2 so j k t t a cos cos a cos sin a sin

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Unformatted text preview: 2 + z 2 = a2 }, as follows. Let U = (0, π ) × (0, 2π ), and W = R3 \ { (x, y, z ) | x ≥ 0 and y = 0 }. Let � : U −→ M ∩ W, g be the function � (φ, θ) = (a sin φ cos θ, a sin φ sin θ, a cos φ). g Let’s calculate the tangent vectors, ∂� g � Tφ = = (a cos φ cos θ, a cos φ sin θ, −a sin φ), ∂φ ∂� g � Tθ = = (−a sin φ sin θ, a sin φ cos θ, 0). ∂θ 2 So � � � ˆ ˆ ı j ˆ k � � � � � Tφ × Tθ = � a cos φ cos θ a cos φ sin θ −a sin φ� � � � a sin φ sin θ a sin φ cos θ − 0 � ˆ = a2 sin2 φ cos θˆ + a2 sin2 φ sin θj + a2 cos φ sin φk. ı...
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