I encountered the following two exercises when I use the book "Differential Topology" by Guillemin and Pollack to study differential forms and integration on manifolds by myself, and I'm wondering whether one of them is actually a particular case of the other? They are listed below:
Background:
(Exercise 4.4.13) Let $S$ be an oriented two-manifold in $\mathbb{R}^3$ and $\overrightarrow{n}(x) = (n_1(x),n_2(x),n_3(x))$ be the outward unit normal to $S$ at $x$ (defined in Exercise 3.2.19). We may define a $2$-form $dA$ on $S$ as follows: $$dA = n_1 dx_2 \wedge dx_3 + n_2 dx_3 \wedge dx_1 + n_3 dx_1 \wedge dx_2$$ where each $dx_i$ above is restricted to $S$. Then we have that if $S$ is the graph of some function $F:\mathbb{R}^2 \rightarrow \mathbb{R}$ (i.e, $S = \{(x_1,x_2,F(x_1,x_2)) \ | \ (x_1,x_2) \in \mathbb{R}^2\}$) with orientation induced from $\mathbb{R}^2$, then $dA$ is the same as that defined in text, i.e $dA = |\overrightarrow{n}|dx_1 \wedge dx_2$
(Exercise 4.4.14) Let $\omega = f_1 dx_2 \wedge dx_3 + f_2 dx_3 \wedge dx_1 + f_3 dx_1 \wedge dx_2$ be an arbitrary $2$-form on $\mathbb{R}^3$, then we have that the restriction of $\omega$ to $S$ is the form $(\overrightarrow{F} \cdot \overrightarrow{n})dA$, where $S$ is the graph of some function mapping from $\mathbb{R}^2$ to $\mathbb{R}$, $\overrightarrow{F}(x) = (f_1(x), f_2(x), f_3(x))$ and $\overrightarrow{n}$ is the vector normal to $S$
I feel that the two questions above have the same assumption, but is it true that Exercise 13 is a particular case of Exercise 14? By picking $f_i =n_i \ (i=1,2,3)$ to be the components of the normal vector, we can reduce Exercise 14 to Exercise 13, right? FYI, I have followed the hint to solve for Exercise 14 and I'm currently unsure whether I can use Exercise 14 to prove Exercise 13 directly.
Any help/hint would be appreciated! Thank you so much!