Let $\Sigma^d\subset \mathbb{R}^{d+1}$ be a smoothly immersed closed (compact without boundary) hypersurface. If $\Sigma$ is orientable, the normal vector field $\mathbf{n}:\Sigma\to \mathbb{R}^{d+1}$ can be written as $$\mathbf{n}(x)=(n_1(x),\dots,n_{d+1}(x))$$ for some functions $n_i:\Sigma\to \mathbb{R}$.
Question: Are the functions $\{n_i\}$ linearly independent?
The example I am thinking of is when $\Sigma$ is the unit sphere. For example, when $d=1$, then $n_1(x)=\cos(x)$ and $n_2(x)=\sin(x)$, which are linearly independent.
On the contrary, if $\Sigma$ is a hyperplane, then $\mathbf{n}$ is constant, so each $n_i$ is constant, and thus the $n_i$ are linearly dependent.