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:
“A higher class, in the estimation and love of this city- building, market-going race of mankind, are the poets, who, from the intellectual kingdom, feed the thought and imagination with ideas and pictures which raise men out of the world of corn and money, and console them for the short-comings of the day, and the meanness of labor and traffic.”
—Ralph Waldo Emerson (1803–1882)
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)