A cylinder of unit radius is charted by through the embedding

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

A cylinder of unit radius is charted by (θ,z)(\theta, z) through the embedding F(θ,z)=(cos⁡θ,sin⁡θ,z)∈R3F(\theta, z) = (\cos\theta, \sin\theta, z) \in \mathbb{R}^3. At the point θ=π/3\theta = \pi/3, z=2z = 2, a tangent vector on the cylinder is v=2 ∂θ−5 ∂zv = 2\,\partial_\theta - 5\,\partial_z. Find the components (vx,vy,vz)(v_x, v_y, v_z) of the pushforward F∗vF_*v in R3\mathbb{R}^3.

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