Understanding a local definition of the gradient In this topic, the tangential gradient on a submanifold $M \subset \mathbb{R}^d$, embedded by a map $F: M\rightarrow \mathbb{R}^d$, is defined as
$$\nabla^M f = \nabla^i f \frac{\partial F}{\partial x_i} = g^{ij} \frac{\partial f}{\partial x_j} \frac{\partial F}{\partial x_i},$$
where $g_{ij}=\langle\frac{\partial F}{\partial x_i}, \frac{\partial F}{\partial x_j}\rangle$.
I'm stack in understanding the embedding $F$ and how to calculate $\frac{\partial F}{\partial x_i}$ even in the basic example of a sphere ($d=3$). Any hint would really be helpful.
 A: Note that for $F\colon M \to\mathbb{R}^d$, we have the differential map $dF_p\colon T_pM\to T_{F(p)}\mathbb{R}^d$ for all $p\in M$, i.e. $dF_p$ transports tangential vectors in $T_pM$ to tangential vectors in $T_{F(p)}\mathbb{R}^d$. The tangential vector they transport here via the differential map is the gradient of a function $f$ on $M$. So the given notation means
$$ (\nabla^M f)_p := dF_p((\nabla f)_p)$$
for all points $p\in M$, where they denote $dF_p((\nabla f)_p)$ with respect to local coordinates.
Looking at an example like the embedding of a 2-sphere into $\mathbb{R}^3$ is certainly a good way to get a better understanding of the definition. Using standard coordinates on $\mathbb{S}^2$, an embedding can (almost everywhere) be given by
$$ F\colon [0,\pi]\times[0,2\pi)\to\mathbb{R}^3,\quad (\theta,\phi)\mapsto\left(\begin{array}{c}\sin\theta\cos\phi \\ \sin\theta\sin\phi\\ \cos\theta\end{array}\right), $$
which yields
$$ dF =  \left(\frac{\partial F}{\partial x_i}\right)_{x_1 = \theta, x_2 = \phi} = \left(\begin{array}{cc}\cos\theta\cos\phi & -\sin\theta\sin\phi \\ \cos\theta\sin\phi & \sin\theta\cos\phi\\ -\sin\theta & 0\end{array}\right).$$
Then, with respect to these coordinates, you can compute the gradient vector $\nabla^{\mathbb{S}^2}f$ in $\mathbb{R}^3$, which is tangential to the sphere embedded into $\mathbb{R}^3$, from the gradient vector $\nabla f$ on $\mathbb{S}^2$ by multiplication (for any differentiable function $f$):
$$ \nabla^{\mathbb{S}^2}f = DF\cdot\nabla f = \left(\begin{array}{cc}\cos\theta\cos\phi & -\sin\theta\sin\phi \\ \cos\theta\sin\phi & \sin\theta\cos\phi\\ -\sin\theta & 0\end{array}\right)\cdot \nabla f. $$
Simple example: $f=\phi$ gives you $ \nabla^i \phi = g^{ij}(df)_j $, hence
$$ \nabla\phi = \left(\begin{array}{cc} 1 & 0 \\ 0 & \frac{1}{\sin^2\theta}\end{array}\right)\left(\begin{array}{c} 0 \\ 1 \end{array}\right) = \left(\begin{array}{c} 0 \\ \frac{1}{\sin^2\theta}\end{array}\right) $$
and thus
$$ \nabla^{\mathbb{S}^2}\phi = \left(\begin{array}{c} -\frac{\sin\phi}{\sin\theta} \\ \frac{\cos\phi}{\sin\theta}\\0\end{array}\right) $$
tangential to $\mathbb{S}^2$ in $\mathbb{R}^3$.
