Question

Calculate the first fundamental forms of the following surfaces:

i. 𝝈(𝑢,𝑣)= (sinh𝑢sinh𝑣,sinh𝑢cosh𝑣,sinh𝑢)

ii. 𝝈(𝑢,𝑣) = (𝑢 − 𝑣,𝑢 + 𝑣, 𝑢^2+ 𝑣^2).

Answer 1

1)

$\sigma = (sinhusinhv, sinhucoshv, sinhu)\\ \sigma_u = (coshusinhv, coshucoshv, coshu)\\ \sigma_v = (sinhucoshv, sinhusinhv, 0)\\$

$E= \sigma_u \cdot \sigma_u = (coshusinhv, coshucoshv, coshu) \cdot (coshusinhv, coshucoshv, coshu) = cosh^2usinh^2v+cosh^2ucosh^2v +cosh^2u = 2cosh^2ucosh^2v$

$G =\sigma_v \cdot \sigma_v = (sinhucoshv, sinhusinhv, 0) \cdot (sinhucoshv, sinhusinhv, 0) = sinh^2ucosh^2v+sinh^2usinh^2v +0 = sinh^2u + 2sinh^2usinh^2v$

$F= \sigma_u \cdot \sigma_v = (coshusinhv, coshucoshv, coshu) \cdot (sinhucoshv, sinhusinhv, 0) = 2sinhucoshusinhvcoshv$

$EG = 2sinh^2ucosh^2ucosh^2v + 4sinh^2usinh^2vcosh^2ucosh^2v$

$F^2 = 4sinh^2usinh^2vcosh^2ucosh^2v$

Let $k_1$ represent the first fundamental form, then

$k_1 = EG-F^2$

So, $k_1 = 2sinh^2ucosh^2ucosh^2v$

2)

$\sigma = (u-v, u+v, u^2+v^2) \\ \sigma_u = (1,1,2u) \\ \sigma_v = (-1,1,2v)$

$E= \sigma_u \cdot \sigma_u = (1,1,2u) \cdot (1,1,2u) = 1+1+ 4u^2 = 2+4u^2$

$G = \sigma_v \cdot \sigma_v = (-1,1,2v) \cdot (-1,1,2v) = 1+1+ 4v^2 = 2+4v^2$

$F = \sigma_u \cdot \sigma_v = (1,1,2u) \cdot (-1,1,2v) = -1+1+ 4uv= 4uv$

$EG = 4 + 8(u^2+v^2) +16u^2v^2$

$F^2 = 16u^2v^2$

$k_1 = EG - F^2 = 4 + 8(u^2+v^2)$