Let $I \in C^1(\mathbb{R}^n,\mathbb{R})$ be an even functional. There is a claim that then the restriction to the sphere has the following form for the Frechet derivative $$I|_{S^{n-1}}'(u) = I'(u)-(I'(u),u)u$$ I don't see why this is true. Could you explain please?
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The Fréchet derivative does not make sense here, because $I|_{\mathbb{S}^{n-1}}$ is not a function defined on an open set of a linear space. I think what is meant here is the gradient of $I|_{\mathbb{S}^{n-1}}$ as a function on the Riemannian manifold $\mathbb{S}^{n-1}$. That is a vector field, i. e. a section of the tangent bundle of the sphere, so embedded in the tangent bundle of $\mathbb{R}^n$, at each point $u \in \mathbb{S}^{n-1}$ it has to be orthogonal to $u$. In fact, it is the orthogonal projection of the gradient of $I$ in $\mathbb{R}^n$ along $u$, and that is what the formula is supposed to mean. |
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