Old Exam Papers June 2012 (Set 2)

7. (a) Show that at a point on a surface, where the Gausian
curvature is negative and equal to K, the torsion of
the asymptotic lines is K .
(b) Derive Mainardi-Codazi equations for surface
theory.
8. (a) Define covariant vector. A covariant vector has
2
components xy, 2y
z , xz in rectangular
coordinates. Determine its covariant components in
spherical polar coordinates.
(b) Show that the fundamental tensor g ij is a covariant
9. (a) If
symmetric tensor of the order two.
R
S
i
Tjk U Vdenotes Christoffel symbol of second kind
W
and g ij is the conjugate fundamental tensor then show
that :
g
x
mk
l
k
g g
il
m
mi ki
li
R S
U
T
VW R S
U
T
VW
(b) State and prove Ricci's theorem.
800 4 M.A./M.Sc.-M.T.-04
c h form
2 2 2 2
(b) Show that the curves du u c dv 0
an orthogonal system on the right helicoid :
b g
r u cos v, u sin v, cv
5. (a) If k and k n are the curvatures of oblique and normal
section's through the same tangent line and be the
angle between these sections, then show that
kn k cos .
2 2
(b) For the hyperboloid 2z 7x 6xy
y , prove that
the principal radii at the origin are 1
8 and 1
, and
2
that the principal sections are x 3y, 3x
y .
6. (a) Show that the necessary and sufficient conditions that
parametric curves be lines of curvature are F 0
and M 0, EN GL 0 .
(b) Prove that the cone
1
1 R
S
U
2 2 2 2
T
c h c hV
W
2 2
kxy z x z y z
passes through a line of curvature of the paraboloid
xy az
.
M.A./M.Sc.-M.T.-04 3 PTO