Annamalai University 2008-1st Year M.Sc Mathematics Differential Geometry and Differential Equations ( ) - Question Paper
Course : M.SC mathematics 1st year.
Subject : Differential Geometry and Differential Equations ( 1st year ) Dec 2008
4
(b) Establish the normal property of a geodesic.
13. (a) State and prove Monges theorem.
(b) Derive Gauss equations of surface theory.
14. (a) Find the general solution of the equation
x2 y" + 2xy' 2y = 0.
(b) Find the series solution of the equation 2x2y" + x(2x+ l)y' - y = 0.
15. (a) Derive Rodrigues formula giving an
expression for Pn(x).
(b) Define Gamma function. Show that
Name of the Candidate :
16 3 7
M.Sc. DEGREE EXAMINATION, 2008
(MATHEMATICS)
(FIRST YEAR)
( PAPER - III)
130. DIFFERENTIAL GEOMETRY AND DIFFERENTIAL EQUATIONS
(Revised Regulations )
May ] [ Time: 3 Hours
Maximum : 100 Marks
SECTION - A (8 x 5 = 40)
Answer any EIGHT questions.
Each question carries FIVE marks.
1, Determine the curvature and the torsion of the cubic curve
?= (u, u2, u3).
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2. Define the terms :
Osculating plane, point of inflexion, inflexional line.
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3. A helicoid is generated by the screw motion of a straight line skew to the axis. Find the curve coplanar with the axis which generates the same helicoid.
4. Explain the concept of angle between curves and state the expressions for the angle between parametric curves.
5. Explain the second fundamental form.
6. What is an umbilic ?-Explain. State a property of a point which is not an umbilic.
7. Find a particular solution of the equation
y" - y' - 6y = e~\
first by undetermined co-efficients and then by variation of parameters.
8. Find the general solution of the equation
9, Locate and classify the singular points of the equation
x3(x-l)y" 2(x - 1) y' + 3xy = 0.
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10. Prove that:
SECTION - B (3 x 20 = 60)
Answer any THREE questions.
Each question carries TWENTY marks.
11. (a) State and prove the fundamental existence theorem for space curves.
(b) Prove that a space curve is a helix if and only if, the ratio of the curvature to torsion is constant at all points.
12. (a) Find a surface of revolution which is isometric with a region of the right helicoid.
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