sborník

Michal Benes
The latter condition is valid if there exist functions : R 2 ! R,
: R 2 ! R, : R 2 ! R such that
Hence
@y
@1 = @r
@1 + @r
@2 ;
dy = @y
@1 d 1+ @y
@2 d 2 =
If
@
@r
@2 @1 + @r
@2
@r
@1 + @r
@2
= @
@y
@2 = @r
@1
@1
d1+
@r
@1
@r
@1
@r
@2 : (11)
@r
@2
@r
@2
d2:
(12)
(13)
then (12) is the total dierential of the vector function y, by
integrating we get the eld y(1; 2).
It can be proved, that if the following partial dierential equations
are fullled
@
@2
@
@1 = 1 11 2 1 12 1 22 ; (14)
@
@1 + @
@2 = 2 11 2 2 12 2 22 ; (15)
b11 2b12 b22 = 0; (16)
where b ij be the coecients of the second fundamental form and i jk
denotes the Christoel's symbol of the surface, then (13) holds.
dz =
y ¢ @r d1 +
y ¢ @r d2 (17)
@1
@2
is the total dierential, we get the eld z(1; 2) by integration.
3 Innitesimal deformations of elliptic
paraboloid
We consider the vector equation of hyperbolic paraboloid
62
r = r(1; 2) = (1; 2; 2 1 2 2 + 1) (18)