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Summary

1. Parametrizações de variedades:
(a) g(x) = (x, x3)
(b) g(t) = (cos t², sen t³)
(c) g(t, x) = (x, cos t, sen t)
(d) g(x, y) = (x, y, x² + y²)
(e) g(θ, φ) = ((3 + cos φ) cos θ, (3 + cos φ) sen θ, sen φ)
(f) g(θ, φ) = (sen φ cos θ, sen φ sen θ, cos φ)
(g) g(θ, y) = (√(y² + 1) sen θ, y, √(y² + 1) cos θ)
(h) g(x) = (x, 1 - x, 2x² - 2x + 1)
2. Espaço tangente: {(x, x, 4x): x ∈R} e espaço normal: {(x, y, -x⁴ - y⁴): x, y ∈R}.
3. Espaço tangente: {(x, 0, z): x, z ∈R} e espaço normal: {(0, y, 0): y ∈R}.
4. Recta tangente: x = 1; z = 2y e plano normal: y + 2z = 0.
5. Recta normal: x + z = 2; y + z = 2 e plano tangente: x + y - z = 1.

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