A sphere of radius m has the metric in the usual colatitude and longitude

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Manifolds, vectors, tensor fields and the metric

Problem

A sphere of radius a=2.0a = 2.0 m has the metric ds2=a2(dθ2+sin⁡2θ dϕ2)ds^2 = a^2(d\theta^2 + \sin^2\theta\,d\phi^2) in the usual colatitude and longitude (θ,ϕ)(\theta,\phi). Work at the point θ0=30∘\theta_0 = 30^\circ. ((a)) The tangent vector V=∂ϕ\mathbf V = \partial_\phi acts on the function f=cos⁡θf = \cos\theta. Evaluate V(f)\mathbf V(f) at θ0\theta_0. ((b)) Find the length of the full circle of latitude at θ0\theta_0. ((c)) Find the angle between the tangent vectors U=∂θ\mathbf U = \partial_\theta and W=∂θ+2∂ϕ\mathbf W = \partial_\theta + 2\partial_\phi at θ0\theta_0. ((d)) At θ0\theta_0 a covector has component ωϕ=8.0 m2\omega_\phi = 8.0\ \text{m}^2. Use the inverse metric gμνg^{\mu\nu} to find ωϕ\omega^\phi. ((e)) How many independent components does a totally antisymmetric tensor with two lower indices have in three dimensions?

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