PGMT-10A [PT/10/XA(i) & XA(ii)] PGMT-10A [PT/10/XA(i) & XA(ii)] 2

PG-Sc.-6553-P [ P.T.O. PG-Sc.-6553-P

POST-GRADUATE COURSE Term End Examination — December, 2013 / June, 2014

MATHEMATICS Special Paper : Pure Mathematics

Paper - 10A(i) : Advanced Differential Geometry Time : 2 Hours Full Marks : 50

( Weightage of Marks : 80% ) Special credit will be given for accuracy and relevance

in the answer. Marks will be deducted for incorrect spelling, untidy work and illegible handwriting.

The marks for each question has been indicated in the margin.

( Notations have their usual meanings.)

Answer Question No. 1 and any four from the rest.

1. Answer any five questions : 2 × 5 = 10

a) Find the Jacobian matrix of :φ IR 2 → IR 2

given by ),(),( uvevu u=φ .

b) If C is a constant function on a differential manifold M and X is a tangent vector to some curve on M, show that X.C = 0

c) Evaluate [ X, X ].

d) Show that df is a 1-form, for every )(MFf ∈ .

e) Verify that abba LRRL = .

f) Define Torsion tensor field on a manifold M.

g) Define a gradient vector field.

2. a) Find a functional relation between the two local coordinate systems defined in the overlap region of any point of a manifold M.

7

b) Let a curve σ

on IRn be given by

tba iii +=σ , ni ...,,1= . Find the tangent vector to the curve σ . 3

3. a) Show that jji

i

ji

i

ji

xxxYX

∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

ξ∂η−

∂

η∂ξ= ∑

,],[

where ii

xX

∂

∂ξ= , jj

xY

∂

∂η= . 7

b) Compute ],[ YX

where 1xX

∂

∂= ,

32 xe

xY

ix

∂

∂+∂

∂= . 3

4. a) Let )2,1(, =iYX ii be two f-related vector

fields on manifolds M and N respectively. Show that the vector fields ],[ 21 XX and

],[ 21 YY are f-related. 6

b) Prove that

( ){ }YYtt

limYX tsss *** )(10],[)( +φ−φ→=φ 4

5. a) Test whether ( ) yyxxxy dd 221 −+=ω is

closed or not. 3

3 PGMT-10A [PT/10/XA(i) & XA(ii)]

PG-Sc.-6553-P

b) If ω is a 1-form, prove that

( ) ( ) [ ]( ){ }21122121 ,)()(21),(d XXXXXXXX ω−ω−ω=ω

7

6. a) Define a linear connection in the sense of Koszul. 5

b) If ),( YXTYY xx −∇=∇ , show that ∇ is a

linear connection. 5

7. a) If R is the curvature tensor of a Riemannian manifold ),( gM , show that

ZYXRfZfYXR ),(),( = ,

where )(,,),( MZYXMFf χ∈∈ . 7

b) Define Riemann curvature tensor of 1st kind on ( M, g ). 3

4 PGMT-10A [PT/10/XA(i) & XA(ii)] PGMT-10A [PT/10/XA(i) & XA(ii)] 5

PG-Sc.-6553-P PG-Sc.-6553-P [ P.T.O.

POST-GRADUATE COURSE Term End Examination — December, 2013 / June, 2014

MATHEMATICS Special Paper : Applied Mathematics

Paper - 10A(ii) : Fluid Mechanics Time : 2 Hours Full Marks : 50

( Weightage of Marks : 80% ) Special credit will be given for accuracy and relevance

in the answer. Marks will be deducted for incorrect spelling, untidy work and illegible handwriting.

The marks for each question has been indicated in the margin.

( Notations and symbols have their usual meaning. )

Answer Question No. 1 and any four from the rest.

1. Answer any five questions : 2 × 5 = 10

a) Define an irrotational motion of a homogeneous liquid. Show that velocity potential for an irrotational motion in a homogeneous incompressible fluid satisfies Laplace's equation.

b) Show that a stream function in a two-dimensional irrotational motion satisfies two-dimensional Laplace's equation and the stream function is constant in a motion on any rigid boundary.

c) For a fluid motion the complex potential w is given by zmw log= where iyxz += . Obtain an equation of eqipotential line and a streamline.

d) Write down the field equation of motion and

boundary condition for a flow of a viscous

fluid along a flat rigid plate.

e) For a simple harmonic progressive wave

given by )(sin),( ntmxatx −=η explain

phase velocity.

f) If for a vortex motion vorticity vector is

given by →Ω show that 0=Ω

→div .

g) Obtain velocity potential due to a point

source of strength m in an infinite liquid.

2. a) Find the velocity potential where a sphere

of radius a is moving in an incompressible

homogeneous liquid at rest at infinity with

uniform velocity U along any fixed direction.

Show that the kinetic energy of liquid is

241 UM ′ where M ′ is the mass of displaced

liquid. 5

b) Define a doublet in three-dimensional

motion and find out the velocity potential

due to a three-dimensional doublet of

strength μ . 5

6 PGMT-10A [PT/10/XA(i) & XA(ii)] PGMT-10A [PT/10/XA(i) & XA(ii)] 7

PG-Sc.-6553-P PG-Sc.-6553-P

3. a) Define strength of a vortex filament and

also show that the strength is constant

along the filament for all time in a

homogeneous incompressible liquid under

conservative field of force. 5

b) If ),( 11 θr , ),( 22 θr , .... ),( kkr θ be polar

coordinates at any time of a system of

rectilinear vortices of strength kkkk ,..., 21

then show that ∑ =2kr constant,

∑∑ π=θ 21

221 kkkr . 5

4. Show that a progressive wave in an infinite liquid

with a free surface, liquid particle describes circle

about its mean position near the free surface. 10

5. a) State and prove Milne-Thomson's Circle

theorem. 5

b) Find the image of a line source in front of a

circular cylinder. 5

6. Obtain the boundary layer approximation of

Navier-Stokes equation for a flow of viscous

incompressible liquid passing over a flat plate

and write down the boundary conditions for this

flow. 10

7. a) Discuss the steady flow of an

incompressible viscous fluid through an

elliptic tube of uniform cross-section and

find the volume rate of flow at any cross-

section. 7

b) Define boundary layer thickness and

explain its significance. 3

.