MA/MSCMT-04
December Examination 2016
M.A. M.Sc. (Previous) Mathematics Examination
Differential Geometry and Tensors
Paper MA/MSCMT-04
Time 3 Hours Max. Marks 80
Note: The question paper is divided into three sections B and C. Use of non-programmable scientific calculator is allowed in this paper.
Section A 8 × 2 16
Note: Section contain 08 Very Short Answer Type Questions. Examinees have to attempt all questions. Each question is of 02 marks and maximum word limit may be 30 words.
Define skew-curvature.
Define osculating plane.
Write formula for Torsion of the involute.
Interpretate envelope of family of curves geometrically.
Define surface of revolution.
Write geometrically significance of second fundamental theorem.
Define trajectory of family of curves.
(viii) Define skew symmetric tensor.
402
MA/MSCMT-04 1600 3 (P.T.O.)
MA/MSCMT-04 1600 3 (Contd.)
402
Section B 4 × 8 32
Note: Section contain 08 Short Answer Type Questions.
Examinees will have to answer any four questions.
Each question is of 08 marks. Examinees have to delimit
each answer in maximum 200 words.
Prove that divergence of Einstein tensor vanishes.
If Aij is the curl of a covarient vector then prove that
Aij, k Ajk, i Aki, j 0
Prove that outer multiplication of tensors is commucative and
associative.
Find the lines that have four point contact at with the surface
x4 3xyz x2 y2 z2 2yz 3xy 2y 2z 1 0
Prove that the principal normals at consecutive points of a curve do
not intersect unless 0
Find the evolutes of circular helix
x a cos y a sin z a q tan a
Prove that the metric of a surface is invariant under parametric
transformation.
State and prove Bonnet's theorem for parallel surface.
MA/MSCMT-04 1600 3
402
Section C 2 × 16 32
Note: Section contain 04 Long Answer Type Questions.
Examinees will have to answer any two questions.
Each question is of 16 marks. Examinees have to delimit
each answer in maximum 500 words.
10) Prove that Christoffel symbols of first kind are not tensor
quantities.
Prove that Riemannian co-ordinates are geodesic co-ordinates
with the pole at P0.
11) Prove that the generators of a developable surface are tangents
to curve.
On the paraboloid x2 y2 z find the orthogonal trajectories of
sections by the planes z constant.
12) Show that the surface ez cos x cos y is minimal surface.
Show that conjugate direction at a point P on a surface are
parallel to conjugate diameters of the indicatrin at point P.
13) Explain geodesic on a surface of revolution and derive expressions
for complete integral of the differential equation of geodesic on the
surface of revolution.