F = ψ \nabla ϕ

, for functions smooth enough on the closure of the bounded piecewise smooth volume. The product rule gives

\nabla \cdot (ψ \nabla ϕ) = \nabla ψ \cdot \nabla ϕ + ψ \nabla^{2} ϕ, (ψ \nabla ϕ) \cdot n = ψ \frac{\partial ϕ}{\partial n} .

\int_{V} (ψ \nabla^{2} ϕ + \nabla ψ \cdot \nabla ϕ) d V = \int_{\partial V} ψ \frac{\partial ϕ}{\partial n} d S .

The normal is outward. This identifies the boundary term and its sign directly from the divergence theorem.

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