----- > [!proposition] Proposition. ([[integration by parts]]) > Let $\Omega \subset \mathbb{R}^{n}$ be a [[bounded set|bounded]] [[Lipschitz boundary|Lipschitz domain]] and let $f,g \in C^{1}(\overline{\Omega})$. Then, as a corollary of the [[divergence theorem]], we obtain $\int _{{ }\Omega} f \partial_{i}g= \int _{\partial \Omega} fg \, n_{i} \, - \int _{\Omega} g \partial_{i} f \, d\mu .$ In particular, if $f$ or $g$ has [[compact]] [[support]] in $\Omega$ then the boundary term vanishes and we are left with $\int _{\Omega} f \partial_{i}g=-\int _{\Omega} g \partial_{i} f \, .$ Relate to [[geometer's integration by parts]]. ^proposition > [!specialization] $\int_{a}^{b} u(x) v'(x)\, dx=[u(x)v(x)] |^{b}_{a}- \int_{a}^{b} v(x)u'(x) \, dx.$ ^specialization > [!proof]- Proof. ([[integration by parts]]) > Let $F:=(0, \dots, \overbrace{ fg }^{ i\text{th slot} }, \dots, 0):\mathbb{R}^{n} \to \mathbb{R}^{n}$. Then $\text{div } F= \partial_{i}F^{i}=f \partial_{i}g + g \partial_{i}f.$ > Also, $F \cdot N=fg \, n_{i}$ for $n_{i}$ the $i$th component of the outward [[orientation of a Euclidean submanifold|unit normal field]] $N$. Thus [[divergence theorem]] gives $\int f \partial_{i}g + g \partial_{i}f= \int _{\partial_{ }\Omega} fg \,n_{i} .$ > Rearrange for the result. > [!basicexample] > Let us compute the [[antiderivative]] of $x\cos x$. Put $u:=x$ and $v:=\sin x$. Then $u'(x)v(x)=\sin x$ and the integration by parts formula gives $\begin{align} \int u(x)v'(x) \, dx = x \sin x + \cos x + C . \end{align}$ ^basic-example ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```