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ANALYSIS OF SURFACES AT SMALL DEFORMATIONS
For this surface we have:
1
11 = 1 22 =
41
1 + 4 2 1 + 42 2 ; 2 11 = 2 22 =
b11 = b22 =
1
12 = 2 12 = 0;
2
p
1 + 4 2 1 + 42
From (14), (15) and (16) we have = and
@
@2
@
@1 + @
@2
For y 1
, respectively y 2
we get
respectively
42
1 + 4 2 1 + 42 2 ;
; b12 = 0:
@
= 0; (19)
@1
= 0: (20)
@y
@1 = @r
@1 + @r
@2 = (; ; 2 1 22);
@y
@2 = @r
@1
Hence it follows
@r
@2 = (; ; 2 1 + 22):
dy = @y
@1 d 1+ @y
@2 d 2 = (; ; 21 22)d1+(; ; 21+22)d2:
By integrating, we get the rotation eld of hyperbolic paraboloid
in the form
y(1; 2) = (y1(1; 2); y2(1; 2); y2(1; 2)).
Now we determine the innitesimal deformations eld of elliptic
paraboloid. It appears, that
dz = y ¢ dr =
e1 e2 e3
y1(1; 2) y2(1; 2) y3(1; 2)
d1 d2 21d1 22d2
= ( 2y2(1; 2)1; y3(1; 2) + 2y1(1; 2)1; y2(1; 2))d1+
+( 2y2(1; 2)2 y3(1; 2); 2y1(1; 2)2; y1(1; 2))d2:
By integrating, we get the innitesimal deformations eld of elliptic
paraboloid
z(1; 2) = (z1(1; 2); z2(1; 2); z3(1; 2)).
=
63