Integration By Parts - Higher Dimensions

Higher Dimensions

The formula for integration by parts can be extended to functions of several variables. Instead of an interval one needs to integrate over an n-dimensional set. Also, one replaces the derivative with a partial derivative.

More specifically, suppose Ω is an open bounded subset of with a piecewise smooth boundary Γ. If u and v are two continuously differentiable functions on the closure of Ω, then the formula for integration by parts is

where is the outward unit surface normal to, is its i-th component, and i ranges from 1 to n.

By replacing v in the above formula with v_i and summing over i gives the vector formula

where v is a vector-valued function with components v₁, ..., v_n.

Setting u equal to the constant function 1 in the above formula gives the divergence theorem

For where, one gets

which is the first Green's identity.

The regularity requirements of the theorem can be relaxed. For instance, the boundary Γ need only be Lipschitz continuous. In the first formula above, only is necessary (where H1 is a Sobolev space); the other formulas have similarly relaxed requirements.