Leibniz Integral Rule - Three-dimensional, Time-dependent Case

Three-dimensional, Time-dependent Case

See also: Differentiation under the integral sign in higher dimensions

A Leibniz integral rule for three dimensions is:

 

  

where:

F ( r, t ) is a vector field at the spatial position r at time t

Σ is a moving surface in three-space bounded by the closed curve ∂Σ

d A is a vector element of the surface Σ

d s is a vector element of the curve ∂Σ

v is the velocity of movement of the region Σ

∇• is the vector divergence

× is the vector cross product

The double integrals are surface integrals over the surface Σ, and the line integral is over the bounding curve ∂Σ.