Tricki

## Integration by parts

### Quick description

If are continuously differentiable functions, then

If have sufficient decay at infinity (e.g. if they are compactly supported), then one can eliminate the boundary terms:

There are multidimensional versions, such as Stokes theorem and Greens theorem. Without boundary terms (and assuming sufficient regularity and decay), we have for instance

for the partial derivatives and scalar fields , as well as vector-valued variants such as

for a scalar field and a vector field .

### Example 1

(Maybe integrate by two applications of integration by parts?)

### Example 2

(Use integration by parts to compute adjoints of a differential operator)

### General discussion

See "use integration by parts to exploit cancellation" for one common application of integration by parts.

Integration by parts can simplify an integral by differentiating one term and integrating another. See "Which integrals are simpler to integrate" for some discussion.