Global analysis. Exercises 4
1) Which of the following distributions on R3
\{(0, 0, 0)} are involutive:
* the distribution generated by the vector fields X = x 
x
+y 
y
+z 
z
, Y = 
x
+ 
y
;
* the distribution generated by the vector fields
X = xyz 
x
+ y2 
y
, Y = x 
x
+ (z + y) 
z
.
2) Prove that any 1-dimensional distribution is involutive.
3) Find the dimensions of the spaces r
V and r
V if dim V = n.
4) Prove that 2
V = S2
V 2
V
5) Let V be a vector space, A  r
V , e1, ..., en and e1, ..., en bases of V and B the
transition matrix from the first basis to the second one. Find the relation between
the components Ai1...ir
and Ai1...ir of the tensor A in these bases.
6) Let A  3
V . Prove that Sym(SymA) = SymA, Alt(AltA) = AltA,
Sym(AltA) = 0, Alt(SymA) = 0.
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