Answer to Question #192891 in Differential Geometry | Topology for Himanshi yadav

Question

Consider a plane with the surface patch σ(u, v) = (1+2u+3v, u-v,-2+u-4v). Verify the Gauss equations for σ.

Answer 1

$K = \dfrac{LN-M^2}{EG-F^2} \\ E = \sigma_{u} \cdot \sigma_{u} \\ F = \sigma_{u} \cdot \sigma_{v} \\ G = \sigma_{v} \cdot \sigma_{v} \\ L = \sigma_{uu} \cdot \mathcal{n} \\ M = \sigma_{uv} \cdot \mathcal{n} \\ N = \sigma_{vv} \cdot \mathcal{n}$

$\sigma_u = (2,1,1) \\ \sigma_v = (3,-1,-4) \\ \sigma_{uu} = (0,0,0) \\ \sigma_{uv} = (0,0,0) \\ \sigma_{vv} = (0,0,0)$

$E = \sigma_{u} \cdot \sigma_{u} = 2(2)+1(1)+1(1) = 6$

$F = \sigma_{u} \cdot \sigma_{v} = 2(3) +1(-1)+1(-4) = 1$

$G = \sigma_{v} \cdot \sigma_{v} = 3(3)+(-1)(-1)+(-4)(-4) = 26$

$L = 0 \\ M =0 \\ N = 0 \\ K = \dfrac{0(0)-0}{6(26)-1} = 0$