Higher Dimensions
The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem:
where is a scalar function, and denote a time-varying connected region of and its boundary, respectively, is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and is the unit normal component of the surface element.
The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in -dimensions is:
where is a time-varying domain of integration, is a -form, is the vector field of the velocity, denotes the interior product, is the exterior derivative of with respect to the space variables only and is the time-derivative of Note that in -space, .
Read more about this topic: Differentiation Under The Integral Sign
Famous quotes containing the words higher and/or dimensions:
“By going one step further back in thought, discordant opinions are reconciled by being seen to be two extremes of one principle, and we can never go so far back as to preclude a still higher vision.”
—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)