Question

Q. Calculate the first fundamental forms of the following surfaces:
1) σ(u,v) =(sinhusinhv, sinhv, sinhucoshv, sinhu)
2) σ(u,v)=(u-v,u+v,u^2+v^2)

Answer 1

Answer on Question #77265, Math / Geometry

$$\sigma(u, v) = (\sinh u \cdot \sinh v, \sinh v, \sinh u \cdot \cosh v, \sinh u)$$ $$\frac{\partial \sigma}{\partial u} = (\cosh u \cdot \sinh v, 0, \cosh u \cdot \cosh v, \cosh u)$$ $$\frac{\partial \sigma}{\partial v} = (\sinh u \cdot \cosh v, \cosh v, \sinh u \cdot \sinh v, 0)$$ $$E = \cosh^2 u \cdot \sinh^2 v + \cosh^2 u \cdot \cosh^2 v + \cosh^2 u = 2 \cosh^2 u \cdot \cosh^2 v$$ $$F = 2 \sinh u \cdot \cosh u \cdot \sinh v \cdot \cosh v$$ $$G = \sinh^2 u \cdot \cosh^2 v + \cosh^2 v + \sinh^2 u \cdot \sinh^2 v$$

Then, $1(x,y) = Ex^2 + 2Fxy + Gy^2$

$$\sigma(u, v) = (u - v, u + v, u^2 + v^2)$$ $$\frac{\partial \sigma}{\partial u} = (1, 1, 2u)$$ $$\frac{\partial \sigma}{\partial v} = (-1, 1, 2v)$$ $$E = 2 + 4u^2$$ $$F = 4uv$$ $$G = 2 + 4v^2$$

Then, $1(x,y) = Ex^2 + 2Fxy + Gy^2$