1.3. Geodesics 9 x y A B Figure 1.5. Hypocycloid with inner radius S AB π the position vector r now takes the form r(r, θ, φ) = r sin θ cos φ i + r sin θ sin φ j + r cos θ k . (1.15) Since this position vector depends on r, θ, and φ, dr = ∂r ∂r dr + ∂r ∂θ dθ + ∂r ∂φ dφ . (1.16) The three partial derivatives on the right-hand side of this equation are vectors tangent to motions in the r, θ, and φ directions. Thus dr = hr dr ˆr + hθ dθ ˆθ + hφ dφ ˆφ , (1.17) where ˆr, ˆθ, and ˆφ are unit vectors in the r, θ, and φ directions and hr = ∂r ∂r = 1 , hθ = ∂r ∂θ = r , hφ = ∂r ∂φ = r sin θ (1.18) are the scale factors for spherical coordinates.

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