We consider the following function

$$f:(\mathbb S^n)^N \rightarrow \mathbb R^{n+1} \text{ such that } f(x_1,...,x_N)= \sum_{i=1}^N x_i.$$

This function can be written in Cartesian coordinates as $$f(x)=(f_1(x),..,f_{n+1}(x))$$ and I would like to know if one can find a simple expression for the derivative

$$\nabla_{x_1} \left(\frac{f_1(x)}{\Vert f(x) \Vert_{\mathbb R^{n+1}}}\right)$$ where $$\nabla_{x_1}$$ is the gradient on $$\mathbb S^n$$ with respect to $$x_1.$$

Can one somehow carry out this differentiation? I am a bit struggeling with computing $$\nabla_{x_1} f_1(x)$$ here.

• If $M$ is a smooth submanifold of a Euclidean space $E$, $U\subset E$ is an open subset of $E$ containing $M$ and $f:U\to\mathbb{R}$ is a $C^1$-function, then for every $p\in M$ we have $\nabla^M f(p)= \mathrm{Proj}_p\nabla^E f(p)$, where $\mathrm{Proj}_p:E \to T_p M$ denotes the orthogonal projection onto the tangent space to $M$ at $p$. Aug 27 '20 at 13:46
