Assume a surface xuv has the rst fundamental forms as E 1 F

Assume a surface x(u,v) has the ?rst fundamental forms as E = 1, F = 0, G = 1 + u^2.

(a) Let ?₁ : u = v, ?₂ : u = ?v, ?₃ : v = 1 be three curves on R2 plane (with coordinates (u,v). Let C₁,C₂,C₃ be the images of ?₁,?₂,?₃ under x (so these curves are on the given surface) respectively. Find the total length of sides of the (generalized) triangle on the surface, which is formed by these three curves.

(b) Find the surface area of the (generalized) triangle on the surface, which is formed by the curve u = v, the curve u = ?v, and the curve v = 1.

(c) Let ?₁ : u = 1/2v², ?₂ : u = ?1/2v², ?₃ : v = 1 be three curves on R2 plane (with coordinates (u,v). Let C₁,C₂,C₃ be the images of ?₁,?₂,?₃ under x (so these curves are on the given surface) respectively. Find the total length of sides of the (generalized) triangle on the surface, which is formed by these three curves.

Solution

^{B surface area of triangle- 1/2 base * height.}

^{Surface area- 1/2*2v*v}

^{Surface area-v2}

^C.

^.