A flat plane has plane polar coordinates with metric

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 flat plane has plane polar coordinates (r,ϕ)(r,\phi) with metric ds2=dr2+r2dϕ2ds^2 = dr^2 + r^2d\phi^2. A particle moves so that at the instant it is at r=2.0r=2.0 m, ϕ=60∘\phi=60^\circ, it has dr/dλ=3.0 m s−1dr/d\lambda = 3.0\ \text{m s}^{-1} and dϕ/dλ=1.0 rad s−1d\phi/d\lambda = 1.0\ \text{rad s}^{-1}, where λ\lambda is time. ((a)) Let f(r,ϕ)=r2cos⁡ϕf(r,\phi) = r^2\cos\phi (in m2\text{m}^2). Using the tangent vector V=(dr/dλ)∂r+(dϕ/dλ)∂ϕ\mathbf V = (dr/d\lambda)\partial_r + (d\phi/d\lambda)\partial_\phi, evaluate V(f)=df/dλ\mathbf V(f) = df/d\lambda at that instant. ((b)) Find the length of the arc traced by keeping r=2.0r=2.0 m fixed while ϕ\phi increases by 60∘60^\circ. ((c)) At r=2.0r=2.0 m a covector has component ωϕ=6.0 m2\omega_\phi = 6.0\ \text{m}^2. Use the inverse metric gμνg^{\mu\nu} to find ωϕ\omega^\phi. ((d)) How many independent components does a totally antisymmetric tensor with two lower indices have in this two-dimensional plane?

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