Question

Q. Compute the first fundamental form and second fundamental form of the elliptical paraboloid σ(u,v)=(u, v,〖 u〗^2+v^2)

Answer 1

Answer on Question #77661 – Math – Differential Geometry | Topology

Question

Compute the first fundamental form and second fundamental form of the elliptical paraboloid $\sigma(u, v) = (u, v, \left\{ \begin{array}{l} u \end{array} \right\} \wedge 2 + v \wedge 2)$

Solution

$$\sigma_u = (1, 0, 2u), \sigma_v = (0, 1, 2v),$$ $$E = \sigma_u^2 = (1, 0, 2u)^2 = 1 + 4u^2$$ $$F = \sigma_u \sigma_v = (1, 0, 2u)(0, 1, 2v) = 4vu$$ $$G = \sigma_v^2 = (0, 1, 2v)^2 = 1 + 4v^2$$

First fundamental form: $I = (1 + 4u^2)du^2 + 8uvdudv + (1 + 4v^2)dv^2$.

$$\sigma_u \times \sigma_v = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 2u \\ 0 & 1 & 2v \end{array} \right| = \left| \begin{array}{cc} 0 & 2u \\ 1 & 2v \end{array} \right| \mathbf{i} - \left| \begin{array}{cc} 1 & 2u \\ 0 & 2v \end{array} \right| \mathbf{j} + \left| \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right| \mathbf{k} = -2u\mathbf{i} - 2v\mathbf{j} + \mathbf{k} = (-2u, -2v, 1)$$ $$n = \frac{\sigma_u \times \sigma_v}{\sqrt{EG - F^2}} = \left(- \frac{2u}{\sqrt{1 + 4u^2 + 4v^2}}, - \frac{2v}{\sqrt{1 + 4u^2 + 4v^2}}, \frac{1}{\sqrt{1 + 4u^2 + 4v^2}}\right)$$ $$\sigma_{uu} = (0, 0, 2), \sigma_{uv} = (0, 0, 0), \sigma_{vv} = (0, 0, 2),$$ $$L = (\sigma_{uu}, n) = \frac{2}{\sqrt{1 + 4u^2 + 4v^2}}$$ $$M = (\sigma_{uv}, n) = 0$$ $$N = (\sigma_{vv}, n) = \frac{2}{\sqrt{1 + 4u^2 + 4v^2}}$$

Second fundamental form: $\mathrm{II} = \frac{2}{\sqrt{1 + 4u^2 + 4v^2}} du^2 + \frac{2}{\sqrt{1 + 4u^2 + 4v^2}} dv^2.$

Answer: $I = (1 + 4u^2)du^2 + 8uvdudv + (1 + 4v^2)dv^2$, $\mathrm{II} = \frac{2}{\sqrt{1 + 4u^2 + 4v^2}} du^2 + \frac{2}{\sqrt{1 + 4u^2 + 4v^2}} dv^2.$

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