# Chain rule in $\mathbb{R}^n$, did I wrote it down correctly?

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

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

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