Transformation - Lebesgue Integration - Exam, Exams for Integrated Case Studies. Bengal Engineering & Science University

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This is the Exam of Lebesgue Integration and its key important points are: Transformation, Parabolic Coordinates, Jacobian Matrix, Positive, Parameterization, Dot Products, Cartesian Space, Constants, Vectors, Vector fie...
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PRIFYSGOL CYMRU/UNIVERSITY OF WALES

ABERYSTWYTH

INSTITUTE OF MATHEMATICS AND PHYSICS

SEMESTER 1 EXAMINATIONS, JANUARY/FEBRUARY 2009

MA37110 - Differential Geometry

Time allowed - 2 hours

• All questions may be attempted.

• Marks gained from section B will be given greater consideration in assessing a first class performance.

• Calculators are permitted, provided they are silent, self-powered, without communications facilities, and incapable of holding text or other material that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorized to remove any suspect calculators.

11/12/2008

MA37110 - Differential Geometry 2 of 5

Section A

1. Parabolic coordinates (u, v) are defined through the transformation

x = 1

2

( u2 − v2

) ,

y = uv,

with u ∈ R and v > 0. Calculate the Jacobian matrix J (x, y;u, v) for this transformation and show that its determinant is everywhere positive.

[5 marks]

2. Let ( ξi )

be curvilinear coordinates related to Cartesian coordinates (xi), with Cartesian position vectors having the parameterization

r ( ξ1, · · · , ξn

) = xi

( ξ1, · · · , ξn

) ei.

Setting Ei = ∂

∂ξi r, and Ni = ∇ξi, show that we have the dot products

Ei.N j = δji .

[5 marks]

3. The position of a point on the surface of a torus embedded in Cartesian space can be written as

r = (a+ b sin θ) (cosϕex + sinϕey) + b cos θez,

where (θ, ϕ) are surface parameters, 0 < b < a are constants, and ex, ey, ez are unit

vectors along the Cartesian coordinate axes. Explain how the vectors Eθ = ∂

∂θ r

and Eϕ = ∂

∂ϕ r may be used to calculate the metric. Show that the element of

arclength ds on the surface is given by

(ds)2 = b2 (dθ)2 + (a+ b sin θ)2 (dϕ)2 .

[3+10 marks]

4. State how the components of vector fields, respectively covector fields, must transform under a change of local coordinates (xi) to (xi

′ ). Show that if (vi) and

(θi) are the components of a vector field v and a covector field θ respectively, then θ (v) = viθi is a scalar field independent of the choice of coordinates.

[5 marks]

5. State the definition of the Lie bracket [U, V ] of a pair of vector fields U, V over a differential manifold. Show that the Leibniz rule

[U, V ] (fg) = [U, V ] (f) g + f [U, V ] (g) ,

holds for all smooth functions f, g on the manifold.

[5 marks]

11/12/2008

MA37110 - Differential Geometry 3 of 5

6. State what is meant for a connection to be compatible with a metric g. Show that if such a metric is torsionless, then its components in a local coordinate system can be written as

Γijk = 1

2 gil (gjl,k + glk,j − gjk,l) .

[5+10 marks]

7. State the definition of an abstract Lie algebra. Consider the set A of vector fields over three-dimensional Euclidean space which are linear combinations of the vector fields

X(1) = y∂z − z∂y, X(2) = z∂x − x∂z, X(3) = x∂y − y∂x.

Prove that A is a Lie algebra with the usual Lie bracket of vector fields. If Y = x∂x + y∂y + z∂z, show that [X, Y ] = 0 for all X ∈ A.

[7+5 marks]

11/12/2008

MA37110 - Differential Geometry 4 of 5

Section B

8. A torus is a two-dimensional Riemannian manifold with metric determined from the line element

(ds)2 = b2 (dθ)2 + (a+ b sin θ)2 (dϕ)2 ,

where (θ, ϕ) are surface coordinates with θ ∈ [0, 2π) and ϕ ∈ [0, 2π), and 0 < b < a. (i) Determine the components of the Riemannian connection corresponding to this metric.

[10 marks]

(ii) For what values of α, β ∈ R will parameterized curves of the form θ (t) = αt+θ0, ϕ (t) = βt+ ϕ0, be geodesics on the torus.

[5 marks]

9. Let gij be the components of a metric on a Riemannian manifold in a fixed system of local coordinates, and let |g| denote the determinant of the matrix (gij). (i) Treating the entries Xij of a matrix X as independent variables, one has the fact

that ∂

∂Xij |X| is the i, j entry of the adjoint of the matrix. Using this show that

1

2 ln |g|,i = Γkki.

[8 marks]

(ii) The covariant divergence of a vector field V is defined in local coordinates by div.V , V k;k. Show that this may be written as

div.V = 1√ |g|

(√ |g|Vk

) ,k .

[7 marks]

11/12/2008

MA37110 - Differential Geometry 5 of 5

10. Let (xi) be a system of local coordinates about an arbitrary point p of a differential manifold M with a fixed connection. Take x̄i = xi (p) to be the coordinates of the point p, and let Γ̄ijk be the components of the connection at the point p in this coordinates system.

(i) Given a second system of local coordinates (xi ′ ) about p, show that the

components of the connection transform according to the non-tensorial rule

Γijk = ∂xi

∂xl′ ∂xr

∂xj ∂xs

∂xk Γl

r′s′ + ∂xi

∂xl′ ∂2xl

∂xj∂xk .

[7 marks]

(ii) Define new local coordinates (xi ′ ) by

xi ′ = xi − x̄i + 1

2 Γ̄ijk ( xj − x̄j

) ( xk − x̄k

) .

Show that the components of the connection in these new coordinates vanish identically at the point p.

Comment briefly on this result.

[8 marks]

11. Let H denote the set of matrices of the form( a b −b̄ ā

) where a and b are complex numbers with at least one non-zero.

(i) Show that H is a Lie subgroup of GL (2,C) . (ii) Deduce that dimH = 4 by showing that a basis is given by {I, Jx, Jy, Jz} where

I =

( 1 0 0 1

) , Jx =

1

2

( 0 i i 0

) , Jy =

1

2

( 0 −1 1 0

) , Jz =

1

2

( i 0 0 −i

) .

(iii) Let X = 2xJx + 2yJy + 2zJz + tI with x, y, z, t ∈ R. Show that X ∈ SU (2) if and only if

x2 + y2 + z2 + t2 = 1.

(iv) Show that there is a one-to-one mapping from SU(2) to the unit sphere in R4. Comment briefly on this fact.

[4,4,4,3 marks]

11/12/2008