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## Main Question or Discussion Point

I am reviewing Jackson's "Classical Electromagnetism" and it seems that I need to review vector calculus too. In section 1.11 the equation ##W=-\frac{\epsilon_0}{2}\int \Phi\mathbf \nabla^2\Phi d^3x## through an integration by parts leads to equation 1.54 ##W=\frac{\epsilon_0}{2}\int |\mathbf \nabla\Phi|^2 d^3x=\frac{\epsilon_0}{2}\int |\mathbf E|^2 d^3x##. My problem is that I tried to derive the result with back to basics integration by parts with ##\int u dv = uv - \int vdu## with $$

u = \Phi, dv = \mathbf \nabla^2\Phi d^3x=\mathbf \nabla \cdot (\mathbf \nabla \Phi) d^3x, du=\mathbf \nabla \Phi d^3x, v = \mathbf \nabla \Phi\\

\int \Phi \mathbf \nabla^2\Phi d^3x=\Phi \mathbf \nabla \Phi - \int \mathbf \nabla \Phi \cdot \mathbf \nabla \Phi d^3x=\Phi \mathbf \nabla \Phi - \int \mathbf |\mathbf \nabla \Phi|^2 d^3x

$$ that is obviously wrong, the term ##\Phi \mathbf \nabla \Phi=-\Phi \mathbf E## shouldn't be there and it is a vector quantity summed to a scalar. How I should proceed?

u = \Phi, dv = \mathbf \nabla^2\Phi d^3x=\mathbf \nabla \cdot (\mathbf \nabla \Phi) d^3x, du=\mathbf \nabla \Phi d^3x, v = \mathbf \nabla \Phi\\

\int \Phi \mathbf \nabla^2\Phi d^3x=\Phi \mathbf \nabla \Phi - \int \mathbf \nabla \Phi \cdot \mathbf \nabla \Phi d^3x=\Phi \mathbf \nabla \Phi - \int \mathbf |\mathbf \nabla \Phi|^2 d^3x

$$ that is obviously wrong, the term ##\Phi \mathbf \nabla \Phi=-\Phi \mathbf E## shouldn't be there and it is a vector quantity summed to a scalar. How I should proceed?