This post is about mathematical formulas for my reference.

Integration by parts

For ordinary functions of one variable, the rule for integration by parts follows immediately from integrating the product rule

\[\begin{align*} \frac{d}{dx}(fg) &= \frac{df}{dx}g + f\frac{dg}{dx}, \\ \int_a^b \frac{d}{dx} (fg) \, dx &= \int_a^b \frac{df}{dx}g \, dx + \int_a^b f \frac{dg}{dx} \, dx, \\ fg |_a^b &= \int_a^b \frac{df}{dx}g \, dx + \int_a^b f \frac{dg}{dx} \, dx. \end{align*}\]

Rearraging, we obtain

\[\int_a^b \frac{df}{dx}g \, dx = fg |_a^b - \int_a^b f \frac{dg}{dx} \, dx.\]

In an analogous way, we can obtain a rule for integration by parts for the divergence of a vector field by starting from the product rule for the divergence

\[\nabla \cdot (f \mathbf{G}) = (\nabla f) \cdot \mathbf{G} + f (\nabla \cdot \mathbf{G}).\]

integrating both sides yields

\[\int \nabla \cdot (f \mathbf{G}) \, d\tau = \int (\nabla f) \cdot \mathbf{G} \, d\tau + \int f (\nabla \cdot \mathbf{G}) \, d\tau .\]

Now use the divergence theorem to rewrite the first term, leading to

\[\int (f \mathbf{G}) \cdot \, d\mathbf{A} = \int (\nabla f) \cdot \mathbf{G} \, d\tau + \int f (\nabla \cdot \mathbf{G}) \, d\tau .\]

which can be rearranged to

\[\int f (\nabla \cdot \mathbf{G}) \, d\tau = \int (f \mathbf{G}) \cdot \, d\mathbf{A} - \int (\nabla f) \cdot \mathbf{G} \, d\tau .\]

which is the desired integration by parts.