Global analysis. Exercises 3
1) Find the Lie brackets of the following vector fields:
* X = sin u 
v
+ cos v 
u
, Y = u 
u
+ v 
v
;
* X = z2 
x
+ xy 
y
, Y = xyz 
x
+ y2 
y
+ x 
z
.
2) Let M be a 1-dimensional manifold, X, Y vector fields on M, Xx = 0 for all
x  M and [X, Y ] = 0. Prove that Y = cX, where c  R is a constant.
3) Find the vector fields defined by the following flows:
* FlX
t (x, y) = (5t + x, 4t + y);
* FlX
 (x, y) = (x cos  - y sin , x sin  + y cos ).
4) Find the integral curves of the vector fields:
* X = x 
x
+ y 
y
;
* X = (x + y) 
x
+ y 
y
;
* X = x2 
x
+ y2 
y
.
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