Notes

$$ \begin{align*} \delta\mathcal{L} = &\left(\frac{\partial\mathcal{L}}{\partial\phi} + \sum_{n=1}^\infty (-\boldsymbol{\nabla})^{\otimes n}\cdot\frac{\partial\mathcal{L}}{\partial(\boldsymbol{\nabla}^{\otimes n}\phi)}\right)\delta\phi\\ &+ \sum_{n=1}^{\infty}\boldsymbol{\nabla}\cdot\left(\sum_{k=0}^{n-1}\left((-\boldsymbol{\nabla})^{\otimes k}\cdot \frac{\partial\mathcal{L}}{\partial(\boldsymbol{\nabla}^{\otimes n}\phi)}\right)\cdot \boldsymbol{\nabla}^{\otimes n-1-k}\delta\phi\right) \end{align*} $$