Global analysis. Exercises 6
1) Let T be a tensor field of type (r, s) on Rn
. Let x1
, ..., xn
and x1
, ..., xn
be two
coordinate systems. Find the relation between the components of the tensor field T
in these coordinates.
2) Let x1
, x2
and x1
, x2
be two coordinate systems related by
x1
=
rx1
(x1)2 + (x2)2
, x2
=
rx2
(x1)2 + (x2)2
,
where r > 0 is a fixed number. Knowing the components of the following tensors
in the first coordinate system, find the components of these tensors in the second
coordinate system.
* a11 = (x1
)2
+ (x2
)2
, a12 = x1
, a21 = x2
, a22 = 1
(x1)2+(x2)2 ;
* a1
1 = x1
, a2
1 = x1
+ x2
, a1
2 = x1
- x2
, a2
2 = x2
.
3) For w = (x2
+ y2
)dx + xzdz and  = zdy  dx + xdz  dx find dw, d, w  w,
  , w  , d(w  w), d(  ), d(w  ).
4) Find dw for
* w = x2
ydy - xy2
dx;
* w = xdy + ydx;
* w = f(x)dx + g(y)dy;
* w = xdy  dz + ydz  dx + zdx  dy.
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