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

$$\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$$Take any $$(x_1,x_2,\dots,x_N)\in(\S^n)^N$$. Let $$(-1,1)\ni t\mapsto X_1(t)\in\S^n$$ be any smooth curve such that $$X_1(0)=x_1.$$ For any $$t\in(-1,1)$$, let $$X(t):=(X_1(t),x_2,\dots,x_N)$$ and $$S(t):=f(X(t))=X_1(t)+x_2+\dots+x_N[\in\R^{n+1}],$$ so that $$X(0)=(x_1,x_2,\dots,x_N)$$ and $$S(0)=s:=x_1+x_2+\dots+x_N.$$ Let $$v:=X'_1(0)$$, so that $$S'(0)=v$$. Let $$S_1(t):=f_1(X(t))=e_1\cdot S(t),$$ where $$\cdot$$ denotes the dot product and $$e_1$$ is the first vector of the standard basis of $$\R^{n+1}$$. So, $$S'_1(0)=e_1\cdot v$$. So, for $$r(t):=\frac{f_1(X(t))}{\|f(X(t))\|}=\frac{S_1(t)}{\|S(t)\|}$$ we have $$r'(0)=\frac1{\|S(0)\|^2}\Big(\|S(0)\|S'_1(0)-S_1(0)\frac{S(0)}{\|S(0)\|}\cdot S'(0)\Big) \\ =\frac1{\|s\|^2}\Big(\|s\|e_1\cdot v-(e_1\cdot s)\frac{s}{\|s\|}\cdot v\Big).$$ Thus, the gradient in question is $$g-(g\cdot x_1)x_1$$, where $$g:=\frac1{\|s\|^2}\Big(\|s\|e_1-(e_1\cdot s)\frac{s}{\|s\|}\Big).$$ $$\big($$Note: $$g-(g\cdot x_1)x_1$$ is the orthogonal projection of the vector $$g$$ onto the tangent hyperplane to the unit sphere $$\S^n$$ at point $$x_1$$.$$\big)$$