Question

Q. Calculate the first fundamental forms of the following surfaces:
1) Sphere: σ(θ,φ) =(cosθcosφ, cosθsinφ,sinθ)
2) A generalized cylinder: σ(u,v) =γ(u)+Va

Answer 1

Answer on Question #77264 – Math – Differential Geometry | Topology

Calculate the first fundamental forms of the following surfaces:

Question

1) Sphere: $\sigma(\theta, \varphi) = (\cos\theta\cos\varphi, \cos\theta\sin\varphi, \sin\theta)$

Solution

$$\sigma(\theta, \varphi) = (\cos\theta\cos\varphi, \cos\theta\sin\varphi, \sin\theta)$$ $$\sigma_{\theta}(\theta, \varphi) = (-\sin\theta\cos\varphi, -\sin\theta\sin\varphi, \cos\theta)$$ $$\sigma_{\varphi}(\theta, \varphi) = (-\cos\theta\sin\varphi, \cos\theta\cos\varphi, 0)$$ $$E = \sigma_{\theta}(\theta, \varphi) \cdot \sigma_{\theta}(\theta, \varphi) = \sin^2\theta\cos^2\varphi + \sin^2\theta\sin^2\varphi + \cos^2\theta = 1$$ $$F = \sigma_{\theta}(\theta, \varphi) \cdot \sigma_{\varphi}(\theta, \varphi) = \sin\theta\cos\theta\sin\varphi\cos\varphi - \sin\theta\cos\theta\sin\varphi\cos\varphi = 0$$ $$G = \sigma_{\varphi}(\theta, \varphi) \cdot \sigma_{\varphi}(\theta, \varphi) = \cos^2\theta\sin^2\varphi + \cos^2\theta\cos^2\varphi = \cos^2\theta$$

Ans.: $d\sigma^2 = d\theta^2 + \cos^2\theta d\varphi^2$

Question

2) A generalized cylinder: $\sigma(u, v) = \gamma(u) + Va$

Solution

$$\sigma_u = y'(u), \sigma_v = a$$ $$\sigma_u^2 = 1, \sigma_v^2 = a^2, \sigma_u\sigma_v = a y'(u)$$

Ans.: $d\sigma^2 = du^2 + 2a y'(u) d u d v + a^2 d v^2$

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