Madras University (UnOM) 2006 M.Sc Mathematics Differential Geometry and Differential Equations - Question Paper

Time: 3 hours
Maximum: 100 marks

part A - [4 x 20 = Marks 80]
ans ALL ques..
every ques. carries 20 marks.

1. (a) (i) Let y be a curve for which b varies differentiably with a arc length. Prove that a necessary and sufficient condition that y be a plane curve is that t = 0 for all points. (10)

(ii) obtain the orthogonal trajectories of the parts by the planes z = constant for the paraboloid X2 - Y2 = Z. (10)

Or

(b) (i) With usual notation prove that [r, r'',r''']= K' t. (10)

(ii) A helicoid is generated by the screw motion of a straight line skew to the axis. obtain the curve coplanar with the axis which generates the identical helicoids. (10)

2. (a) State and prove Minding's Theorem. (20)

Or

(b) (i) Derive the Rodrique's formula Kdr + dN = 0. (10)

(ii) Prove that a necessary and sufficient condition for a surface to be a developable is that its Gaussian curvature shall be zero. (10)

3. (a) (i) Let X1 (t), X2(t), … xn (0 be linearly independent solutions of a0 Wx (n) + a, Wx (n - 1) + - +

an (0 x = 0, t e I. Then show that any solution x (t) on
I is of the form
X (0 = Ci X1 (0 + C2 X2 (t) + "' + Cn Xn (t), t E
where C1 I C21 ... Cn are a few constants. (10)

(ii) obtain the power series solution for the formula (10)

x' - 2tx' + 2x = 0.

Or

(b) (i) State and prove Abel's formula for Wronskian. (10)

(ii) find the series solution for the formula x' + tx' + x = 0. (10)

4. (a) (i) Prove that (det (D) (tr A) (det, (D) where A (t) is a n x n matrix which is continuous on I and (D is the matrix that satisfies the formula V = AW (D - (10)

(ii) obtain L, K and h for the initial value issue (10)
X,(t) = X2 + t2,X (0) = 0, R= f(t, x) < 1,

Or

(b) State and prove Picard's theorem for the existence of a unique solution for a class of nonlinear initial value issues. (20)

part B - [10 x two = Marks 20]
ans any TEN ques..
every ques. carries two marks.

5. State Serret-Frenet formulae.

6. describe curvature and torsion of a curve.

7. describe osculating sphere.

8. State Gauss-Bonnet Theorem.

9. describe Helicoids.

10. Write down the canonical equations for Geodesics.

11. Write the Liouville s formula for Kg.

12. describe minimal surfaces and ruled surfaces.

13. Show that e"' and e 121 where M, # M2 are linearly independent in any interval I.

14. obtain the general solution of
x' + x' - 6x 0, t c= I.
15. obtain the singular points of the differential formula


(t - 1) X, + tX' + one X = 0.
t

16. describe fundamental matrix of a system of differential equations.

17. State Cauchy-Peano existence theorem for nonlinear equations.

18. describe Lipschitz condition.

19. State Gronwall Inequality.

Earning:   Approval pending.