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:
“Good and evil are so close as to be chained together in the soul. Now suppose we could break that chain, separate those two selves. Free the good in man and let it go on to its higher destiny.”
—John Lee Mahin (1902–1984)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)