by the closed curve , is a vector element of the surface , is a vector element of the curve , is the velocity of movement of the region , is the vector divergence, is the vector cross product, The double integrals are surface integrals over the surface , and the line integral is over the bounding curve . 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 R3 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 n dimensions is where is a time-varying domain of integration, ω is a p-form, is the vector field of the velocity, denotes the interior product with , dxω is the exterior derivative of ω with respect to the space variables only and is the time derivative of ω. However, all of these identities can be derived from a most general statement about Lie derivatives: Here, the ambient manifold on which the differential form lives includes both space and time. is the region of integration (a submanifold) at a given instant (it does not depend on , since its parametrization as a submanifold defines its position in time), is the Lie derivative, is the spacetime vector field obtained from adding the unitary vector field in the direction of time to the purely spatial vector field from the previous formulas (i.e, is the spacetime velocity of ), is a diffeomorphism from the one-parameter group generated by the flow of , and is the image of under such diffeomorphism. Something remarkable about this form, is that it can account for the case when changes its shape and size over time, since such deformations are fully determined by . Measure theory statement Let be an open subset of , and be a measure space. Suppose satisfies the following conditions: is a Lebesgue-integrable function of for each . For almost all , the partial derivative exists for all . There is an integrable