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Let $(M,g)$ be a Riemannian manifold with boundary, we know that for one-form $A$ we have $$\Delta A=\nabla^*\nabla A+Ric(A).$$

Q Assume $A(\nu)=0$, for the normal vector field $\nu$. How to show that $$\int_M (|dA|^2+|d^*A|^2)dvol_M=\int_M(| \nabla A|^2+Ric(A,A))dvol_M$$

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