# Weitzenbock formula for manifold on boundary

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 $$\int_M (|dA|^2+|d^*A|^2)dvol_M=\int_M(| \nabla A|^2+Ric(A,A))dvol_M$$