In Do Carmo's Differential Geometry of Curves and Surfaces, I'm having a quite hard time trying to solve Excersise 14 on pages 101-102. He defines the gradient of a differentiable function $f:S\to \Bbb{R}$ as a differentiable map $\text{grad}f:S\to\Bbb{R}^3$ which assigns to each point $p\in S$ a vector $f(p)\in T_p(S)\subset \Bbb{R}^3$ such that
$$\qquad \langle \text{grad}f(p),v\rangle_p=df_p(v)\qquad\text{for all }v\in T_p(S)$$
The question is to express $\text{grad}f$ of $\phi(U)$ in terms of the coefficients $E,F,G$ of the first fundamental form in a parametrization $\phi:U\subset\Bbb{R}^2\to S$. The solution is
$$\qquad \text{grad}f=\frac{f_uG-f_vF}{EG-F^2}\phi_u\,+\,\frac{f_vE-f_uF}{EG-F^2}\phi_v$$
I'm not sure how to start. Sure that if $gradf$ sends points into vectors of the tangent plane, it must be a linear combination of $\phi_u$ and $\phi_v$, but I don't know how to use the information that $\langle \text{grad}f(p),v\rangle_p=df_p(v)$ $\,$for all $v\in T_p(S)$ to get the desired result. The $\dfrac{1}{EG-F^2}$ makes me think about the inverse matrix of the first fundamental form, but again, I don't see how can I relate one thing to the other. Any hints that can point me in the right direction would be appreciate. Thanks in advance!
Best Answer
Let $X\colon U\subset \mathbb{R}^2\longrightarrow S$ be a parametrization of a regular surface $S$ and let $p=X(u,v)$. If $f\colon S\rightarrow \mathrm{R}$ is a differentiable function then $\mathrm{grad} f(p)\in T_pS$. Thus, $$\mathrm{grad} f (p)=\alpha X_u+\beta X_v,\ \ \ (\dagger)$$ where $\alpha,\beta $ are functions defined on $U$. From $(\dagger)$ you obtain the following two equations $$f_u=\alpha E+\beta F\ \ \&\ \ f_v=\alpha F+\beta G.$$ By solving the system and substituting into $(\dagger)$ you obtain your desired result.