Question 1.

A geometry is given by the line element ds² = e^2ydx² + e^4xdy².

The curve C₁, parametrized by σ, is defined by:

x(σ) = σ, y(σ) = 2σ.

Compute the length of C₁ between σ = 1 and σ = 4. A second curve, C₂, parametrized by Σ is defined by:

x(Σ) = e^Σ, y(Σ) = 2e^Σ.

Compute the length of C₂ between Σ = 0 and Σ = ln 4 [6 marks]




Question 2.

A geometry is given by the line element ds² = e^2ydx² + e^4xdy².

Compute the Christoffel symbols for this geometry in these (x, y) coordinates. [9 marks]

Question 3.

A geometry has the following Christoffel symbols Γ^xy = 1, Γxyy = −2e4x−2y, Γ^yxy = 1, Γyxx = −2e2y−4x,

and a vector field exists on the geometry with components V^x=1, V^y=0. Compute ∇_y∇_xV x. [5 marks]

Question 4.

It is claimed that the following line element describes the flat geometry of R2 ds² = (sinh² A cosh² B + cosh² A sinh² B) dA² +4sinhAcoshAsinhBcoshB dA dB

+ (cosh² A sinh² B + sinh² A cosh² B) dB².

Find the explicit co-ordinate transformation to standard Cartesian (x, y) coordinates on R2 to decide if this claim is true. [3 marks]

Question 5:

Alice and Bob are examining a two-dimensional geometry with (x, y) coordinates that has the following non-zero Christoffel symbols;

Γ^xy = −e^y−x, Γ^yxx = −e^x+y, Γ^yxy = −e^y.

Alice claims that the vector with components V a = (ex, e2x) is parallel transported along the curve y = 0, while Bob claims that the vector with components Wa = (e2x, ex) is parallel transported along the curve y = 0. Perform a calculation to find out who is correct [9 marks]