By Lefschetz S
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Example text
10) Given. a sufficiently small n-cell ~in ir0 containing M, §.. trajectory passing through §.. point M' of ~\intersects -rr1 in§.. single point N1 such ·chat M' ~ N1 defines §.. topological mapping e of ~ onto §.. similar Bf," C 1T1, containing N and of course N = eM. §™. and Kg~ small enough, let I\ be the arc M'N 1 of the traiectm. /\(t) defines §.. pp~ cl> ·;f the cylinde~ I x ·~ such that cl> (I • M) = MN, cl> (Ix M1 ) = I\. This last result embodies essentially the so-called "field" theorem for minimizing arcs in the Calculus of Variations.
The argument could be extended without particular difficulty to the non-analytical case. 4. 2) dt AX. = Written out explicitly it takes the form dxih(t) dt = 2: aij(t)xjh(t). 3). 3). 4) I X(t) I t S0 I X(t 0 ) I exp = (tr A )dt ) , t for all t 0 , t EI. x J I where the terms,unwritten in each determinant are as in X itself. In the last determinant there are proportional rows unless i = j. XI. 4) follows by integration. An immediat_e consequence of ( 4. 5) If IX(t)l -=f o for some t EI then it is -=f o for all t E I.
X 1 , ••• , ~,t ), admitting generally all real values (more exceptionally all complex values) as the range of functions and variables other than t. It is for such systems alone that sufficiently extensive existence theorems are available. The independent variable t will often be referred to as the time. This is justified on the ground that many systems of differential equations arise from problems in dynamics or other branches of mathematical; physics. 3 .. 1) let (x 1 , •.. ,~)be considered as a vector 1 in a _spdce 7/x.