Integration by parts is basically an integral version of the product rule.12
By beginning with the product rule, and integrating by , we get the following:
which can be rearranged to get the product rule
Or in function notation
It applies to the situation in which you are called upon to integrate the product of one function () and the derivative of another ();
it says you can transfer the derivative from to ,
at the cost of a minus sign and a boundary term.
Higher order derivatives
It follows
Generalizations
By exploiting different generalisations of the Fundamental theorem of calculus
along with different generalisations of the product rule,
we can generalise integration by parts for vector valued functions.[^2013]
Divergence
Scaling
Curl
Scaling
\begin{align*}
\iiint_{\Omega}f(\vab{\nabla}\times\vab G),d\tau &= - \oiint_{\partial\Omega} f \vab G \times d\vab a - \iiint_{\Omega} (\vab{\nabla}f) \times \vab G ,d\tau
\end{align*}