The document provides solutions to tutorial problems on differential geometry. It first shows that the differential of a function from a surface to 3D space is linear. It then calculates the Gauss map, Weingarten map, and principal curvatures for a sphere, surface of revolution, and other surfaces. The solutions involve parametrizing the surfaces and computing derivatives of the parametrizations.
An analysis of the symmetries of Cosmological Billiards;
Talk presented at
Fourteenth Marcel Grossmann Meeting - MG14, University of Rome "La Sapienza" , Rome, July 12-18, 2015,
Parallel Session Exact Solutions (Physical Aspects) on 14 July 2015
Date:
Tuesday 26th June 2012 - 16:30 to 16:45
Venue:
INI Seminar Room 1
Event:
[BSMW05] String Phenomenology 2012
Abstract:
Quantum Cosmology tackles the quantum description of the early universe. It is aimed as an accessible primer that covers the basics, critically discussing ideas and concepts that comprise our current knowledge. The scope for analyzing quantum cosmological models within a supersymmetric framework is pointed.
As much as possible, it summarizes what we know, what we think we know and what we think we do not know on an equal footing. It is focused for ‘young’, inquisitive minds eager to embark on in-depth research in this field. It is hoped to suggest the tools researchers will need to go on their own, pushing them to ask the right questions rather than seek definitive answers.
In this paper, we consider the scaling invariant spaces for fractional Navier-Stokes in the
Lebesgue spaces ( ) p n L R and homogeneous Besov spaces
, ( ) s n
p q B R respectively.
An analysis of the symmetries of Cosmological Billiards;
Talk presented at
Fourteenth Marcel Grossmann Meeting - MG14, University of Rome "La Sapienza" , Rome, July 12-18, 2015,
Parallel Session Exact Solutions (Physical Aspects) on 14 July 2015
Date:
Tuesday 26th June 2012 - 16:30 to 16:45
Venue:
INI Seminar Room 1
Event:
[BSMW05] String Phenomenology 2012
Abstract:
Quantum Cosmology tackles the quantum description of the early universe. It is aimed as an accessible primer that covers the basics, critically discussing ideas and concepts that comprise our current knowledge. The scope for analyzing quantum cosmological models within a supersymmetric framework is pointed.
As much as possible, it summarizes what we know, what we think we know and what we think we do not know on an equal footing. It is focused for ‘young’, inquisitive minds eager to embark on in-depth research in this field. It is hoped to suggest the tools researchers will need to go on their own, pushing them to ask the right questions rather than seek definitive answers.
In this paper, we consider the scaling invariant spaces for fractional Navier-Stokes in the
Lebesgue spaces ( ) p n L R and homogeneous Besov spaces
, ( ) s n
p q B R respectively.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
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Marine and Agriculture engineering,
Aerospace Engineering.
Some aspects of the oldest nearby moving cluster (Ruprecht 147)Premier Publishers
Based on the membership data retrieved from the Two Micron All Sky Survey (2MASS), we have computed some parameters of the moving open cluster Ruprecht 147, like, vertex, velocity, distance, distance modulus, and center of the cluster. All of these aspects were carried out using an algorithm due to Sharaf et al. (2000), into which error estimates of these parameters will be established in closed analytical forms (e.g. standard and probable errors of the vertex coordinates, angular distance, velocity of the cluster, parallaxes of member stars, and distance of the cluster).
Finally, we compared our results with other published values, which is in good agreement.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
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On Some Double Integrals of H -Function of Two Variables and Their ApplicationsIJERA Editor
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variables. A multiple integral and a multiple half-range Fourier series of the H -function of two variables are
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A Ring-Shaped Region Containing All or A Specific Number of The Zeros of A Po...IJERDJOURNAL
ABSTRACT: According to a Cauchy’s classical result all the zeros of a polynomial n j j j P z a z 0 ( ) of degree n lie in z 1 A , where n j j n a a A max0 1 . In this paper we find a ring-shaped region containing all or a specific number of zeros of P(z). Mathematics Subject Classification: 30C10, 30C15.
CONVEYOR BLOCKAGE DETECTION - In order to ensure the safe and reliable operation of belt conveyor, detecting blockage and control is very necessary. The main objective of this proposal is to detect the blockage/fault occurring in the conveyor using Programmable Logic Controllers -PLC. Conveyor blockages are caused by the faults such as belt tear up faults, oil level reduction fault, fire occurrence faults in the belt conveyor are not identified properly and thus leading to serious damage to the belt conveyor.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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1. Sample Assignment For Reference Only
Tutorial 1
1. Show that the differential pdG of a function G , from a surface S to 3
is linear.
(proof) Let be , ( )G F C S
. It is sufficient that for any , ,
( )d F G dF dG .
Assume that , p pp S X T S . Then
( ) ( )
( )
p p p p p
p p p p p
d F G X X F G X F X G
dF X dG X dF dG X
(end)
2. Calculate the Gauss map, the Wiengarten map and the principal curvatures for
(a) A sphere of radius, R ,
(solution) cos cos , cos sin , sinx R u v y R u v z R u !
sin cos , sin sin , cos
cos sin , cos cos , 0
u u u
v v v
x R u v y R u v z R u
x R u v y R u v z
2 2 2
, 0, cosE R F G R u
2 2
cosH EG F R u
2
2
cos cos , cos sin , sin
sin sin , sin cos ,0
cos cos , cos sin ,0
u
uv
v
r R u v R u v R u
r R u v R u v
r R u v R u v
2 2
3 3 3
( , , ) cos , ( , , ) 0, ( , , ) cosu v u v uv u vu v
r r r R u r r r r r r R u
2
2
3
2
3 3
2
2
1 cos
( , , )
cos
1
( , , ) 0
1 cos
( , , ) cos
cos
u v u
u v uv
u v v
R u
L r r r R
H R u
M r r r
H
R u
N r r r R u
H R u
-weingarten map:
3 2
2 4 2 3 2
1
0
cos 01 1
1cos 0 cos
0
GL FM GM FN R u R
FL EM FM ENEG F R u R u
R
A
-Gauss map:
2. Sample Assignment For Reference Only
2 2 2
2
2 2
2
(cos cos ,cos sin ,sin cos ),
(cos cos ,cos sin ,sin cos )
cos
u v
u v
u u
r r R u v u v u v
r r R
n u v u v u v
R ur r
-principle curvature:
1 2
1
k k
R
.
(b) A surface of revolution given by the curve ( )x f z rotated about the z axis, and
(solution)
( )cos
( )sin
x f z
y f z
z z
( ( )cos , ( )sin , )r f z f z z
2
2
2 2
2 2 2 2
( ( )cos , ( )sin ,1)
( ( )sin , ( )cos ,0)
( ) 1, 0, ( ) , ( ) ( ) 1
( ( )cos , ( )sin ,0)
( ( )sin , ( )cos ,0)
( (
z
z
z
z
r f z f z
r f z f z
E r f z F G r f z H EG F f z f z
r f z f z
r f z f z
r f z
)cos , ( )sin ,0)f z
2 2
2
( , , ) ( ) ( ), ( , , ) 0, ( , , ) ( )z z z zz
r r r f z f z r r r r r r f z
2 2
2
2 2 2
( ) ( ) ( )
,
( ) ( ) 1 ( ) 1
0 ( ) ( )
0,
( ) ( ) 1 ( ) ( ) 1 ( ) 1
f z f z f z
L
f z f z f z
f z f z
M N
f z f z f z f z f z
-weingarten map:
2
2
2 2
2
2
2 3/ 2
2 1/ 2
( )
( ) 0
( ) 11
( )( )( ( ) 1)
0 ( ( ) 1)
( ) 1
( )
0
( ( ) 1)
0 ( ( ) 1) ( )
f z
f z
f z
f zf z f z
f z
f z
f z
f z
f z f z
A
-Gauss map:
3. Sample Assignment For Reference Only
2
1
( )cos , ( )sin , ( ) 9 )
( ) ( ) 1
n f z f z f z f z
f z f z
-principle curvature:
1 22 3/ 2
( )
, ( ) ( ) 1
( ( ) 1)
f z
k k f z f z
f z
(c) The surface of revolution about the z axis of a circle in the xz plane with center
( ,0,0)d with radius r d .
(solution)
2 22 2 2
( cos )cos
( cos )sin
sin
sin cos , sin sin , cos
( cos )sin ,( cos )cos ,0
, 0, ( cos ) , ( cos )
( cos )cos cos , ( cos
u
v
u v
u v
x d r u v
y d r u v
z r u
r r u v r u v r u
r d r u v d r u v
E r r F G r d r u H EG F r d r u
r r r d r u u v r d r
2
2
2
)cos sin , ( cos )sin
( cos cos , cos sin , sin )
( sin sin , sin cos ,0)
( ( cos )cos , ( cos )sin ,0)
( cos ) ( cos )cos
, 0, cos
( cos ) ( cos )
u
uv
v
u u v r d r u u
r r u v r u v r u
r r u v r u v
r d r u v d r u v
r d r u r d r u u
L r M N u
r d r u r d r u
-weingarten map:
2
2 2 2
2
1
0
( cos ) ( ) 01
cos( cos ) 0 cos 0
( cos )
d r u r r
ur d r u r u
d r u
A
-Gauss map:
(cos sin ,cos sin ,sin )n u v u v u
-principle curvature:
1 2 2
1 cos
,
( cos )
u
k k
r d r u
(d) The surface parametrized by
3 2 3 2 2 2
( , ) /3 , /3 ,r u v u u uv v v vu u v .
(solution)
4. Sample Assignment For Reference Only
2 2
2 2
2 2
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2
2
(1 ,2 ,2 )
(2 ,1 , 2 )
(1 ) , 0, (1 ) , (1 )
( 2 ,2 ,2), (2 ,2 ,0), (2 , 2 , 2)
2 ( 1),2 ( 1),1 ( )
4(1
u
v
u
uvu v
u v
r u v uv u
r uv v u v
E r u v F G u v H u v
r u v r v u r u v
r r u u v v u v u v
u v
L
2
2 2 2 2 2
2 2
2 2 2 2 2
) 4
, 0,
(1 ) 1
4(1 ) 4
(1 ) 1
M
u v u v
u v
N
u v u v
-weingarten map:
2 2 32 2
2 2 4 2 2
2 2 3
4
0
(1 )4(1 ) 01
4(1 ) 0 4(1 )
0
(1 )
u vu v
u v u v
u v
A
-Gauss map:
2 2 2 2 2 2
2 2 2
2 2
2 2 2 2 2 2
1
( 2 (1 ),2 (1 ),1 ( ))
(1 )
2 2 1
, ,
1 1 1
n u u v v u v u v
u v
u v u v
u v u v u v
-principle curvature:
1 22 2 3 2 2 3
4 4
,
(1 ) (1 )
k k
u v u v
Tutorial 2
1. Show that the second fundamental form II p is symmetric.
(proof) we take two arbitrary tangent vectors , pT S and two arbitrary real number
, . Then we have, first of all:
1
2 1 1 1 2
self-adjoint symmetrical
( , ) ( ( ), ) ( , ( )) ( ( ), ) ( , )
A
A A A
,
which means that 2 is symmetrical. (end)
5. Sample Assignment For Reference Only
2. Show that the elementary symmetric functions 1 1( , , )i nS k k are the coefficient of
i
x in
the expansion of 1 1(1 ) (1 )nk x k x .
(proof) When 2n ,
2 2
1 2 1 2 1 2 1 2(1 )(1 ) 1 ( ) 1k x k x k k x k k x S x S x .
Therefor 0 1 1 2 1 2 2 1 2 1 21, ( , ) , ( , )S S k k k k S k k k k .
When 3n ,
1 2 3 1 2 3
2 3
1 2 2 3 1 3 1 2 3
(1 )(1 )(1 ) 1 ( )
( )
k x k x k x k k k x
k k k k k k x k k k x
So 0 1 1 2 3 2 1 2 2 3 1 3 3 1 2 31, , ,S S k k k S k k k k k k S k k k .
…
1 2
1 2
2
1 2
1
1 2
(1 )(1 ) (1 ) 1
n
n i i i
i i i
n
n
k x k x k x x k x k k
x k k k
so 1 2
1 2
0 1 2 1 1
1
1, , , ,
n
i i i n n
i i i
S S k S k k S k k
.
3. Calculate the frames for the sphere based on
(a) the standard parameterization
(solution)
(cos cos , cos sin , sin )
( sin cos , sin sin , cos )
( cos sin , cos cos , 0)
u
v
r u v u v u
r u v u v u
r u v u v
(b) stereographic projection
4. Calculate frames for
(a) The torus
(solution) the equation
( cos )cos
( cos )sin
sin
x a b u v
y a b u v
z b u
.
( , , ) ( sin cos , sin sin , cos )
( , , ) ( ( cos )sin ,( cos )cos ,0)
u u u u
v v v v
r x y z b u v b u v b u
r x y z a b u v a b u v
(b) The catenoid
6. Sample Assignment For Reference Only
cosh cos , cosh sin ,
sinh cos ,sinh sin ,1
cosh sin , cosh cos ,0
u
v
u u
x a v y a v z u
a a
u u
r v v
a a
u u
r a v a v
a a
Tutorial 3
1. Calculate the first fundamental form for
(a) The sphere of radius, R ,
(solution) ( cos cos , cos sin , sin )r R u v R u v R u
,
sin cos , sin sin , cos
cos sin , cos cos ,0
u
v
r
r R u v R u v R u
u
r
r R u v R u v
v
2 22 2 2
11 12 22, 0, cosu u v vg r R g r r g r R u
.
Therefor the first fundamental form is the following:
2 2 2 2 2
cosR du R udv .
(b) The torus with inner radius, r and outer radius, R
(solution) the equation
( cos )cos
( cos )sin
sin
x a b u v
y a b u v
z b u
Where ,
2 2
R r R r
a b
. Therefor
cos cos
2 2
cos sin
2 2
sin
2
R r R r
x u v
R r R r
y u v
R r
z u
.
i.e.
7. Sample Assignment For Reference Only
sin cos , sin sin , cos
2 2 2
cos sin , cos cos , 0
2 2 2 2
u u u
v v v
r R r R R r
x u v y u v z u
R r R r R r R r
x u v y u v z
2
2 2 2
11 22 12
( ) 1
, [( ) ( )cos ], 0
4 4
u u u
R r
g x y z g R r R r u g
Therefor the first fundamental form is 2 2 2 21
( ) [( ) ( )cos ]
4
R r du R r R r u dv .
2. Use your answers to the previous question to find the length of
(a) A curve from the north pole of the sphere that winds twice around the sphere before
ending up at the south pole
(b) A curve that winds three times around the small randius for each time around the
major radius
3. In lectures we calculated 1E and shows that for the inertial frame
1 1 2
1 2 2 1 2
1 2
( , )
( , ) ( , )
( , )
x r x x
X x x x s x x
q x x
then 11(0,0) 0r . By calculating 2 1 2 1, , ,E F F G , and 2G , show that all the second
derivatives of r and s are zero at (0,0).
4. Show that 2
12 22 11 11 22 12
1 1
2 2
F F G q q q .
Tutorial 4
1. Construct an atlas for
(a) The torus
(b) the cylinder
from the charts for the circle from the lectures.
(solution) (a) Let
1
S be the circumference and
1 1
M S S .
1 1
,U S V S
: [ 1,1] ( , )U U
: [ 1,1] ( , )V V
8. Sample Assignment For Reference Only
the atlas is {( , )}U V .
(b) Let
1
S be the circumference and I be the open interval,
1 1
M S S .
1
,
: [ 1,1]
: ( , )
U S V I
U
V a b
{( , )}U V
2. Show that the function on the sphere that outputs the z coordinate of the point is
differentiable.
(proof) the spherical co-ordinates
cos cos
cos sin
sin
x a u v
y a u v
z a u
cos
dz
a u
du
. Therefor the function is differentiable.
3. Show that function on the real projective plane given by the angle the line makes with the
xy plane is differentiable.
(proof)
2 2 2
:( , , ) arcsin ( 0)
a
f x y z xyz
x y z
2 2 2 2 2
2 2 2 2 2
2 2
2 2 2
( )
( )
f xz
x x y x y z
f yz
y x y x y z
x yf
z x y z
Because 0xyz , the function is differentiable.
Tutorial 5
1. Write the coordinate vector-fields for cartesian coordinates on 2
,
x
and
y
in terms of the polar coordinate vector fields
9. Sample Assignment For Reference Only
r
and
(solution) 2 2( , ) cos
, , , tan , arctan
( , ) sin
x x r x r y y
r x y
y y r y r x x
2 2
cos
cos , sin
sin cos
,
r x r r
x r yx y
x r y r
Therefor
sin
cos
cos
sin
r
x x r x r r
r
y y r y r r
2. Calculate the vector-field transformation between stereographic coordinates and the
angular coordinates on the sphere,
2
S
Tutorial 6
Let , ,A y z B x z C x y
z y z x y x
.
1. Calculate the Lie derivative of B with respect to A.
(solution)
10. Sample Assignment For Reference Only
2 2 2 2
2
2
[ , ]AL B A B AB BA
y z x z x z y z
z y z x z x z y
y x z z x z
z z x y z x
x y z z y z
z z y x z y
z
yx y yz zx z
z z x z x y z
2 2 2 2
2
2
y x
z
xy x xz zy z
z z y z y x z x y
y x
x y
2. Show Af Bf Cf for
2 2 2
( , , )f x y z x y z .
(proof)
2 2 2
( , , )f x y z x y z
2 2 2
2 2 2
2 2 2
( ) ( 2 ) 2 0
( ) ( 2 ) 2 0
( ) 2 2 0
Af y z x y z y z z y
z y
Bf x z x y z x z z x
z x
Cf x y x y z x y y x
y x
3. Use that fact to sketch the curves of the one parameter groups associated with ,A B , and
C .
(solution) curve C :
( )
( )
x x z
y y z
(where z is auxiliary variable)
The one parameter groups are 1 2: , :f z y f y x .
1 2 1 1 2( ), ( ( )) ( )y f z x f f z f f z
2 2 2
( , , ) ( ) ( )f x y z x z y z z
( 2 ) 2 0
f f dy
Af y z y z z y
z y dz
…………………………………..(1)
( 2 ) 2 0
dx
Bf x z z x
dz
…………………………………..(2)
2 2 0
dy dx
Cf x y y x
dz dz
…………………………………..(3)
11. Sample Assignment For Reference Only
From (3),
dy dx
dx dz
.
(1), (2) 1
dy dx
dx dy dz
dz dz
The tangent vector of the curve
( )
( )
x t
y y
z t
, (1,1,1)
1
2
1
1
dy
y z c
dz
dx
x z c
dz
1
2
x t c
y t c
z t
When 1 2 0c c , the curve C is the line parallel to
and passing (0,0,0).
In general, the curve C is the line parallel to
and passing 1 2( , ,0)c c .
Tutorial 7
Let M be a two-dimensional manifold with coordinates 1x and 2x . The Christoffel symbols for a
connection are identically zero except for
1 1 2
12 21 2 11 2 2tan , cos sinx x x
1. Calculate XY for
1 2 1
1
,
cos
X Y
x x x
.
(solution) j i
X i j
Y Y x
x
1 2 1 2
2
1
1, 0, , 0
cos
X X Y Y
x
j
j j k
i iki
Y
Y Y
x
12. Sample Assignment For Reference Only
1
1 1 1 1 2
1 11 12 21
2
2 2 1 2 2
1 11 12 2 2 21
2
1
1 1 1 1 2
2 21 22 22 2
2 2
2
2 2 1 2 2
2 21 222
0 0 tan 0 0
1
0 cos sin 0 sin
cos
1 1
( tan ) 0 0
cos cos
0 0 0 0
Y
Y Y Y x
x
Y
Y Y Y x x x
x x
Y
Y Y Y x
x x x x
Y
Y Y Y
x
1 2
1 2 22 2 2
1 sin sini j
X i j
Y X Y X Y x x
x x x x
.
2. Write down the equations for parallel transport for this connection.
(solution) for parallel transport, 2 2
0 sin 0X Y x
x
Assume that the vector field ( )Y t parallel transport according to the curve r .
:r 1 1
2 2
( )
( )
x x t
x x t
!
2 2
2
2 22
1 11
2
sin
0
( )cos
( )
sin 0
x dx
x t cx dt
x t cdx
x
dt
3. Combine them into a single equation and write down the solution.
(solution)
4. Pick a starting point and vector and solve for the coefficients in the solution.
5. Calculate the torsion of this connection.
(solution) k k k
ij ij jiT : torsion tensor
0k
ij . torsion=0
Tutorial 8
1. Write the standard metric for the sphere in terms of the coordinates and .
(solution)
cos cos
cos sin
sin
x
y
z
the standard metric:
( sin cos , sin sin ,cos )
( cos sin ,cos cos ,0)
r
r
2 2 2 2 2
11 12 221, 0, cos , cosg g g dS d d
2. Write the standard metric for the torus in terms of the toroidal and poloidal angles.
13. Sample Assignment For Reference Only
(solution)
2 2 2
11 12 22
2 2 2 2 2 2
( cos )cos
( cos )sin
sin
(sinh cos ,sinh sin ,1)
( cosh sin , cosh cos ,0)
sinh 1, 0, cosh
(sinh 1) ( cosh )
u
v
x a b u v
y a b u v
z b u
u u
r u v
a a
u u
r a u a v
a a
u u
g g g a
a a
u u
dS du a dv
a a
3. Consider the metric
3 2 2
g dw dt dz and the coordinate transformations
( , ) cosh( )cos( )
( , ) cosh( )sin( )
( , ) sinh( )
z x y x y
t x y x y
w x y x
(a) Calculate
2 2 2
z t w
(b) Express g in the new coordinates
(solution)
cosh( )sin( )sinh( )cos( )
sinh( )sin( ) , cosh( )cos( )
cosh( ) 0
yx
x y
x y
z x yz x y
t x y t x y
w x w
(a)
2 2 2
cosh(2 )z t w x
(b)
x y
x y
x y
dz z dx z dy
dw w dx w dy
dt t dx t dy
2 2 2
2 2 2 22 2 2 2
2 22 2
2 2 2 2 2 22 2
2 2 2 2
2 ( 2 )
( 2 )
( ) 2( ) ( )
(cosh ( ) sinh ( ))
x x y y x x y y
x x y y
x x x x y x y x y y y y
g dw dt dz
w dx w w dxdy w dy t dx t t dxdy t dy
z dx z z dxdy z dy
w t z dx w w t t z z dxdy w t z dy
x x dx
2 2 2
2 2 2 2 2 2
cosh ( )
[cosh ( ) sinh ( )] cosh ( )
x dy
x x dx x dy
4. Express the metric for Minkowski space
2 2 2 2
0 0 0 0g cdt dx dy dz in terms of new
coordinates
14. Sample Assignment For Reference Only
0
0
0
cos( )
sin( )
t t
x r t
y r t
z z
(solution)
2 2 2 2
0 0 0 0g cdt dx dy dz
0 00
00 0
0 0 0
0 00
0 10
sin( )0 sin( )
, ,
0 cos( ) cos( )
1 00
tz
z t
z t
z t
t tt
x r tx x r t
y y r t y r t
z zz
0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
sin( ) sin( ) sin( )( )
cos( ) cos( ) cos( )( )
t z
t z
t z
t z
dt t d t dt t dz dt
dx x d x dt x dz r t d r t dt r t d dt
dy y d y dt y dz r t d r t dt r t d dt
dz z d z dt z dz dz
2 2 2 2
0 0 0 0
2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2 2
sin ( )( 2 ) cos ( )( 2 )
( 2 )
( ) 2
g cdt dx dy dz
cdt r t d d dt dt r t d d dt dt dz
cdt r d d dt dt dz
c r dt r d r d dt dz