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If we have non-vanishing boundary terms, then the map $\lambda \mapsto I[\Phi_\lambda^i]$ is not differentiable in the following sense. Using somewhat less sophisticated notation, let $$I[\Phi^i_\lambda:\eta] := \int_{\mathcal M} \mathcal L\left(\Phi^i_0(x)+\lambda\cdot \eta(x),\partial\Phi_0^i(x)+\lambda\cdot\partial\eta(x)\right) d^4x$$ for some ...