113.
Algebraic Geometry, la Rabida, Spain 1981: Proceedings by J. M. Aroca, R. Buchweitz, M. Giusti, M. Merle