VALLIAMMAI ENGEINEERING COLLEGE
(S.R.M.NAGAR, KATTANKULATHUR-603 203)
DEPARTMENT OF CIVIL ENGINEERING
QUESTION BANK
I SEMESTER
MA 5151- ADVANCED MATHEMATICAL METHODS
Regulation – 2017
Academic Year 2018- 2019
Prepared by
Mr. N. SUNDARAKANNAN
VALLIAMMAI ENGNIEERING COLLEGE
SRM Nagar, Kattankulathur – 603203.
DEPARTMENT OF MATHEMATICS
Year & Semester
: I /I
Subject Code/ Name : MA 5151- ADVANCED MATHEMATICAL METHODS
UNIT I - LAPLACE TRANSFORM TECHNIQUES FOR PARTIAL DIFFERENTIAL
EQUATIONS
Laplace transform : Definitions – Properties – Transform error function – Bessel’s function – Dirac delta
function – Unit step functions – Convolution theorem – Inverse Laplace transform : Complex inversion
formula – Solutions to partial differential equations : Heat equation – Wave equation.
Part A
Bloom’s
Taxonomy
Level
S.No
Domain
1.
State the existence of Laplace Transform
BTL1
2.
Find the Laplace transform of unit step function
BTL4
Rememb
er
Analyze
3.
If L( f (t )  F (s) , then prove that L[e at f (t )]  F ( s  a)
BTL3
Apply
4.
2
Find L[t sin t ]
BTL4
Analyze
5.
Evaluate
BTL4
Analyze
6.
4 t
Find L[t e sin 3t ]
BTL4
7.
State Initial and Final value theorems for Laplace transform
BTL1
Analyze
Rememb
er
8.
1  et
Find L[
]
t
BTL4
Analyze
9.
State and prove that change of scale property in Laplace transforms
BTL3
10.
Define Laplace transform of Error function
BTL1
Apply
Rememb
er
11.
Find L1 (

12.
13.
14.
e t sin t
0 t dt
1 2
)
s 1
Determine L{3t  2e 3t }
2s  5
Find L1 [ 2
]
9s  25
u
 2u
 a2 2
What is a 2 in
t
t
sa
]
sb
15.
Find the inverse Laplace transform of log[
16.
Define Laplace transform of unit step function and find its Laplace
transform
17.
Find the inverse Laplace transform of log[
18.
Solve the integral equation y   3 y  2 ydt  t , given that y(0)=0
s 1
]
s 1
BTL4
Analyze
BTL3
Apply
BTL4
Analyze
BTL2
Understa
nd
BTL4
Analyze
BTL1
Rememb
er
BTL3
Apply
BTL3
Apply
BTL3
Apply
t

0
19.
Using Laplace transform method, solve
dy
 y  2 ,given that y(0)=2
dt
20.
Find Laplace transform of t e 2t cos t
BTL4
Analyze
Part B
3
Find (i) L[t sin 5t cos 2t ] (ii) L[sin t ]
1
(iii) L[t e t sin 2t ] (iv) L[
cos 2t  sin 3t
]
t
BTL3
Apply
(i) Verify Initial Value Theorem and Final Value Theorem for
f (t )  1  e  t (sin t  cos t )
2
1
BTL4
Analyze
BTL5
Evaluate
BTL4
Analyze
BTL5
Evaluate
(ii)Find the Laplace Transform of erf (t 2 )
(i) Derive the Laplace transform of error function and use it, find
L(erf ( t )
3
t
 b , for 0  t  b
(ii)Find the Laplace Transform of f (t )  
 2b  t , for b  t  2b
 b
given that f(t+2b)=f(t) for t>0
(i) State and prove Convolution theorem in Laplace trans forms,
(ii) Using Convolution theorem, find the inverse Laplace Transform of
4
s
2
( s  9)(s 2  25)
(i) Find the Laplace transform of (i ) J 0 (t ) (ii) t J 0 (t )
5
6

s
1 
t
L
;


2
2 2
(ii) Apply the convolution theorem to evaluate
 (s  a ) 
1 bt
(i) Find the Laplace transform of (1) erf ( t ) and (2)
(e  e  at )
t
s
)
(ii) Using Convolution theorem, evaluate L1 (
( s  a)(s 2  1)
BTL4
&
BTL3
Analyze
&
Apply
Using Laplace transform, solve the IBVP

7
PDE : ut  3u xx , BCs : u ( , t )  0, u x (0, t )  0,
2
ICs : u ( x,0)  30 cos5x.
Using Laplace transform, solve the IBVP
8
utt  u xx 0  x  l , t  0, u ( x,0)  0 , ut ( x,0)  sin(
x
l
) ,0  x  l ,
BTL5
Evaluate
BTL5
Evaluate
BTL5
Evaluate
BTL4
Analyze
BTL5
Evaluate
u (0, t )  0 and u (l , t )  0, t  0
Using Laplace transform, solve the IBVP
9
PDE : utt  u xx 0  x  1, t  0, BCs : u ( x,0)  u (1, t )  0, t  0
ICs : u ( x,0)  sin x, ut ( x,0)   sin x, 0  x  1
A string is stretched and fixed between two fixed points (0,0) and (l,0).
Motion is initiated by displacing the string in the form u   sin(
10
11
x
l
) and
released from rest at time t=0.Find the displacement of any point on the
string at any time t.
Using Laplace transform method solve the initial boundary value problem
described by
u tt  c 2 u xx , 0  x  , t  0 , subject to the
conditions u (0, t )  0 , u ( x,0)  0,
u t (0, t )  l for 0  x   and u is bouded as x  .
12
13
State and prove Initial and Final value theorems in Laplace trans forms
Derive the Laplace transform of Bessel functions
(i ) J 0 ( x) (ii) J 1 ( x)
s2 1
[log
]
L
(ii) Find
s2
BTL4
Analyze
BTL4
Analyze
BTL5
Evaluate
BLT5
Evaluate
BLT4
Analyze
BLT5
Evaluate
BLT5
Evaluate
1
14
A infinitely long string having one end at x=0 is initially at rest along xaxis. The end x=0 is given transverse displacement f(t), when t>0. Find the
displacement of any point of the string at any time.
Part C

1.
2.
 f (t ) 
(i) If L f (t ) : s   F ( s) , then prove that L 
: s    F ( s)ds (ii)
 t
 s
 cos 5t  cos 6t 
Find L 

t


Derive the Laplace transform of error function and use it find Laplace
transform of cos t / t
Using Laplace transform solve the following IBVP
PDE : kut  u xx ,0  x  l , 0  t  ,
3.
BCS : u (0, t )  0 , u (l , t )  g (t ) , 0  t  ,
IC : u ( x,0)  0, 0  x  l.
Using Laplace transform method, solve the IBVP described as
1
u tt  cost ,0  x   , 0  t  ,
c2
BCS : u (0, t )  0 , u is bounded as x  
IC : u t ( x,0)  u ( x,0)  0
PDE : u xx 
4.
UNIT II - FOURIER TRANSFORM TECHNIQUES FOR PARTIAL DIFFERENTIAL
EQUATIONS
Fourier transform : Definitions – Properties – Transform of elementary functions – Dirac delta
function – Convolution theorem – Parseval’s identity – Solutions to partial differential equations :
Heat equation – Wave equation – Laplace and Poisson’s equations.
Part A
1
Define Fourier transform
BTL1
Remember
2
State Fourier Integral theorem
BTL1
Remember
3
Find the Fourier transform of f ( x)  
BTL4
Analyze
BTL3
Apply
BTL1
Remember
BTL4
Analyze
BTL3
BTL4
BTL1
BTL5
BTL4
Apply
Analyze
Remember
Evaluate
Analyze
sin x, in 0  x   

0, otherwise

5
Write the existence conditions for the Fourier transform of the function
f(x).
Give Fourier transform pairs
6
Find the Fourier Transform of
7
8
9
10
11
State and prove modulation property in Fourier transform
State and prove change of scale property in Fourier transform
State and prove shifting property in Fourier transform
4
f ( x)   1 , in x  a
0, in x  a
a x
Find the Fourier Sine transform of e
Find the Fourier Sine transform of 1/x
,  x  
12
13
14
 x
Find the Fourier Sine transform of e
Does the function f(t)= 1 (--∞ < x < ∞) have a Fourier transform
representation?
Determine the frequency spectrum of the signal g (t )  f (t ) cos  c t
If F(α) is the Fourier Transform of f(x), then the Fourier Transform of
1
F ( / a )
a
BTL4
Analyze
BTL1
Remember
BTL3
Apply
BTL3
Apply
15
f(ax) is
16
17
18
Find the Fourier transform of Dirac Delta function
State convolution Theorem on Fourier Transform
State Parseval’s identity on Fourier Transform
BTL4
BTL1
BTL1
Analyze
Remember
Remember
19
Find the Fourier cosine Transform of
BTL4
Analyze
20
If F(s) is the Fourier transform of f(x), then find the Fourier transform of
f(x) cos ax
BTL5
Evaluate
BTL4
Analyze
BTL4
Analyze
f ( x)   x , 0  x  a
0, x  a
Part B
(i) Find the Fourier Transform of f ( x )  e
1
2
 x2
2
 x, in 0  x  1

(ii)Find the Fourier Transform of 2  x, in 1  x  2
0, in x  2

Find the Fourier Cosine and Sine Transform of e 2 x and evaluate the
integrals

(i)

cosx
 sinx
0  2  4d (ii) 0  2  4 d
(i) State and prove Parseval’s identity the Fourier transforms
(ii) Find the Fourier Transform of
3

that
(
0
f ( x)   1, if x  a
0, else
sin t

)dt 
2
t
(i)If the Fourier sine transform of f(x) is
4
(ii)Find the Fourier Transform of

that
(
0
hence deduce
BTL3
&BTL4
Apply &
Analyze

, find f(x).
1  2
f ( x)   1  x , if x  1
0, if
x 1
hence deduce
BTL5
Evaluate
BTL4
Analyze
BTL5
Evaluate
sin t 4

) dt 
t
3
(i) Find the Fourier Transform of f(x) defined by

f ( x)  1  x 2 , if x  1
0, if
5

(
0
x 1
hence evaluate
x cos x  sin x
x
) cos dx .
3
2
x
(i) Find the Fourier Transform of
6
f ( x)   1, if x  a
0, if
x a
hence deduce
that




sin t
sin t 2
0 ( t )dt  2 and 0 ( t ) dt  2


x 2 dx

2
2

(i) Using Fourier Sine Transform, prove that ( x  9)( x  16) 14
0
.
(ii) Find the Fourier transform of e

 (x
2
0
8
9
a x
if a  0 , deduce that
BTL5
Evaluate
BTL5
Evaluate
BTL5
Evaluate
BTL4
Analyze
BTL4
Analyze
BTL4
Analyze
BTL6
Create
BTL5
Evaluate
BLT5
Evaluate

dx
 3 , if a  0
2
 a ) 4a
A one-dimensional infinite solid,    x   ,is initially at temperature
F(x). For times t>0, heat is generated within the solid at a rate of g(x ,t)
units, Determines the temperature in the solid for t > 0
A uniform string of length L is stretched tightly between two fixed points
at x=0 and x=L. If it is displaced a small distance  at a point x=b, 0<b<L,
and released from rest at time t=0, find an expression for the displacement
at subsequent times.
Using the method of integral transform, solve the following potential
problem in the semi-infinite
Strip described by
10
PDE : u xx  u yy  0 , 0  x   , 0  y  a
Subject to BCs: u(x,0)=f(x) , u(x,a)=0, u(x,y)=0 ,
0  y  a , 0  x   and
u
tend to zero as x  
x
Using the finite Fourier transform, solve the BVP described by PDE:
V  2V
,0  x  6 , t  0 , subject to BC: Vx (0, t )  0  Vx (6, t ) ,

t x 2
11
IC :V(x,0)=2x.
Solve the heat conduction problem described
12
13
14
u
 2u
 k 2 , 0  x   , t  0 u (0, t )  u 0 t  0,
t
x
u
u ( x,0)  0 0  x  , u and
both tend to zero
x
as x  
Compute the displacement u(x ,t) of an infinite string using the method of
Fourier transform given that the string is initially at rest and that the initial
displacement is f(x),    x   .
Solve the following boundary value problem in the half-plane y > 0
described by
PDE : u xx  u yy  0 ,    x   , y  0
BCs : u( x,0)  f ( x) ,    x  
u is bounded as y  ; u and
u
both vanishas x  
x
Part C
(i)Using the Fourier cosine transform of e  ax and e  bx , show

that
 (x
0
1.
2
dx


2
2
 a )( x  b ) 2ab(a  b)
2
1, x  a 
 and
0, x  a 
(ii) Find the Fourier transform of f(x) defined by f ( x)  

hence evaluate


sin a cosx

d
a  x , x  a 
.
x  a 
0,
Find the Fourier transform of f(x) if f ( x)  
2.

2

 sin t 
Hence show that (i)  
 dt 
t 
2
0
4


 sin t 
(ii)  
 dt 
t 
3
0
BLT5
Evaluate
BLT4
Analyze
Solve the heat conduction problem described by
3.
 2u
u
 k 2 , 0  x   , t  0, u (0, t )  u 0 , t  0
t
x
u
u ( x,0)  0 0  x  , u and both tend to zero as x  
x
Find the steady state temperature distribution u(x,y) in a long square bar of
side  with one face maintained at constant temperature u 0 and other faces
BLT4
analyze
at zero temperature.
UNIT III - CALCULUS OF VARIATIONS
Concept of variation and its properties – Euler’s equation – Functional dependant on first and
higher order derivatives – Functionals dependant on functions of several independent variables –
Variational problems with moving boundaries – Isoperimetric problems – Direct methods – Ritz and
Kantorovich methods.
4.
Part A
1
2
3
4
Define a functional
Write Euler’s equation for functional.
Define isoperimetric problems.
State any two different forms of Euler’s equation
BTL1
BTL3
BTL1
BTL1
Remember
Apply
Remember
Remember
BTL3
Apply
BTL3
Apply
BTL1
BTL1
Remember
Remember
BTL3
Apply
BTL3
Apply
BTL4
Analyze
BTL3
Apply
BTL1
Remember
BTL3
Apply
BTL5
Evaluate
BTL3
BTL1
Apply
Remember
Test for an extremum the functional
5
1
I [ y( x)}   ( xy  y 2 2 y 2 y ) dx , y(0)  1 , y(1)  2.
0
6
7
8
9
Prove that the commutative character of the operators of variation and
differentiation.
Define ring method.
State Brachistochrone problem.
.Find the curves on which the functional
1
 ( y
2
 12xy) dx with y(0)  0 and y(1)  1 can be extremised.
0
Write the ostrogradsky equation for the functional
10
d
and  commutative
dx
11
Show that the symbols
12
Write Euler-Poisson equation.
Define moving boundaries.
13
Find the transversality condition for the functional
14
v
x1
 A( x, y).
x0
dy 2 

1  ( dx )  dx
1
15
16
17

2
Test for an extremum the functional I [ y ( x)}  ( xy  y  2 y 2 y ' ) dx
0
y(0)=1, y(1)=2.
Write down biharmonic equation.
State Hamilton’s Principle
18
19
20
State du Bois-Reymond
Explain direct methods in variational problem
Explain Rayleigh-Ritz method.
BTL1
BTL2
BTL2
Remember
Understand
Understand
BTL4
Analyze
BTL3
Apply
BTL4
Analyze
BTL4
Analyze
BTL4
Analyze
BTL5
Evaluate
BTL5
Evaluate
BTL3
Apply
BTL4
Analyze
BTL6
Create
BTL5
Evaluate
BTL5
Evaluate
BTL5
Evaluate
Part B
(i) Find the extremals of
1
(ii) Find the extremals of
(i)Solve the extremal
2
3
(ii)Find the extremal of the function
(i)Show that the straight line is the sharpest distance between two points in a
plane.
) and (
) is revolved about the x(ii) A curve c joining the points (
axis. Find the shape of the curve, so that the surface area generated is a
minimum.
(i)State and Prove Brachistochrone problem
.(ii) Show that the curve which extremize
4
5
given that
Find the shortest distance between the point (-1,5) and the parabola y 2  x
(i)Find the plane curve of a fixed perimeter and maximum area.
1
6
(ii)Find the extremals of the functional
 (1  y 
2
) dx.
0
Subject to y(0)  0 , y (0)  1 , y(1)  1 , y (1)  1.
Find the extremal of the functional

 (y
'2
 y 2  4 y cos x) dx. y(0)  0 , y( )  0.
0
7
Find the extremasl of the functional
1
I ( y, z )   ( y '  z ' 2 2 y ) dx; y (0)  1, y (1) 
2
0
z (0)  0, z 1)  1
8
9
10
3
,
2
Solve the boundary value problem
by Rayleigh Ritz method.
To determine the shape of a solid of revolution moving in a flow of gas with
least resistance.
Derive the Euler’s equation and use it, find the extremals of the
x1

functional V [ y ( x)}  ( y 2  y ' 2 2 y sin x) dx.
x0
Find an approximate solution to the problem of the minimum, of the
1
11
functional
J ( y)   ( y  2  y 2  2 xy ) dx. y(0)=0=y(1) by Ritz method
0
and compare it with the exact solution.
12
Find the solid of maximum volume formed by the revolution of a given
surface area.
Find the extremals of the functional
 2
13
v[ y ( x), z ( x)} 
 ( y
2
 z ' 2 2 yz) dx. Given that
0
y(0)  0 , y( 2)  1 , z (0)  0 , z ( 2)  1
14
Find the path on which a particle in the absence of friction will slide from
one point to another in the shortest time, under the action of gravity
BTL6
Create
BLT6
Create
BLT6
Create
BLT5
Analyze
BLT5
Analyze
Part C
1.
2.
3.
(i) Derive the Euler’s equation
(ii) Find the geodesic on a plane are straight lines
Derive Euler’s- Poisson equation and give some examples
Use Ritz method to find an approximate solution in the form y(x)=cx(x-1)
of the problem of extremising
of v[ y( x) 
 y
1
2

 y 2  2 yx dx ; y(0)  0  y(1)
0
4.
Write a short notes about Kanrovich Method
UNIT IV - CONFORMAL MAPPING AND APPLICATIONS
Introduction to conformal mappings and bilinear transformations – Schwarz Christoffel transformation –
Transformation of boundaries in parametric form – Physical applications : Fluid
flow and heat flow problems.
Part A
1
2
3
4
5
Define Conformal Mapping and give an example.
Define Isogonal trandformation
State the condition for the function f(z) to be conformal
Define Mobius transformation
Define cross ratio of Z 1 , Z 2 , Z 3 , Z 4
BTL1
BTL1
BTL1
BTL1
BTL2
Remember
Remember
Remember
Remember
Understand
6
7
8
Define Schwarz-Christoffel transformation.
Define fixed and invariant point of the transformation w=f(z)
BTL3
BTL1
BTL3
Apply
Remember
Apply
BTL4
Analyze
BTL4
Analyze
BTL5
Evaluate
BTL1
BTL1
BTL2
BTL1
BTL1
BTL1
BTL3
Remember
Remember
Understand
Remember
Remember
Remember
Apply
BTL2
BTL1
Understand
Remember
9
10
11
12
13
14
15
16
17
18
19
20
2z  5
z4
2
Find the critical point of the transformation W  ( Z   )(Z   )
z 1
Find the fixed points of w 
z2
Find the fixed points of the transformation w 
Let C be a curve in the z-plane with parametric equations x=F(t),
y=G(t).Show that the transformation z= F(  )+i G(  ) maps the curve C on
to the real axis of the w-plane.
Discuss the transformation defined by w= coshz.
Define equipotential lines.
Give the formula for Schwarz Christoffel transformation
Define Heat flux of heat flow.
State the basic assumptions of fluid flow.
Define streamline
Show that for a closed polygon, the sum of exponents in Schwarz
Christoffel transformation is -2.
Define complex temperature of heat flow
Define velocity potential, source and sink
Part B
1
(i) Prove that the bilinear transformation transforms circles of the z BTL2
plane into circles of the w plane, where by circles we include circles of
infinite radius which are straight lines
Understand
(ii) Define Bilinear transformation and find the bilinear transformation
z1  2 , z 2  0 , z 3  2 on to the po int s
w1   , w2  1 / 2 , w3  3 / 4.
(u, v) ( x, y)
(i) Prove that ( x, y) (u, v)  1
which maps
2
(ii) Discuss the Schwarz- Christoffel transformation
BTL3
Apply
3
(i) Define cross ratio of z1 , z 2 , z3 and Critical points.
(ii)Find a BLT
BTL-1
Remember
Find the bilinear transformation which maps the points
z=1, i, -1 into the points w=2, i, -2.Find the fixed and critical points of the
transformation.
(i) Prove that a harmonic function ( x, y) remains harmonic under the
transformation w=f(z) where f(z) is analytic and f ( z)  0
(ii) Discuss the motion of a fluid having complex
BTL2
Understand
BTL2
Understand
BTL4
Analyze
Find the complex potential due to a source at z=-a and a sink at z=a equal
strengths k, Determine the equipotential lines and streamlines and represent
graphically and find the speed of the fluid at any point.
(i)Find the complex potential for a fluid moving with constant speed V 0 in a
direction making an
angle  with the positive x axis
(ii) Determine the velocity potential and stream function
(iii) Determine the equations for the streamlines and equipotential lines.
BTL5
Evaluate
BTL3
Apply
a2
( z )  V0 ( z  ) where
The complex potential of a fluid flow is given by
z
V 0 and  are positive constants.(a) Obtain equations for the streamlines
BTL5
Evaluate
BTL4
Analyze
BTL3
Apply
BTL4
Analyze
BTL3
Apply
BTL4
Analyze
z1  0 , z 2  i , z  1 in to the po int s w1  1, w2  1 w3  0
4
5
6
7
8
9
10
potential ( z)  ik log z where k  0
Find the transformation which maps the semi-infinite strip bounded by v=0,
u=0 and v=b into the upper half of the
z-plane.
and equipotential lines, represent them graphically and interpret
physically.(b) Show that we can interpret the flow as that around a circular
obstacle of radius a.(c) Find the velocity at any point and determine its value
far from the obstacle
(d) Find the stagnation points.
Fluid emanates at a constant rate from an infinite line source perpendicular to
the z plane at z=0
(a) Show that the speed of the fluid at a distance r from the source is V=k/r,
where k is a constant(b) Show that the complex potential is
( z)  k ln z. (c) What modification should be made in (b) if the line
source is at z=a? (d) What modification is made in (b) if the source is
replaced by a sink in which fluid is disappearing at a constant rate?
11
(i)
(ii)
12
13
Find a bilinear transformation which maps points
z= 0, -i, -1 into w= i, 1, 0 respectively
Find the complex potential for a fluid moving with constant speed
V0

in a direction making an angle with the positive x-axis. Also determine
the velocity potential and stream function.
Find the transformation which transforms the semi infinite strip bounded by
v= 0, v= π and u=0 onto the upper half z-plane.
3 z

w
Show that the transformation
z  2 transforms circle with center(5/2, 0)
and radius 1/2 in the z-plane into the imaginary axis in w-plane and the
interior of the circle into the right half of the plane.
14
(i) Find the transformation that maps an infinite strip
w-plane onto the upper half of z-plane.
0  v   of the
(ii) Give the procedure for transformation of boundaries in parametric
form and hence find a transformation which maps the
1.
2.
x2 y2

 1 in the z-plane onto the real axis of the w-plane.
ellipse a 2 b 2
Part C
Find the bilinear transformation which maps the points z=1, i, -1 on
to the points w= i, 0,-i. Hence find (i) The image of |z|<1 (ii) The
invariant points of this transformation.
( z  i)
(i)Show that under the transformation w 
, real axis in the z-plane
( z  i)
BLT4
Apply
BLT5
Evaluate
BLT4
Analyze
BLT4
Analyze
is mapped into the circle|w|=1.Which portion of the z-plane corresponds to
the interior of the circle?
(ii) Find the critical points of w 
3.
4.
2z  5
z2  4
(i)Find the transformation which maps the semi infinite strip in the w-plane
into the upper half the z-plane.
(ii)Explain Schwarz-Christoffel transformation
(i) Write a short note about Fluid flow
(ii) Write a short note about Heat flow
UNIT V - TENSOR ANALYSIS
Summation convention – Contravariant and covariant vectors – Contraction of tensors – Inner
product – Quotient law – Metric tensor – Christoffel symbols – Covariant differentiation – Gradient Divergence and curl.
Part A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Define Tensor and Summation convention
Define Kronecker delta function of tensor
Define Covariant and Contravariant vectors
Define contraction of a tensor.
If Ai is a covariant and B i
an invariant
Define  i i
Define reciprocal tensors
Prove tha  pr, q   qr, p  
i
is a contravariant vector, show that Ai B is
g pq
x r
.
Define Symmetric and Skew-symmetric tensors
Define divergence of a contra variant vector
Prove that Kronecker delta (  ) is a mixed tensor of order 2.
Define Metric tensor and Conjugate tensor
State the Christoffel symbol of first and second kind.
If  is a function of the n quantities x i , write the differential of  using
the summation convention
i j
Write the terms contained in S  a ij x x , taking n=3.
If (ds)  (dr )  r (d )  r sin  (d ) ,find the value of [22.1] and
[13,3]
2
2
2
Prove that divAi 
i
i
prove that 
i
2
2
2
2
k
1 ( g A )
.
x k
g
 
(log g )

j  x i
i
Given A and B j are contra variant vectors, show that A B
variant tensor of rank 2
j
is contra
BTL1
BTL1
BTL1
BTL3
BTL3
Remember
Remember
Remember
Apply
Apply
BTL3
BTL1
BTL1
Apply
Remember
Remember
BTL1
BTL1
BTL4
BTL3
BTL2
BTL3
Remember
Remember
Analyze
Apply
Understand
Apply
BTL3
Apply
BTL6
Create
BTL4
Analyze
BTL3
Apply
BTL3
Apply
20
Prove that the property of the Christoffel symbols
ij , k  kj ,i 
g ik
BTL4
Analyze
BTL5
Evaluate
BTL4
Analyze
q j
Part B
1
dx dy
, in rectangular
dt dt
(i)In a contravarant vector has components
dr
d
in polar coordinates.
and
dt
dt
i
(ii) If A is a contravariant vector and B j is a covariant vector, prove that
coordinates, show that they are
Ai B j is a mixed tensor of order 2.
2
(i) State and prove Quotient law of tensor.
ij
(ii) Find g and g corresponding to the metric
(ds) 2  5(dx1 ) 2  3(dx 2 ) 2  4(dx 3 ) 2  6dx1dx 2  4dx 2 dx 3
3
Find the components of the first and second fundamental
tensor in spherical coordinates.
BTL6
Create
4
Find the components of the metric tensor and the conjugate tensor in the
cylindrical co-ordinates..
(i)Define Christoffel’s symbols. Show that they are not tensors
BTL3
Apply
BTL4
Analyze
BTL5
Evaluate
7
2
A covariant tensor has components x  y, xy, 2 z  y in rectangular coordinates. Find its covariant components in cylindrical co-ordinates.
Given the covariant components in rectangular co-ordinates
2 x  z , x 2 y, yz . Find the covariant components in
(i) Spherical polar co-ordinates (r, , ) (ii) Cylindrical co-ordinates(  ,  , z )
BTL4
Analyze
8
Show that the tensors g ij , g ij and the Kroneckor delta  j are constants with
BTL3
Apply
9
respect to covariant differentiation.
ij
(i) Find the g and g for the metric
BTL5
Evaluate
BTL5
Evaluate
BTL5
Evaluate
BTL5
Evaluate
BTL5
Evaluate
BTL5
Evaluate
5
ij
(ii) Prove that the covariant derivative of g is zero.
6
i
ds 2  5(dx1 ) 2  3(dx 2 ) 2  4(dx 3 ) 2  6dx1 dx 2  4dx 2 dx 3  0dx 3 dx1 (ii)
ij
Prove that the covariant derivative of g is zero.
10
If (ds)  (dr )  r (d )  r sin  (d ) , find the value of (i) [22,1]
2
2
12
13
2
2
1 
 and
22
(ii) 
and [13,3]
11
2
2
2
3 
 
13
Determine the metric tensors g ij , g ij
in cylindrical
co-ordinates.
If x and yare the components of a contra variant vector in rectangular coordinates, determine them in polar co-ordinates.
Prove tat
g ij
(i) x k
 [ik , j ]  [ jk , i ]
g ij
jl i 
im  j 
(ii)



g
g
 


x k
mk 
lk 
14
Given the metric
ds  (dx )  ( x ) (dx )  ( x ) sin ( x )(dx )
2
1 2
1 2
2
2
1 2
2
2
3 2
find the Christoffel symbols
[22 ,1] , [32 , 3]
2 
3

 and 
2 1
3


2
Part C
1.
2.
3.
4.
Explain Covariant, Contravariant and Invariant with suitable examples
Write a short note about metric tensor with suitable example.
State and prove Christoffel symbols of first and second kind
A covariant tensor has componets xy , 2 y  z 2 , xz in rectangular
coordinates. Find its covariant components in spherical coordinates
**********************
BLT4
BLT4
BLT6
BLT5
Analyze
Analyze
Create
Evaluate