everyone. I had to use the chain rule and I am a bit confused by the notation, could you please tell me if I wrote it down correctly ?

I have a map $H : \mathbb{R}^{2n} \rightarrow \mathbb{R}$ and I take two smooth periodic functions $x,y \in C^{\infty}(S^1,\mathbb{R}^{2n})$, i.e. $x(t),y(t) \in \mathbb{R}^{2n}$.

Now I have to evaluate

$\frac{d}{d \epsilon} \mid_{\epsilon = 0} \int_0^1 H(x+\epsilon y) dt$.

Here's my attempt

$H(x+\epsilon y)=H(z_1,...,z_{2n})$ and hence

$\frac{d}{d \epsilon} \mid_{\epsilon = 0} \int_0^1 H(z_1,...,z_{2n}) dt = \int_0^1 \sum_{i=1}^{2n} \frac{\partial H}{\partial z_i} \frac{d z_i}{d \epsilon} \mid_{\epsilon =0} dt= \int_0^1 < \nabla H(x),y> dt$

  • $\begingroup$ Looks right to me $\endgroup$ – Daniel Gendin Mar 4 at 22:34
  • $\begingroup$ Thanks :) I'm sometimes a bit confused with the notation for the chain rule. $\endgroup$ – Alain Mar 5 at 10:32

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