The differential $i∗ : TpS_{2} → TpR_{3 }$ maps $ ∂/∂u|p,∂/∂v|p $ into $TpR_{3}. $ Find $(α_{i}, β_{i}, γ_{i})$

Question

Hi! This was my homework. Prof. sent its answer. But I didnt understand how can this answer be reached? Please can someone explain this?

Answer 1

Note that $i(u,v) = (u,v,\sqrt{1-u^2-v^2})$. Now we calculate:

$$i_*\left(\left.\frac{\partial}{\partial u}\right|_p\right)(a) = \frac{\partial}{\partial u}(a\circ i) = \frac{\partial}{\partial u}(u) = 1$$ $$i_*\left(\left.\frac{\partial}{\partial u}\right|_p\right)(b) = \frac{\partial}{\partial u}(b\circ i) = \frac{\partial}{\partial u}(v) = 0$$ $$i_*\left(\left.\frac{\partial}{\partial u}\right|_p\right)(c) = \frac{\partial}{\partial u}(c\circ i) = \frac{\partial}{\partial u}(\sqrt{1-u^2-v^2}) = \frac{-u}{\sqrt{1-u^2-v^2}} = -\frac{a}{c}$$

The pushforward of $\frac{\partial}{\partial v}$ is similar.