Anna University Chennai 2007 M.Sc Computer Science Mathematics - exam paper

Mathematics

4

(b) Derive Gauss equations of surface theory.

14. (a) Solve the equation

xy" - (2x-l)y^/ + (x- 1) y = e^x.

(b) Find the series solution of the equation.

2 x² y" + x (2x + l)y'-y = 0.

15. (a) Define gamma function. Show that

(b) Obtain Rodrigues formula which gives an expression for P (x).

Name of the Candidate :

7 7 0 2 M.Sc. DEGREE EXAMINATION, 2007

( MATHEMATICS )

( FIRST YEAR )

( PAPER - III )

130. DIFFERENTIAL GEOMETRY AND DIFFERENTIAL EQUATIONS

( Revised Regulations )

May ]    [ Time : 3 Hours

Maximum : 100 Marks

SECTION-A (8x5 = 40)

Answer any EIGHT questions.

All questions carry equal marks.

1. Define the curvature and torsion of the curve. Find the curvature and torsion of the cubic curve

7 = (u, u², u³).

2.    If the radius of spherical curvature is constant, prove that the curve either lies on a sphere or has constant curvature.

3

V

3.    Prove that the curves of the family - = constant

u

are geodesics on a surface with metric

v² du² - 2 u v du dv + 2u² dv² ; (u > 0, v > 0).

4.    State and prove the normal property of geodesics.

5.    Find the Gaussian curvature of the surface

x = u + v, y = u - v, z = uv.

6.    If there is a surface of minimum area passing through closed space curve, show that it is necessarily a minimal surface.

7.    Solve the equation y" + 4y = 4tan2x using variation of parameters method.

8.    Find the general solution of

2x² y" + x (2x + l)y' - y = 0.

9.    Locate and classify the singular points of

(3x + 1) xy" - (x + l)y' + 2y = 0.

(i) (X%(x)) =

SECTION-B (3 x 20 = 60)

Answer any THREE questions.

All questions carry equal marks.

11. (a) Find the arc length of the curve given as the intersection of the surfaces

_x2

-r r = 1, x = a cos h

3l    hr

(b) State and prove the fundamental existence theorem for space curves.

12. (a) Obtain Liouvilles formula for K .

g

(b) State and prove Tissots theorem.

13. (a) Prove that a necessary and sufficient condition for a surface to be a developable is that its Gaussian curvature is zero.

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