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 vi and summing over i gives the vector formula
where v is a vector-valued function with components v1, ..., vn.
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.
Read more about this topic: Integration By Parts
Famous quotes containing the words higher and/or dimensions:
“The idea was to prove at every foot of the way up that you were one of the elected and anointed ones who had the right stuff and could move higher and higher and even—ultimately, God willing, one day—that you might be able to join that special few at the very top, that elite who had the capacity to bring tears to men’s eyes, the very Brotherhood of the Right Stuff itself.”
—Tom Wolfe (b. 1931)
“Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
—bell hooks (b. c. 1955)